# Beta source vs TCT: Theory

In the lab I am currently characterizing silicon detectors using two types of sources to excite them: 1) a radioactive Sr-90 source and 2) a pulsed laser in a TCT setup. I want to do a theoretical comparison between these two ways of testing the detectors to improve my understanding of what happens in each case, find out what do they have in common and what not, to know how should the results in each case should be interpreted and compared between themselves, etc. Furthermore, we use the TCT because it is very convenient to characterize spacial and temporal resolution due to the fact that we can precisely choose where to shine the laser and we can also produce a precise delay between two pulses. But the real application of the detectors will be with MIP particles. That said, I find it important to understand what goes on inside the silicon in each case.

## Interaction of particles with matter

I will start by a short research on how do particles interact with matter. In my case I am interested in two types of particles:

1. Minimum ionizing particles. These are charged particles with $$\beta \gamma \approx 3-3.5$$[1]Kolanoski, Hermann, and Norbert Wermes. Particle Detectors: Fundamentals and Applications. Particle Detectors. Oxford University Press, 2020. https://cds.cern.ch/record/2721300. Sec. 3.2.1.3. where $$\beta$$ and $$\gamma$$ are the Lorentz factors. It is common to use the term MIP for any relativistic charged particle with $$\beta \gamma > 3-3.5$$ because the interaction with matter does not change too much. This will apply for the radioactive Sr-90 source.
2. Infrared photons. This will apply for the TCT.

and only in one type of “matter”, which is silicon. The fact that there are important differences between these two scenarios should be clear from the following image:

### Minimum ionizing particles (MIPs)

As said before, a MIP is a charged particle with an energy such that $$\beta \gamma \approx 3$$. In words of Wikipedia:

A minimum ionizing particle (or MIP) is a particle whose mean energy loss rate through matter is close to the minimum. In many practical cases, relativistic particles (e.g., cosmic-ray muons) are minimum ionizing particles.

Wikipedia

When a MIP enters into a piece of depleted silicon, it randomly collides with the electrons in the valence band and, in each collision, it transfers some random energy to the corresponding electron. If the energy transfer is enough it will move the electron to the conduction band and produce an $$e^- h^+$$ (electron-hole) pair. This process is depicted below:

The total energy lost by the MIP particle in this process is given by[3]Kolanoski, Hermann, and Norbert Wermes. Particle Detectors: Fundamentals and Applications. Particle Detectors. Oxford University Press, 2020. https://cds.cern.ch/record/2721300. Eq. (3.50).

$E_\text{lost by MIP} = \sum_{n=1}^{N_\text{MIP collisions}} E_n$

where $$E_n$$ is the energy transferred from the MIP to the electron in the $$n$$-th collision and $$N_\text{MIP collisions}$$ is the number of collisions. Note that here both $$N_\text{MIP collisions}$$ and $$E_n$$ (for all $$n$$) are random variables. As a consequence, $$E_\text{lost by MIP}$$ is a random variable. Now, what we measure through the electric current is not this energy, but the number of electron-hole pairs produced. Obviously these two quantities are related: The bigger $$E_\text{lost by MIP}$$ the bigger the number of charge carriers created. This relation, however, is not trivial. We could be tempted to say that the number of electron-hole pairs created is just $$N_\text{MIP collisions}$$, however it is possible that, for some $$n$$, $$E_n$$ is bigger than twice the energy required to produce a single electron-hole pair (which is $$\sim E_g \approx 1.1 \text{ eV}$$) and this will produce an electron with enough energy to create further electron-hole pairs.

The relation between the number of electron-hole pairs created $$N_{e^- h^+}$$ and $$E_\text{lost by MIP}$$ is probably random at some extent (I don’t know). It is reasonable to assume, however, that their relation is approximately proportional

$N_{e^- h^+} \overset{\sim}{\propto} E_\text{lost by MIP}$

where $$\overset{\sim}{\propto}$$ means “approximately proportional”. So we can relate the collected charge $$Q$$ (the integral of the measured current) with the energy lost by the MIP

$Q\overset{\sim}{\propto}E_{\text{lost by MIP}}.$

Even though this is probably not exact, it seems to work fine. The mean values, in fact, satisfy a proportionally relation[4]Kolanoski, Hermann, and Norbert Wermes. Particle Detectors: Fundamentals and Applications. Particle Detectors. Oxford University Press, 2020. https://cds.cern.ch/record/2721300. Sec. 8.4.

$\left\langle N_{e^{-}h^{+}}\right\rangle =\left\langle \frac{dE}{dx}\right\rangle \frac{\text{thickness}}{w_{i}}$

where $$\left\langle \frac{dE}{dx}\right\rangle$$ is the mean energy lost by a particle in a material medium (given by the Bethe-Bloch formula[5]Kolanoski, Hermann, and Norbert Wermes. Particle Detectors: Fundamentals and Applications. Particle Detectors. Oxford University Press, 2020. https://cds.cern.ch/record/2721300. Sec. 3.2.1.), $$\text{thickness}$$ is the thickness of the material sample and $$w_i$$ is the average energy to create an electron-hole pair, which for silicon at room temperature is[6]Kolanoski, Hermann, and Norbert Wermes. Particle Detectors: Fundamentals and Applications. Particle Detectors. Oxford University Press, 2020. https://cds.cern.ch/record/2721300. Table 8.2.

$w_i^{\text{Silicon},T=300\text{ K}} = 3.65 \text{ eV}.$

Taking this into account it seems reasonable to assume that

$Q \approx e \frac{E_\text{Lost by MIP}}{w_i}$

where $$e = 1.6 \times 10^{-19} \text{ C}$$ is the charge of the electron. I did not read this from any trusted source, I am just proposing this based on the previous stuff. So it might be wrong, use with caution. Now, $$E_\text{Lost by MIP}$$ follows a so called Landau distribution[7]Kolanoski, Hermann, and Norbert Wermes. Particle Detectors: Fundamentals and Applications. Particle Detectors. Oxford University Press, 2020. https://cds.cern.ch/record/2721300. Sec. 3.2.3.2. and, according to the previous relation, so it will $$Q$$.

### Photons

The behavior of photons is different from that of MIPs. Photons can interact through a variety of different processes with matter. The way they interact is a strong function of the photon energy and also depends on the material. The plots below show the cross section for the different processes for carbon and lead[8]Kolanoski, Hermann, and Norbert Wermes. Particle Detectors: Fundamentals and Applications. Particle Detectors. Oxford University Press, 2020. https://cds.cern.ch/record/2721300. Fig. 3.37..

In the TCT I have an infrared laser with an energy of $$\sim 1 \text{ eV}$$. Though none of these plots is for silicon and $$1 \text{ eV}$$ is outside the plots, they strongly suggest that the only process that is relevant in my case is $$\sigma_\text{p.e.}$$ which is “atomic photoelectric effect”. This is the case for silicon detectors indeed, and when an electron is taken from the valence band to the conduction band it is called internal photoelectric effect[9]Kolanoski, Hermann, and Norbert Wermes. Particle Detectors: Fundamentals and Applications. Particle Detectors. Oxford University Press, 2020. https://cds.cern.ch/record/2721300. Sec. 10.1.1.[10]As opposed to external photoelectric effect which is when the electron is removed from the material. This happens e.g. in a photo cathode and requires higher photon energies.. The photoelectric effect is[11]Kolanoski, Hermann, and Norbert Wermes. Particle Detectors: Fundamentals and Applications. Particle Detectors. Oxford University Press, 2020. https://cds.cern.ch/record/2721300. Sec. 3.5.3.

$\gamma + \text{atom} \to \text{atom}^+ + e^-$

meaning this that the photon is completely absorbed in a single interaction by an electron and as a result the electron becomes unbounded from the atom. In a semiconductor this means that the electron goes from the valence band to the conduction band, and the positive ion becomes a hole. For this process to occur there is a minimum threshold energy for the photon, which is basically the binding energy of the electron (to the atom). This is different for each material, in silicon this is

$E_\gamma > E_g^\text{Silicon} \approx 1.12 \text{ eV} ~~~~~ (\lambda_\gamma < 1100 \text{ nm})$

where $$E_g^\text{Silicon}$$ is the band gap energy for silicon. The real story is more complicated since silicon is an “indirect band gap semiconductor” and the absorption of a photon requires the absorption of a phonon to provide momentum[12]Kolanoski, Hermann, and Norbert Wermes. Particle Detectors: Fundamentals and Applications. Particle Detectors. Oxford University Press, 2020. https://cds.cern.ch/record/2721300. Sec. 10.1.3.. This makes the absorption of optical and infrared photons in silicon to be dependent on temperature, but we can ignore all these details for now.

When a beam of photons enters in a material medium, its intensity (the number of photons in the beam) follows an exponential decay[13]Kolanoski, Hermann, and Norbert Wermes. Particle Detectors: Fundamentals and Applications. Particle Detectors. Oxford University Press, 2020. https://cds.cern.ch/record/2721300. Eq. (3.111).:

$N \sim \exp \left(- \frac{x}{\text{absorption length}} \right)$

wehre $$x$$ is the distance traveled within the medium and $$\text{absorption length}$$ is a quantity that is usually tabulated. For silicon[14]At room temperature? in the optical region it is given in the plot below:

Assuming that the absorption of each photon in the beam is an independent process[16]I think this makes sense as long as the intensity is not too high to fully ionize the material. this means that the probability for each photon of being absorbed in a distance $$x$$ (or before) is

$\mathbb{P}\left(\gamma\text{ absorbed in }x<x_{0}\right)=1-\exp\left(-\frac{x_0}{\text{absorption length}}\right).$

Here we can consider $$x_0$$ as the thickness of the silicon detector and so this probability is basically the probability of detecting the photon:

$\mathbb{P}\left(\text{detecting }\gamma\right)=1-\exp\left(-\frac{\text{detector thickness}}{\text{absorption length}}\right) .$

If the energy of the photon is low enough (as in my case, I have $$E_\gamma \gtrsim 1.12 \text{ eV}$$) then its absorption is the only random process. It can be absorbed, or not. If it is absorbed, then it will produce one and only one electron-hole pair without doubt because the energy of the photon is not enough for more.

### Summary/comparison between MIP and photon

Now we can compare each of the two scenarios: The detection of a single MIP particle vs a single infrared photon. Both processes have a random nature, but the origin and characteristics of this randomness is very different. In the case of a MIP particle there are many random electron-hole pairs produced as the particle traverses the silicon. Two identical MIP particles will inherently produce different signals. In the case of an infrared photon there is one and only one electron-hole pair produced if the photon interacts, otherwise there is no signal at all. Thus, two infrared photons will produce the same signal, if detected.

## Sr-90 beta source

Now let’s focus on the Sr-90 beta source setup. In the lab we (and everybody) use it to test our detectors with MIP particles. In this post I wrote a review of our setup, I left a picture of it here below.

I will start with a brief research on the properties of this type of source, because I have no prior experience with them. Below we can see the chain of decays[17]Buang, S.; Subri, E. D.; Kandaiya, S.; Razak, N. N. A. N. A. & Yahaya, N. Z. Gafchromic XRQA-2 film for Strontium-90/Yttrium-90 (Sr-90/Y-90) DetectionJournal of Physics: Conference Series, IOP … Continue reading from 90Sr to 90Zr, passing through 90Y, and the energy spectrum[18]Arfaoui, S; CERN and Joram, C; CERN and Casella, C; ETH Zurich, Characterisation of a Sr-90 based electron monochromator, PH-EP-Tech-Note-2015-003. Link. of the emitted particles.

There are two decays until a stable configuration is reached. These are beta decays in which a neutron converts to a proton and in the process it emits an electron and a neutrino. We obviously do not see the neutrino, only the electron. If we plot the $$\beta \gamma$$ factor for electrons in the range of energies up to $$2.5 \text{ MeV}$$ we get this:

For these calculations I proceeded as follows[19]Review of formulas for relativistic motion, Barletta, Spentzouris, Harms, link.:

$\gamma=\frac{E_{\text{total}}}{E_{\text{rest}}}=\frac{E_{\text{total}}}{m}$

where $$m \approx 0.51 \text{ MeV}/c^2$$ is the mass of the electron (in units of energy), and then $$\beta=\sqrt{1-\gamma^{-2}}$$. As can be seen $$\beta\gamma$$ ranges from $$0$$ to $$6$$. For energies of around $$1.1 \text{ MeV}$$ we have $$\beta \gamma \approx 3$$, which is the range of MIP energies (see above in the beginning of the post). So, according to the previous calculation, the beta source is emitting as follows:

As can be seen there are many electrons that are not MIP, I still don’t know how this affects our results, if it does. As a quick analysis I think that since the energy lost by a charged particle below the MIP range increases very rapidly[20]Kolanoski, Hermann, and Norbert Wermes. Particle Detectors: Fundamentals and Applications. Particle Detectors. Oxford University Press, 2020. https://cds.cern.ch/record/2721300. Fig. 3.8. it may be the case that many of these electrons are stopped by the metallic cap on top of the detector. This cap is made of some unknown metal, but it has a hole of about $$5 \text{ mm}$$ of diameter which is covered by a single layer of adhesive aluminum tape. The density of aluminum is $$\rho \approx 2.7 \text{ g} \text{ cm}^{-3}$$, assuming a thickness of $$300 \text{ µm}$$ for this tape, considering $$\beta \gamma = 1$$ and using the energy loss $$\left\langle \frac{dE}{dx}\right\rangle _{\beta\gamma=1}\approx2\text{ MeV}\text{ cm}^{2}\text{ g}^{-1}$$[21]Kolanoski, Hermann, and Norbert Wermes. Particle Detectors: Fundamentals and Applications. Particle Detectors. Oxford University Press, 2020. https://cds.cern.ch/record/2721300. Fig. 3.8. I get an average energy lost for these electrons of about $$0.16 \text{ MeV}$$. If this calculation is approximately correct, then the metallic cap is able of partially blocking these low energy electrons before they reach the detector.

As I stated before, the collected charge (integral of measured current) should be more or less proportional to the energy deposited by the MIP particle in the silicon. Knowing that the deposited energy follows a Landau distribution, this should also be the case for the collected charge with the beta source:

$\boxed{ Q_\text{beta source} \sim \text{Landau distributed} }$

## TCT laser

In our lab we have a Particulars Large Scanning TCT. This setup is equipped with a 1064 nm pulsed laser[22]See this link in the section “1.) Laser”.. The energy of 1064 nm photons is

$E_\text{photons from the TCT laser} \approx 1.16 \text{ ev}$

which is slightly above the band gap energy of silicon. As discussed before, this allows each photon to create one and only one electron-hole pair in the silicon. The absorption length for these photons in silicon, according to the previous discussion, is

$\text{absorption length TCT photons in silicon} \approx 1 \text{ mm}.$

The relative number of photons in the beam as a function of the distance traversed within silicon, thus, looks like this:

According to the previous discussion, the probability of one of these photons to produce an electron-hole pair in a 50 µm[23]Our detectors usually have a thickness around 50 µm. thick detector is around

$\mathbb{P}\left(\text{producing one }e^-h^+\text{ pair}\right)\approx5\ \%$

It is possible to do a rough estimation of the number of photons in each laser pulse. The laser in our setup is a Thorlabs LPS-1060-FC which has a rated power of $$50 \text{ mW}$$. On the other hand the minimum pulse width that can deliver the driver is of around $$400 \text{ ps}$$[24]See this link in the section “1.) Laser”.. The number of photons for such a pulse is

$N=\frac{E}{h\nu}=\frac{Pt\lambda}{hc}\sim\frac{50\text{ mW}\times400\text{ ps}\times1064\text{ nm}}{hc}\approx10^{8}$

This is a huge number of particles thrown “at the same time” to the detector, as opposed to the case with the beta source in which a single particle is radiated at each time.

If the absorption of each photon is independent of the detection of the other photons[25]One example in which it is NOT independent is when the number of photons is high enough such that all the atoms in the silicon become ionized. In this scenario a new incoming photon will not find … Continue reading, the number of electron-hole pairs should follow a binomial distribution with parameters $$p \approx 5\text{ %}$$ and $$n \sim 10^8$$. For these values of $$n$$ and $$p$$ the Gaussian approximation to the binomial distribution should be valid, so

$N_{e^-h^+ \text{ pairs}} \sim \text{Gaussian}\left(\mu = np, \sigma = \sqrt{np(1-p)}\right).$

Finally, the collected charge, which is the integral of the measured current, should be proportional to $$N_{e^-h^+ \text{ pairs}}$$, specifically $$Q = e N_{e^-h^+ \text{ pairs}}$$ where $$e$$ is the charge of the electron. So the collected charge should follow a Gaussian distribution too:

$\boxed{ Q_\text{TCT} \sim \text{Gaussian} \left( \mu = enp, \sigma = e \sqrt{np(1-p)} \right) }.$

## Conclusions

I have presented a brief research on how the electron-hole pairs are produced in silicon in the two cases 1) with a charged MIP particle and 2) with infrared photons. I have found that both processes have a strong random component, but that it is very different in each case. When using the radioactive beta source there is a single particle that produces many electron-hole pairs with a Landau distribution. When using the infrared laser from the TCT there are, in turn, many particles (photons) and each can produce only one electron-hole pair. The distribution for the number of electron-hole pairs produced in this case should be binomial, and should be very well approximated by a Gaussian given the high number of photons in each pulse.

In a future post I will compare the beta setup and the TCT with real measurements, in order to perform the “MIP calibration” in the TCT.

References

↑1 Kolanoski, Hermann, and Norbert Wermes. Particle Detectors: Fundamentals and Applications. Particle Detectors. Oxford University Press, 2020. https://cds.cern.ch/record/2721300. Sec. 3.2.1.3. Kolanoski, Hermann, and Norbert Wermes. Particle Detectors: Fundamentals and Applications. Particle Detectors. Oxford University Press, 2020. https://cds.cern.ch/record/2721300. Fig. 3.2. Kolanoski, Hermann, and Norbert Wermes. Particle Detectors: Fundamentals and Applications. Particle Detectors. Oxford University Press, 2020. https://cds.cern.ch/record/2721300. Eq. (3.50). Kolanoski, Hermann, and Norbert Wermes. Particle Detectors: Fundamentals and Applications. Particle Detectors. Oxford University Press, 2020. https://cds.cern.ch/record/2721300. Sec. 8.4. Kolanoski, Hermann, and Norbert Wermes. Particle Detectors: Fundamentals and Applications. Particle Detectors. Oxford University Press, 2020. https://cds.cern.ch/record/2721300. Sec. 3.2.1. Kolanoski, Hermann, and Norbert Wermes. Particle Detectors: Fundamentals and Applications. Particle Detectors. Oxford University Press, 2020. https://cds.cern.ch/record/2721300. Table 8.2. Kolanoski, Hermann, and Norbert Wermes. Particle Detectors: Fundamentals and Applications. Particle Detectors. Oxford University Press, 2020. https://cds.cern.ch/record/2721300. Sec. 3.2.3.2. Kolanoski, Hermann, and Norbert Wermes. Particle Detectors: Fundamentals and Applications. Particle Detectors. Oxford University Press, 2020. https://cds.cern.ch/record/2721300. Fig. 3.37. Kolanoski, Hermann, and Norbert Wermes. Particle Detectors: Fundamentals and Applications. Particle Detectors. Oxford University Press, 2020. https://cds.cern.ch/record/2721300. Sec. 10.1.1. As opposed to external photoelectric effect which is when the electron is removed from the material. This happens e.g. in a photo cathode and requires higher photon energies. Kolanoski, Hermann, and Norbert Wermes. Particle Detectors: Fundamentals and Applications. Particle Detectors. Oxford University Press, 2020. https://cds.cern.ch/record/2721300. Sec. 3.5.3. Kolanoski, Hermann, and Norbert Wermes. Particle Detectors: Fundamentals and Applications. Particle Detectors. Oxford University Press, 2020. https://cds.cern.ch/record/2721300. Sec. 10.1.3. Kolanoski, Hermann, and Norbert Wermes. Particle Detectors: Fundamentals and Applications. Particle Detectors. Oxford University Press, 2020. https://cds.cern.ch/record/2721300. Eq. (3.111). At room temperature? Kolanoski, Hermann, and Norbert Wermes. Particle Detectors: Fundamentals and Applications. Particle Detectors. Oxford University Press, 2020. https://cds.cern.ch/record/2721300. Fig. 10.4. I think this makes sense as long as the intensity is not too high to fully ionize the material. Buang, S.; Subri, E. D.; Kandaiya, S.; Razak, N. N. A. N. A. & Yahaya, N. Z. Gafchromic XRQA-2 film for Strontium-90/Yttrium-90 (Sr-90/Y-90) DetectionJournal of Physics: Conference Series, IOP Publishing, 2018, 1083, 012063. Link. Arfaoui, S; CERN and Joram, C; CERN and Casella, C; ETH Zurich, Characterisation of a Sr-90 based electron monochromator, PH-EP-Tech-Note-2015-003. Link. Review of formulas for relativistic motion, Barletta, Spentzouris, Harms, link. Kolanoski, Hermann, and Norbert Wermes. Particle Detectors: Fundamentals and Applications. Particle Detectors. Oxford University Press, 2020. https://cds.cern.ch/record/2721300. Fig. 3.8. See this link in the section “1.) Laser”. Our detectors usually have a thickness around 50 µm. One example in which it is NOT independent is when the number of photons is high enough such that all the atoms in the silicon become ionized. In this scenario a new incoming photon will not find electrons in the valence band to interact with, so the absorption probability will be lower than for the previous photons.