Energy Loss

The principle of solid state detectors is based on the energy loss of traversing particles. Free electron-hole pairs are generated, which move towards opposite electrodes under the influence of an electric field. The energy loss of heavy particles in matter was described by H.A.BETHE^2.1and F.BLOCH^2.2 [3].

$$-\frac{1}{\rho}\frac{dE}{dx}=4\pi N_{A} r^2_e m_e c^2 z^2 \f... ... \right) -\beta^2-\frac{\delta(\gamma)}{2}-\frac{C}{Z} \right]$$ (2.1)

Eq.  represents the differential energy loss per mass surface density $[\rm MeV\,(g\,cm^{-2})^{-1}]$, where $ze$ is the charge of the incident particle, $N_A$, $Z$ and $A$ are Avogadro's number, the atomic number and the atomic mass of the material, $m_e$ and $r_e$ are the electron mass and its classical radius ( $\frac{e^2}{4 \pi \epsilon_0 m_e c^2}$). $T_{\rm max}$ is the maximum kinetic energy which is still detected in the material, $I$ is the mean excitation energy, $\beta=v/c$, $\gamma=(1-\beta^2)^{-1/2}$ and $\delta(\gamma)$ is a correction for the shielding of the particle's electric field by the atomic electrons, the density effect caused by atomic polarization. At very low incident particle energies, the basic assumption of static atomic electrons is violated, which is taken into account by the shell correction term $C$.

However, in thin layers, the deposited energy is less than expected because a fraction of the lost energy is carried off by energetic knock-on electrons (also known as $\delta$ electrons). These considerations lead to the restricted energy loss, which is expressed by an additional term in the Bethe-Bloch equation [9],

$$-\frac{1}{\rho}\frac{dE}{dx}=4\pi N_{A} r^2_e m_e c^2 z^2 \f... ...}}\right) -\frac{\delta(\gamma)}{2}-\frac{C}{Z} \right] \quad,$$ (2.2)

where $T_{\rm upper}=\inf(T_{\rm cut},T_{\rm max})$ with $T_{\rm cut}$ depending on the material and the incident particle momentum.

Figure: Energy deposition of pions in silicon. While the standard Bethe-Bloch theory covers thick layers, restrictions apply to thin layers as shown for $300\,\rm \mu m$ to account for energy carried off by energetic knock-on electrons.

Fig.  compares the standard Bethe-Bloch theory to the restricted form for a pion traversing $300\,\rm\mu m$ of silicon in terms of the incident particle momentum. In the low energy range, there is no difference between standard and restricted forms, since knock-on electron production is improbable. However, in the regime of a few hundred $\rm MeV/c$, there is already considerable deviation: The standard theory predicts a minimum ionizing particle (MIP) at $450\rm\, MeV/c$, while the restricted energy loss states $750\rm\, MeV/c$. Moreover, the relativistic rise at high energies is quite flat in the restricted model due to energy carried off by knock-on electrons.

The statistical fluctuation of the energy loss in thin layers was described by L.D.LANDAU^2.3 [10]. The Landau distribution resembles a distorted normal distribution with a long upper tail due to rare, but highly ionizing knock-on electrons.

The tail of an ideal Landau distribution extends to infinite energies, which is unrealistic. In practice, the measurement range is always limited, which leads to a truncated Landau curve. As a result of its asymmetry, the mean energy loss is higher than the most probable (MP). However, the latter is much easier to obtain from measured data and therefore usually stated in experimental results. The scale factor between MP and mean is typically around 1.3 but depends on particle energy and measurement range.

With particle energies far below the MIP energy, corresponding to thick layers, knock-on electrons are improbable, the Landau tail vanishes and thus the resulting distribution is Gaussian^2.4.

The restricted energy loss model has been confirmed by experiments, e.g. by a dedicated test with silicon pad sensors at BNL performed in 1998 by HEPHY and MIT, where excellent agreement between measurement and theory was found [11]. Fig.  shows the most probable energy loss of pions in the range of minimum ionization in a silicon detector of $300\,\rm\mu m$ thickness.

Figure: Calculated and measured most probable energy deposition of pions in a silicon detector of $300\,\rm \mu m$ thickness, compared to the standard Bethe-Bloch theory.

The energy deposited in the detector material flows into the creation of free electron-hole pairs. The number of pairs $n$ depends on the total energy loss $E_{\rm loss}$ and the ionization energy $E_{eh}$, which is necessary for a pair production,

$$n=\frac{E_{\rm loss}}{E_{eh}} \quad .$$ (2.3)

In silicon, $E_{eh}=3.6\,\rm eV$, which results in an most probable charge of about $n=22000\,\rm pairs$ for a MIP in $300\,\rm\mu m$ of silicon. Different values between $20000$ and $25000\,\rm pairs$ (corresponding to a charge between $2 \times 3.2$ and $4\,\rm fC$) are given in literature. Within this thesis, a number of $n=22500\,\rm pairs$ shall be defined as the MIP charge.

Figure: Measured MIP signal distribution in a silicon detector of $300\,\rm \mu m$ thickness.

The measured energy loss distribution of MIPs in a typical silicon sensor ($300\,\rm\mu m$ thick) is shown in fig.  in terms of the collected charge [12]. The measured data have been fitted by a Landau distribution, convoluted with a narrow normal distribution due to electronic noise and intrinsic detector fluctuations [13]. This results in a minor broadening of the shape and a slight increase of the peak position compared to pure Landau. As stated in the boxes on the right, the pure Landau $\rm MP=22250\, e$, but after convolution with a Gaussian distribution of $\sigma=3136\,\rm e$, the position of the $\rm peak=23512\,\rm e$. Since the convoluted peak is the measured property, we will implicitly refer to this value when stating experimental MP signal values.

Fig.  also illustrates why the MP is obtained much easier than the mean value: Cuts on either end do affect the mean, but not the MP. The low edge depends on the pedestal threshold, and the high edge is defined by the range of the readout electronics. Moreover, nonlinearity and saturation occur when measuring very high signals of the tail.

More experimental results, which confirm the theory of energy loss, are found in section , p. .