\chapter{Detection of gamma radiation}

This chapter discusses interaction of ionizing radiation with matter and is followed by an overview of common detector types. Primary focus is on scintillation detection systems. \cite{knoll}, \cite{ahmed}, \cite{ullmann}

\section{Interaction of radiation and matter}

In order to detect ionizing radiation, it has to somehow interact with matter. The measurement is based on detecting consequences of those interactions. The particles interact on atomic level. The quantum of radiation looses energy during its passage through matter in atomic and subatomic interactions and leaves free electrons and cations behind. Fate of the particle depends on its character, in case of electromagnetic radiation the photon ceases to exist after passing its energy. The situation is more complicated for particles of corpuscular radiation.

If the energy of the primary particle is high enough, it generates secondary radiation, which produces more particles and causes cascade interaction. There are many types of secondary radiation, for example Compton scattering, Cherenkov, Bremsstrahlung (breaking), annihilation or secondary gamma emission, it enriches field of primary radiation.

Excitation and ionization represent the most important interaction of a charged particle with matter. The particle passes its kinetic energy mostly by electromagnetic interaction with electrons in atoms of matter. If the energy is not high enough the electron is only transferred to higher energy level (excitation) from where it returns (de-excitation) and emits a photon. If the passed energy is high enough the electron becomes free and the atom is therefore ionized. The freed electron may have enough energy to cause secondary ionization. Ionizing effects of the particle along its track can be described by the Bragg curve. As the particle slows down its ionizing effects increase because it affects surrounding electrons for a longer period. Just before the particle is fully stopped, there is a peak in Bragg curve.

Beside those principles, there are other phenomena which usually produce secondary radiation. One of them is scattering – it can be either inelastic or elastic. The first one alters wavelength of the radiation while the latter does not. Another important phenomenon is Bremsstrahlung – breaking radiation. It is emitted during scattering of charged particles with great angle during which the vector of velocity of the particle changes rapidly. According to Maxwell equations, it leads to emission of electromagnetic radiation in form of X-ray and gamma photons. This radiation has a continuous spectrum. The phenomenon occurs mostly close to the nucleus (the shorter distance is the greater energy Bremsstrahlung photon has). X-ray radiation is created also in form of the so-called characteristic radiation with discrete spectrum dependent on material. When the particle unbinds electron on low energy level, its place is filled by electron on higher energy level along with emission of photon of characteristic radiation.

Positronic radiation beta-plus interacts the same way as an electron and unbinds electrons from surrounding atoms. However, after it stops, it annihilates with electron. They both cease to exist and are converted into two quanta of gamma radiation with energy 511 keV \cite{knoll}. The quanta starts to move in an opposite direction.

\subsection{Interaction of gamma radiation}

This thesis is focused on detection of gamma radiation, and therefore the mechanisms of interaction of gamma photons and matter should be described well. These effects also apply to X-ray radiation because it has the same physical principle as gamma radiation.

Quanta of gamma radiation – photons – do not have electrical charge and therefore cannot affect charged particles by electrical forces. Oscillation of electric and magnetic field by which the photon is accompanied can however pass energy to particle and cause excitation or ionization. The nucleus can also be excited by interaction with photons. For lower energies the radiation has rather wave character and affects the whole atom while for higher energies the radiation has rather corpuscular character and interacts mainly with individual electrons.

Some interactions taking place between gamma radiation and matter do not have ionizing character – namely Thompson-Rayleigh scattering, Thomson scattering and electron excitation. It is possible that there is no interaction at all when light material and hard radiation are considered. These phenomena have very little importance regarding the radiation physics.

There are four basic types of interactions. Three of them are showed in figure \ref{fig.interactions}. The first one is a photoelectric effect during which the photon passes all its energy to electron in an electron cloud of atom. This electron is emitted with kinetic energy given by difference between passed energy and bond energy. The unbound electron will act like particle of beta-minus radiation and may ionize other atoms. Another electron de-excites to created free place and emits characteristic radiation which can further interact with electron cloud and free the so-called Auger electron. Probability of photoelectric effect is proportional to fifth power of proton number and inversely proportional to energy of the photon. Therefore, the effect does not occur much for energies greater than 1 MeV.

\begin{figure}[!h]
  \begin{center}
    \includegraphics[scale=1]{images/interactions}
  \end{center}
  \caption{Interactions of gamma radiation \cite{medical}}
  \label{fig.interactions}
\end{figure}

When the photon collides with free or lightly bounded electron, the Compton scattering takes place. It is inelastic scattering during which the photon looses only part of its energy and alters its direction by angle $\theta$. The electron is accelerated by this collision and is able to further ionize. Probability of the interaction is proportional to a density and proton number and inversely proportional to energy of photon.

If the energy of photon is greater than 1.022 MeV, it can be converted to electron and positron in result of electromagnetic interaction with strong field of nucleus. The threshold for this interaction is given by a sum of invariant energy of created particles, that is 511 keV \cite{knoll} for both of them. Under some circumstances, the pair generation may take place in electron cloud as well. Positron eventually annihilates and secondary gamma radiation is emitted. For high energies of photon this is the dominant interaction.

When the energy is even higher (at least 2.23 MeV – threshold energy for deuterium \cite{ullmann}), it is possible that nuclear photoelectric effect occurs. During the process, nuclear reaction is initiated and either proton or neutron is unbound from the nucleus. The resulting nucleus may be radioactive due to so-called gamma-activation.

\section{Inverse square law}

The intensity of radiation may be characterized by its flux – a number of particles passing through a unit area per unit time. It is inversely proportional to the square of distance from a point source. There are no point sources in fact but common sources can be approximated by them if the distance is high in relation with dimensions of the source. This law is extremely important for radiation detection. All quantities used for description of radiation intensity, such as dose rate, are dependent on flux and therefore are inversely proportional to square of distance. The dose rate produced by a known gamma ray source in distance $d$ from it can be simply calculated from the following equation \cite{exp-rate}:

\begin{equation}
\label{eq.drate}
\dot{D} = \frac{\Gamma}{3,7\cdot 10^8}\frac{A}{d^2} \hspace{20mm} [\unit{mGy\cdot h^{-1}; R\cdot cm^{2}mCi^{-1}h^{-1}, Bq, m}]
\end{equation}

$A$ stands for activity, it is a number of radioactive transformations per second and $\Gamma$ is exposure rate constant, its value for different radionuclides can be found in \cite{exp-rate}. The inverse square law is very limiting, especially in case of radiation map measurement and source localization.

It is important to mention that this law is generally valid only in vacuum. In other environments the interaction of radiation with matter takes place and the radiation is attenuated. However, attenuation coefficient in air is nearly equal to one.


\section{Detector types}

There are three basic types of detectors that are able to measure instantaneous intensity of ionizing radiation and are widely used. First, there are gas-filled detectors, then semiconductor detectors and finally scintillation detectors. A separate section is dedicated to scintillation detectors, first two types are described in this section.

\subsection{Gas-filled detectors}

Ionizing chambers, proportional detectors and Geiger-Müller tubes belong to this category. There are also spark and corona detectors \cite{gaseous}. Their common principle is a generation of ion pairs in gas. Ionizing chambers usually work in current mode. They are based on a collection of charge generated by a passage of radiation quantum. Cations and free electrons are charge carriers. They recombine quickly and therefore the collection has to be fast. High intensity electric field is used to do so. The particles are attracted to electrodes, and generate electric current. Its size is proportional to the intensity of incident radiation. Ionizing chambers are typically used in standard handheld dosimeters and are sensitive   to energies greater than 100~keV. They are also suitable for calibration of radiation sources due to their good time stability.

Proportional detectors (counters) work almost exclusively in pulse mode. They take advantage of an avalanche effect and therefore are able to detect even a low intensity of radiation compared to ionizing chambers. If the electric field is intense enough, the freed electrons may gain enough energy to cause secondary ionization. The secondary electrons create tertiary ones and the avalanche goes on until all free electrons are collected by anode. A resistor is in series with detector and the voltage drop on it causes the avalanche to stop. A voltage pulse is measurable. These detectors work in proportional region of a current–voltage characteristic when amplitude of the pulse is proportional to energy of the radiation. Therefore, they can be used for spectroscopy. Typically, the detector is formed of cylindrical cathode with a thin anode wire in its axis. The advantage of this form is a narrow localization of the avalanche – it leads to shorter dead time in comparison to GM tubes. Proportional counters are suitable for detection of gamma radiation with energy lower than 100~keV. With increasing energy the probability of interaction rapidly drops.

Geiger-Müller tubes use the avalanche effect as well and also have a similar geometric form. High internal gain is their advantage because often it is not necessary to use external amplifier which would only cause additional noise. In GM tube, one primary particle usually invokes more than one avalanche due to a chain reaction. The chain reaction is stopped after enough cations are present to reduce the electric field. Number of needed cations is rather constant and therefore the amplitude of measured pulse is always the same. It means that Geiger-Müller tubes do not have spectrometric abilities. It takes quite a long time until cations are collected, and during this period avalanches are not created, and therefore another ionizing particle cannot be detected. The phenomenon is called dead time. Geiger-Müller tubes operate  in region of current–voltage characteristic where plateau occurs. GM tubes are used in dosimetry application for example in planar form as a detector of surface contamination.

\subsection{Semiconductor detectors}

Utilization of solid state for detection of ionizing radiation has many advantages. They are superior to gas filled detectors in case of high energy radiation due to their higher density. Scintillation detectors also have this advantage but compared to semiconductor detectors they have worse energy resolution due to lower quantum efficiency.

Electron--hole pairs serve as charge carriers in semiconductor detectors. They are generated by a passage of primary charged particle or by a secondary charged particle generated by interaction with photon. The movement of charge carriers in applied electric field creates output signal. Besides better energy resolution, semiconductor detectors are also more compact and have fast response.

For a description of these detectors, ionizing energy is an important parameter. It determines an average amount of energy which is necessary to deliver to electron to jump from valence to conduction band. Typically, it has value 3 eV (ten times less in comparison to gas filled detectors). With this knowledge one is able to calculate the energy of incident radiation from measured charge. Therefore, semiconductor detectors are suitable for spectroscopy. Ionizing energy depends on temperature. It is almost independent on the type of radiation. 

The depleted region of PN junction serves as sensitive volume for detection. The junction  operates in reverse mode to enlarge the depleted region and collect generated charge carriers before recombination. Thermal generation of electron-hole pairs is a considerable issue. Silicon detectors may operate  in room temperature but germanium detectors have to be always cooled due to their lower band gap.

It is possible to use detectors in current mode and it is convenient especially in high rate situations. They have good linearity up to several amperes. There is another operation mode – the so-called photovoltaic. The detector in this mode works without voltage supply, it uses internal electric field only. But the efficiency is very low.

For the detection of gamma radiation PIN diode is more suitable. Major part of its volume is made of intrinsic semiconductor I in which electron-hole pair generation takes place. Beside standard semiconductor detector, there are also avalanche detectors. They are analogic to proportional counters.


\section{Scintillation detectors}

Scintillators are based on converting energy of incident radiation particles into photons in visible spectrum. These photons are then measured using photodetectors which will be described later in this chapter. There are two categories of scintillation detectors – inorganic and organic – their mechanism of conversion slightly differs. In general, a particle in the detector is excited to higher energy level and during its relaxation a photon of visible light is emitted. This principle of detection is widely used and even suitable for spectroscopy. Although the efficiency of scintillation typically ranges from 10 \% to 15 \%, it is possible to develop highly efficient detector thanks to the photodetectors.

\subsection{Scintillation mechanism}

In organic scintillators, the whole molecules are excited and therefore they can be found in solid, liquid and also gaseous states. Basic energy level of electrons is $S_0$ ground level ({\it S} for singlet). Absorption of energy causes their transition to $S_1$ band, i.e. $S_1$ ground state and its vibrational levels. Through radiationless transitions all electrons eventually end in $S_1$ ground state. Then they fall back to the $S_0$ ground and vibrational levels, causing emission of photons. The whole process takes only a few nanoseconds and it is called fluorescence. A typical energy diagram is shown in Fig.~\ref{orgflu}. Due to the energy loss of electrons transitioning from vibrational levels, Stoke’s shift takes place causing a shift of peak of absorption and emission spectrum. Consequently, the re-absorption of scintillation photons in material is lower.

\begin{figure}[!h]
  \begin{center}
    \includegraphics[scale=0.86]{images/org_fluorescence}
  \end{center}
  \caption{Fluorescence in organic scintillators \cite{ahmed}}
  \label{orgflu}
\end{figure}

Electrons in $S_1$ band can also relax to metastable $T_1$ band ({\it T} for triplet). Eventually all electrons fall back to the $S_0$ band with emission of light. But this process known as phosphorescence typically lasts more than 100 milliseconds and therefore creates a parasitic signal. Energy diagram can be found in Fig. \ref{orgpho}.

\begin{figure}[!h]
  \begin{center}
    \includegraphics[scale=0.86]{images/org_phosphorescence}
  \end{center}
  \caption{Phosphorescence in organic scintillators \cite{ahmed}}
  \label{orgpho}
\end{figure}

Inorganic scintillators are found in crystalline structure and only individual electrons are excited. Generally, they have higher density and atomic number, therefore they are more suitable for detection of gamma radiation. 

These scintillators are basically insulators with wide gap between valence and conduction band. Inside the gap, luminescence and recombination centers can be found. Their presence is caused by dopants (activators) which are required for emission of photons in visible spectrum. If the absorbed energy is greater than band gap, the electron moves to the conduction band leaving a hole in valance band. After some short amount of time, the electron jumps to the energy level of luminescence center (essentially excited level of activator). Following de-excitation to the recombination center (ground level of activator) can produce emission of photon. Alternatively, energy is dissipated by non-radiation transfer into heat. That is undesirable because it leads to an information loss – quenching. The described process of instant light emission is called fluorescence (analogy with organic detectors).

Another possibility for the electron in the conduction band is to jump to a~meta-stable energy state called electron trap. Presence of traps is caused by impurities and defects of the crystal. For a trapped electron it is impossible to recombine with the hole but is able to return to the conduction after receiving typically thermal energy. Then electron may cause delayed light emission – phosphorescence. Energy diagram of scintillation in inorganic material is shown in Fig. \ref{inorg}.

\begin{figure}[!h]
  \begin{center}
    \includegraphics[scale=0.9]{images/inorg_scint}
  \end{center}
  \caption{Diagram of inorganic scintillators \cite{ahmed}}
  \label{inorg}
\end{figure}

\subsection{Basic parameters}

The most important parameter of  the scintillator is light yield. Low light yield causes insufficient signal-to-noise ratio and is critical especially in spectroscopic applications. It is usually measured in number of emitted photons per MeV of absorbed radiation. Tens of thousands of photons per MeV is not an uncommon value. Light yield depends mostly on the scintillation material, type of incident particles, energy of particles and temperature. The energy dependence worsens energy resolution of the system and is caused by secondary excitation of electrons.

Other important parameters are rise and decay times. Both scintillation rise and decay have exponential character but their time constants may differ rapidly. While typical rise time is less than one nanosecond, decay time ranges from a few nanoseconds to several milliseconds. Long decay time is causing lower efficiency of the system, especially in high rate situations. Decay time constant varies for different types of incident particle. For example, alpha-particles due to their high stopping power cause that more electrons end in the metastable energy states and therefore cause significant delayed emission. Usually, the average decay constant can be found in technical literature. 

\subsection{Plastic scintillators}

Plastic detectors are fabricated by homogenously mixing base monomer material and scintillator with concentration about 1 \%. The mixture is then polymerized. In result it is chemically stable and optically homogenous. It also can be easily fabricated in any shape. Because the photons emitted by scintillator usually have short attenuation lengths (typically few millimeters), a combination of more scintillation materials is used. The main scintillator called primary fluor emits ultraviolet photons with larger attenuation length. The secondary fluor works as a wavelength shifter and converts ultraviolet photons typically to blue light.

Although these detectors have rather varied properties, generally they are not suitable for spectroscopy. Usually they have high light yield but it is dependent on type and energy of radiation and climatic conditions. The major problem is the nonlinearity of light output. The material is also unsuitable for measurement of high radiation fields because of extensive delayed glow effect. The scintillation and optical properties can be altered by changing concentrations of primary and secondary fluors.


\subsection{Inorganic scintillators}

The most typical material in this category is thallium doped sodium iodide – NaI(Tl). It is also the most widely used scintillator thanks to its features. It has light yield higher than 40,000 photons per MeV and low self-absorption of scintillation photons. Photons of blue light with wavelengths coinciding with common photomultiplier tubes are emitted. Large area crystals can be produced and the production cost is low. It is also suitable for spectroscopy. The biggest disadvantage of NaI(Tl) is hygroscopicity which makes material vulnerable to moisture. Therefore, the crystal has to be sealed well.

Other typical inorganic scintillators are sodium doped cesium iodide (CsI(Na)), thallium doped cesium iodide (CsI(Tl)) and bismuth germanate (BGO). The first one has similar light yield and emission like NaI(Tl) but is less hygroscopic. CsI(Tl) has light yield over 65,000 photons per MeV, is mechanically stable and non-hygroscopic. Its emission coincides with common photodiodes. Bismuth germanate has large energy resolution, short decay time and is mechanically stable. It also has high absorption efficiency for photons of gamma radiation and is therefore commonly used for gamma spectroscopy.

\subsection{Light guides}

It is not always feasible to attach the scintillator directly to a photodetector with large enough area. Therefore, light guides are employed. Their purpose is to transmit scintillation photons to the photodetector with minimal loss. Light guides are based on either simple reflection or total internal reflection. The latter is more convenient because it offers lower photon loss. Typical guide materials have refractive index about 1.5 and are surrounded usually by air. That gives critical angle larger than 42°. Fulfilling of this condition for total reflection is guaranteed by proper geometry, usually the so-called fish-tail shape is used. If the path from detector to photodetector is large, optical fibers are utilized.

\section{Photodetectors}

There are two basic types of photodetectors – photodiodes (PDs) and photomultiplier tubes (PMTs). The latter are widely used because of good signal-to-noise ratio and fast response. On the other hand, they are more mechanically sensitive. 

\subsection{Photomultiplier tubes}

The amount of photons emitted by the scintillator is not very large, and without some form of amplification signal-to-noise ratio would be very poor. Photomultiplier tube serves as an amplifier, it converts photons into a large number of electrons. Typical PMT consists of photocathode, electron multiplication structure composed from dynodes and readout anode. Good sensitivity, time response and signal-to-noise ratio are advantages of these devices. Schematic picture can be found in Fig.~\ref{fig.pmt}

\begin{figure}[!h]
  \begin{center}
    \includegraphics[scale=0.8]{images/pmt}
  \end{center}
  \caption{Scheme of a photomultiplier tube \cite{photonics}}
  \label{fig.pmt}
\end{figure}

Photocathodes are able to emit electrons thanks to photoelectric effect. The maximal wavelength of photons to be converted is given by work function of photocathode material. The quantum efficiency of conversion usually ranges from 10 \% to 30 \%. It depends strongly on the wavelength of photons. The dependency has a plateau and beyond it the efficiency drops rapidly. Therefore, scintillation spectrum of the detector has to overlap with this plateau, its peak should be located somewhere in the middle of the plateau. There are two ways to make photocathode~– it can be either transmissive or reflective. The first one allows transmission of photoelectrons through it and is usually constructed as a thin layer on the entrance window. Some materials have good quantum efficiency but are opaque to photoelectrons. However, it is possible to use them in reflection mode. Such photocathode is constructed as a thin layer on a metal electrode inside the tube. Bialkali materials (e.g. SbRbCs) are the most common because they are sensitive to blue light emitted by NaI(Tl) scintillators. Their quantum efficiency reaches 30 \% and they are also thermally stable. Other materials for photocathodes are for example AgOCs, GaAs(Cs) and SbCs$_3$.

The electrons are then collected by the first dynode. Due to low efficiencies, it is essential to collect as many photoelectrons as possible. When fast timing is required all electrons should have the same transit time. To fulfill those requirements, electron focusing structure is often utilized in photomultiplier tubes.

The electron yield is then increased by multiplication structure. Cascade of metal electrodes, called dynodes, is used. They have an ability to emit electrons after interaction with an electron that have enough energy. The amount of emitted electrons is larger than the amount of incident electrons. The conversion process has a high linearity. To provide electrons with energy, they are accelerated by electric field. Each dynode has higher potential than the previous one. The potential difference between photocathode and anode is typically hundreds of volts. Geometry of dynodes is crucial because it is needed to focus electron flux to the next dynode. Some photomultiplier tubes are very sensitive to mechanical jitters which causes a non-linearity. There are many different dynode geometries, some of them are even less prone to mechanical instability, for example: linear, box and grid, circular cage, venetian blind, mesh, micro-channel plate and others. High potential gradient is provided by voltage divider. The photocathode is usually connected to the ground. Stabilization capacitors are usually connected to the last few dynodes because large signal current could alter inter-dynode voltage.

The anode has a form of metallic grid located very close to the last dynode. Collection of electrons before they reach the last dynode by anode is not considerable due to the spaces in its grid. Secondary electrons from the last dynode are not accelerated to next one, and therefore create a space charge which is quickly collected by anode. Output current can be measured at the anode but it is preferable to convert it into voltage at a load resistor. The voltage is then amplified to achieve good signal-to-noise ratio.

To prevent loss of electrons, the multiplication structure has to be kept in vacuum. The enclosure is usually made of glass. The material of entrance window is important because it has to be well transparent for wavelength produced by the scintillator. Fused silica is commonly used, other possible materials are lime glass, lithium fluoride and sapphire.

There are numerous parameters to describe efficiency of photomultiplier tube. Overall detection efficiency is composed of quantum efficiency of the photocathode and collection efficiency of dynodes. It is defined as ratio of average number of measured pulses to average number of incident photons. Some types of sensitivities are also often quoted by manufacturers. First there is radiant sensitivity – ratio of photoelectric current to incident radiant power [A/W]. Cathode luminous sensitivity is ratio of photoelectric current to flux from a tungsten filament operated at 2856~K, similarly anode luminous sensitivity is defined as ratio of anode current to the flux [A/lm].

Photomultiplier tube is also characterized by gain $\mu$ which carries information of how effectively the photoelectrons are converted into output signal. It is defined as ratio of the anode current to the photoelectric current. Every dynode is characterized by secondary emission ratio defined by equation \cite{ahmed}:

\begin{equation}
\delta = AV_{dy}^\alpha
\end{equation}

$V_{dy}$ is the dynode voltage, $\alpha$ is geometrical and material constant, its typical value ranges from 0.7 to 0.8. Finally $A$ is a proportional constant. Assuming the tube has $n$ dynodes with the  same secondary emission ratio and even voltage distribution with supply voltage $V$, overall gain can be computed as \cite{ahmed}:

\begin{equation}
\mu=\frac{A^n}{(n+1)^{\alpha n}}V^{\alpha n}
\end{equation}

Typical gain of PMTs is in order of $10^5$ or $10^6$. When dynodes have different parameter, the overall gain has value \cite{ahmed}:

\begin{equation}
\mu = \prod_{i=1}^{n} \delta_i
\end{equation}

Another desired feature is the uniformity of photocathode. Usually it is needed to cover large area and the response should not depend on position of incident photon. Generally the sensitivity of photocathode drops with increasing distance from center. For timing applications time and frequency response is important. Frequency response is characterized by cutoff frequency which is dependent primarily on the external circuitry. Time constant is defined by effective load resistance and total capacitance of the output. Time response is given by the electron multiplication structure because the secondary electron emission is a statistical process. Even if the photon pulse is rectangular, the resulting anode pulse has exponential spread. Linear focused tubes have superior time response over other types.

In general, energy resolution of photomultiplier tubes is poor. Again it is implied by Poisson distribution of secondary electron emission. The spread in the number of electrons is therefore inversely proportional to the square root of average number of emitted electrons. The only way to improve energy resolution is to provide dynodes with very high gain $\delta$. Scintillators themselves have good energy resolution, therefore semiconductor photodetectors are often utilized to improve resolution of the whole detection system.

There are two different operating modes for photomultiplier tube. In the first one – digital mode – individual pulses are counted. That requires incident photons to be well separated in time. When the pulses overlap, it is necessary to work in the analog mode in which average anode current is measured. The measurement is always burdened by noise. Among noise sources it is possible to find thermally agitated electron emission, glass scintillation, leakage current or mechanical instability. Photocathodes usually have low work function and therefore they are able to emit electrons after absorbing thermal energy. This noise can be minimized by cooling or by decreasing photocathode area. Thermal emission along with leakage current caused by imperfection in insulations are the main sources of the so-called dark current. It is current measurable even if no photons are entering the photomultiplier tube. The resulting signal noise is also affected by readout electronics. The noise is a problem mainly when measurement of pulses’ amplitude is needed – in analog mode and in spectroscopic applications. If the device works purely in a counting mode, the noise is suppressed by discriminators. PMTs are also sensitive to a magnetic field which can alter the trajectory of electrons in the multiplication structure.

\subsection{Photodiode detectors}

In some circumstances, using a semiconductor photodetector can be more convenient. These photodetectors have very high quantum efficiency compared to PMTs and are also more compact, mechanically stable and insensitive to magnetic field. Usually, they have worse time response. However, modern technologies improve parameters of photodiodes. Several types of diodes such as PIN diode, heterojunction diode or Schottky diode can be used as photodiodes. They just need to be optimized to work in visible spectrum.

PIN diode is made of thin entrance P layer, followed by depleted I region and closed by N layer. Incident photon in depleted region produces an electron--hole pair which is collected by preamplifier thanks to the applied voltage. The generated and measured charge is of the same order as number of incident photons, so it is much lower than in photomultiplier tubes. Photodiodes have therefore worse signal-to-noise ratio. There are two types of noise – serial and parallel. The first one appears at the input of preamplifier and its magnitude is proportional to capacity of the diode. The smaller dimensions of diode are the higher its capacity is. Diodes with area lower than 1 cm$^2$ are unsuitable. The parallel noise corresponds with the leakage current and increases with the increasing dimensions of the diode. The noise is affected also by a time constant of the readout circuitry. As the time constant increases, the serial noise is decreasing but the parallel noise is increasing and vice versa. Because of these conflicting requirements, it is always necessary to search for a compromise. Dark current is also significant parasitic effect, it has identical meaning as dark current in PMTs. Especially silicon diodes have higher noise level than a similar photomultiplier tube, its utilization in higher temperature is almost impossible. More suitable material for photodiodes is for example mercury iodide. The noise also makes photodiodes unsuitable for large area scintillators, matrix semiconductor photodiodes have not proved good. Another problem is that the particle of ionizing radiation could interact directly with photodetector’s sensitive volume and cause significant parasitic signal. Therefore, the diode has to be as thin as possible – again it is conflicting requirement.

\subsection{Avalanche photodetectors}

An even better solution is to use avalanche photodiodes (APDs). They use similar multiplication effect as proportional gas detectors. Incident photon generates electron--hole pair and the electron is accelerated by applied high voltage. If the energy of electron is at least three times greater than the band gap of the material, it is able to produce secondary charge pairs. The secondary electrons are again accelerated and may produce tertiary charge pairs and so on. In result, the number of produced charge pairs is much higher than in conventional photodiodes. The current is proportional to energy of incident photons and the signal-to-noise ratio is also good, therefore avalanche detectors are suitable for spectroscopic applications.

There are some desirable features that APDs should have. Small leakage current is required. It causes dark current which can cause unwanted avalanches that may induce non-linearity to response. High and stable gain is also wanted to achieve good signal-to-noise ratio and linearity. Typical APDs have gain of the order of 10$^4$ but can reach up to 10$^8$. Finally, good frequency response comparable to the response of photomultiplier tubes is desirable especially in high rate situations.

The gain is proportional to the applied high voltage and is very sensitive to any change of this voltage, therefore well stabilized voltage source is required. The applied voltage is higher than in case of PIN diodes, however must not be too high because it could cause breakdown of the diode. In a typical form the entrance window is made of thin P layer to which negative voltage is applied. Bulk made of lightly doped semiconductor of P type follows, the charge multiplication takes place here. The structure is closed by P layer and N layer to which positive voltage is applied.

The avalanche diode can work either in linear or Geiger mode. In the latter, the amplitude of charge pulse is not proportional to incident photon energy and therefore is used only as counting device, similarly as the photomultiplier tube in digital mode.

\section{Direction-sensitive detection}

When a lost radiation source is searched, it can be assumed that information about direction would be helpful. To get that type of information, a direction-sensitive detection system is needed. One possible way to achieve this goal is to assemble the system from multiple independent detectors. The idea is inspired by \cite{directional}. An option with two detectors is discussed further below.

The basic idea is that the source located outside of axis of symmetry of two-detector system has larger distance from one detector than from the other. For some angles, also collimation takes place. The radiation is attenuated in closer detector before reaching the further. It is possible to even intensify the effect by placing shielding between the detectors.

The angular dependence of distance can be expressed analytically. Geometry of the system can be seen in Fig.~\ref{fig.geometry}. Ratio $R$ of total count measured by both detectors should carry information about source direction. Measured total count is inversely proportional to square of distance. Assuming detectors have same sensitivity $K$, the ratio can be expressed as:

\begin{equation}
R = \frac{\frac{K}{d_1^2}}{\frac{K}{d_2^2}} = \frac{d_2^2}{d_1^2}
\end{equation}

Square of distance can be expressed from the law of cosines as a function of anglular distance of the source from detectors' axis of symmetry $\phi$.

\begin{figure}[!h]
  \begin{center}
    \includegraphics[scale=0.45]{images/geometry.pdf}
  \end{center}
  \caption{Geometry of the two-detector system}
  \label{fig.geometry}
\end{figure}

\begin{equation}
d_1^2 = s^2 + \left( \frac{b}{2} \right)^2 - 2s\left( \frac{b}{2} \right) \cos{(90-\phi)} = s^2 + \left( \frac{b}{2} \right)^2 - 2s\left( \frac{b}{2} \right) \sin{\phi}
\end{equation}

\begin{equation}
d_2^2 = s^2 + \left( \frac{b}{2} \right)^2 - 2s\left( \frac{b}{2} \right) \cos{(90+\phi)} = s^2 + \left( \frac{b}{2} \right)^2 + 2s\left( \frac{b}{2} \right) \sin{\phi}
\end{equation}

Expressions containing $s$ or $b$  can be replaced by contants $c_0$ and $c_1$. The ratio dependence of angular distance has therefore shape of function:

\begin{equation}
R = \frac{d_2^2}{d_1^2} = \frac{c_0 + c_1\sin{\phi}}{c_0 - c_1\sin{\phi}}
\end{equation}

A comparison of the shape of a function for different values of parameter $s$ is in Fig.~\ref{fig.shape}. It is obvious that with increasing distance of source the shape is more similar to sine wave and its amplitude is decreasing. The low amplitude present a problem because of bad signal-to-noise ratio of measurement of distant sources. Utilization of shielding between detectors could be helpful.

\begin{figure}[!h]
  \begin{center}
    \includegraphics[scale=0.62]{images/shape.pdf}
  \end{center}
  \caption{Parametrized shape of the ratio dependence of angle}
  \label{fig.shape}
\end{figure}