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4.6.1 Energy loss of charged particles in silicon

When a charged particle traverses a medium, it interacts with the electrons and the nuclei of the atoms of the surrounding material. In the case of heavy charged particles, these interactions are almost exclusively collisions, which may simply excite the atom (“soft” collisions) or cause the ejection of secondary electrons with

considerable kinetic energy (ô-rays from “hard” collisions). The mean rate of energy loss per unit path length due to collision interactions is the linear collision stopping

power. In the case of mono-energetic particles, this is approximated by the Bethe-

Bloch equation as f o l l o ws [ A t t i x , 1986]: dx Anz^Ze^N In

(i-VV

-P'

ze = charge of the primary particle (with e = electronic charge) Z = Atomic number of the medium

N = number of atoms/cm^ of the medium = pNa/A, being p the medium density (g/cm^ ), A the medium atomic weight and Na the Avogadro number

moc^ = electron rest mass (= 0.511 MeV)

/?= velocity of the primary particle in units of c

I = mean ionisation and excitation potential of the absorber (MeV)

In the form of (4.11), the Bethe-Bloch equation is valid for heavy particles only (e.g. protons or pions) and is almost independent of particle type. In the case of light charged particles (such as electrons) the collision stopping power is a similar expression, where modifications must be included as the incident electrons and the atomic electrons in the medium are indistinguishable particles. In order to show the typical behaviour of the energy loss rate curves, the collision stopping powers for electrons and protons in silicon (for which 7 = 1 7 3 eV) are plotted as an example in Fig. 4.17 as a function o f the product py (where p = v/c and y = (1-p^)'^^^). As can be seen from the graph, the energy loss due to the passage o f an electron is reduced with respect to the energy loss due to the passage of a heavy particle. This is caused by the

For simplicity the terms for density and shell correction have not been included in equation (4.11). The complete expression o f the Bethe-Bloch approximation can be found for example in [Skyrme,

Chapter 4 Performance Measurements o f Bonded Silicon Strip Detectors

identical nature of the interacting electrons, which affects the maximum allowed energy transfer.

In practical cases, most relativistic particles have stopping powers close to the minimum of the Bethe-Bloch curve and for this reason are said to be “minimum ionising particles” (or '"mips"). The average energy deposit of 1 mip in 100 pm of silicon is approximately 39 keV [Physics Letters, 1990].

20 1

electrons in silicon protons in silicon

0.1 1 10 100 1000 10000

Py

Fig. 4.17: Mass stopping power in silicon (i.e. the stopping power dE/dx divided by the medium density p in MeVcm^/g), plotted for both electrons and protons as a function o f the product Py.

Energetic electrons may also be subject to energy loss by production of bremsstrahlung radiation. The total linear stopping power is the sum of the collision and radiation terms:

The ratio of the two terms is often expressed by the following relationship [Attix, 1986]:

{ d E / d x \ ^ _ E Z

{deE/dx)^^i 700

The energy loss mechanism by radiation production gains importance with increasing electron energy and atomic number of the medium, as can be seen from relationship (4.13). For electrons such as beta particles (the maximum energy of which is typically in the order of a few MeV) the fraction of energy loss due to bremsstrahlung production is relatively low in a material like silicon. Furthermore, most of the bremmstrahlung photons are very energetic and they are likely to escape the silicon detector without interacting. Therefore, the energy loss by emission of radiation does not contribute significantly to the energy deposition recorded by the passage of beta particles in a silicon detector.

4.6.2 The “L andau peak”

If we assume that the thickness of the material is small so that the stopping power does not change along the path, the mean energy lost by a particle in the absorber will be the rate of loss of energy (given by the Bethe-Bloch approximation) multiplied by the thickness of the absorber. However, large fluctuations o f the energy loss will occur with respect to this average value. These are not simply associated with the statistical nature o f the collision process, but also with the probability of &ray production, which may cause energy losses that are considerably larger than the average value. The ionisation produced by a charged particle results in an energy-loss distribution, which is characteristic of the passage of the particle in a given thickness o f a specific medium.

A discussion of the energy loss distributions associated with the passage of charged particles through silicon can be found for example in [Skyrme, 1967], [Hall, 1984].

Chapter 4 Performance Measurements o f Bonded Silicon Strip Detectors

The first theoretical study of the energy loss fluctuations was performed by Landau. The description of the energy loss straggling was subsequently made more accurate by Vavilov, who introduced a limit to the maximum energy transfer in a single collision (which is assumed to be infinite in the original Landau theory). The Landau- Vavilov distribution is asymmetric and skewed towards high energies. The average value of the distribution is therefore larger than the most probable value, which occurs at the so called ‘Landau peak’. If the medium is particularly thick (~ cm) the number of collisions becomes large and by the Central Limit Theorem the distribution approximates a Gaussian function. The average value of energy loss in this case approaches the most probable value.

The pulse height spectrum recorded by a charged particle detector is described in its general features by the Landau-Vavilov distribution. However, some important differences need highlighting. Firstly, the distribution o f the energy lost by a particle does not necessarily correspond to the distribution of the energy deposited by the particle in the absorber. A large number of ^rays may in fact escape from the sensitive volume of the detector. This is particularly the case for the passage of relativistic particles in thin and relatively light media, such as silicon detectors. Therefore, due to the finite geometry and the finite absorption efficiency of the detector, the distribution of the energy deposited will not exhibit such an extreme tail at high energies as a pure Landau-Vavilov distribution. Moreover, experimental studies have observed that the measured distributions tend to be broader than expected on a theoretical basis. The energy loss spectrum that is measured experimentally is in fact a convolution of a Landau-type curve with a Gaussian distribution. This phenomenon is not simply due to the noise produced by the readout electronics. The discrepancy in the width of the distribution is also caused by electron binding effects, as the Landau-Vavilov theory is based upon free electron scattering [G. Hall, 1984], [HancocketaL, 1983].