<figure><img src="image_219.png" /><figcaption>**Figure 35.22:** (left) the form of the light yield and the charge yield vs electric field redrawn from [309]; (right) electron drift velocity vs electric field redrawn from [302].</figcaption></figure>

<figure><img src="image_220.png" /><figcaption>**Figure 35.23:** Schematic of an event in the ICARUS 3 t LAr TPC; Top (from right to left): field lines from drift region to collection wires, electron flow & arrangement of wire-planes, signal shapes on the wires, and wave-forms from the continuous readout and digitization; Bottom: the passage of a muon. The gray scale on the track indicates the ionization density.</figcaption></figure>
the 2D-to-3D approach convolution neural networks (CNNs) are applied on 2D images to identify objects which can be matched across planes [343] [344]. In the 3D approach CNNs, graph neural networks (GNNs), etc., are used in several steps to de-ghost the original 3D image and perform full event reconstruction including calorimetry, particle ID, etc. [345]. (3) The reconstructed tracks and showers are then passed to the calorimetry and particle ID modules. These modules utilize the charge deposition per unit length, after calibrations, to compute track and/or shower energy and particle identification [346].

### 35.7.6 Developments

The successful operation [347] of the ICARUS T600 TPC served as clear demonstration of the feasibility of the liquid argon TPC technology on a massive scale. Since this milestone, the liquid argon TPC community has expanded and the technology has continued to evolve. Notable developments include in-liquid electronics, the adoption of Silicon Photo-Multipliers and new light collection technologies, the use of dopants in the argon to improve the light performance, improved techniques for wavelength shifting, the adoption of a cryostat technology which permits detector masses of tens of kilotons and, in a major change to the standard single-phase topology, the implementation of pixelated readout to replace projective geometry in busy environments [348]. Technical R&D is continuing; as examples, the vertical-drift single-phase design for the second DUNE far-detector that uses strips on perforated printed circuit boards as charge sensors [349], a dual-phase TPC readout [350] using fast cameras to read out the scintillation light produced in the gas region in place of a charge readout; a light detection system powered and readout using optical fiber allowing the readout to operate at any potential and therefore anywhere in the TPC, and efforts to build up the HV directly in the liquid argon with a voltage multiplier (Greinacher design) [351].

An abundance of data from recent experiments is encouraging insights into the liquid argon medium as in [301,352] and revealing new phenomena. For a taste, see [353]. The ground is fertile, busy, and full of opportunity.

## 35.8 Semiconductor detectors

Revised August 2023 by N. Wermes (Bonn U.).

Semiconductor detectors provide outstanding detection opportunities in terms of position, energy and also time resolution, often in combination. In accelerator experiments, they are most widely used as position-sensing devices, e.g. in tracking detectors, or as photodetectors (Sec. 35.2), for example in calorimeters (Sec. 35.10) or in imaging Cherenkov detectors (Sec. 35.5). In recent years also precise time measurement ($O(10-30 \text{ ps})$) with silicon detectors has come into the focus of development. When comparing semiconductor detectors particularly with gaseous detectors, the main features are high density and low ionisation threshold, providing in high resistivity substrates comparatively large signals even without intrinsic amplification. Silicon detectors with active layers only 100–300 µm thick provide adequately large and fast signals on a typical time scale<fnref n="4" /> of 10–50 ns. Challenges are the purity of the semiconductor material, characterised by its mobility-lifetime ($\mu\tau$) product, and its radiation resistance. The development of modern semiconductor detectors is strongly interconnected with integrated circuit technology. Microprocessing and micro-electronics technologies enable fabrication of high-density micron-scale electrodes on large wafers (6–8 in ≈ 15–20 cm diameter for sensor wafers) and allow for high-density amplification and readout circuits connected to them (chip wafer sizes are up to 12 in ≈ 30 cm with typical reticles of 26×33 mm²).

Some important material properties of common semiconductors used as detectors are summarised in Tab. 35.10. While for particle tracking the excellent position resolution is the main (but not the only) interest, high stopping power and high energy resolution are key parameters in X-ray, gamma-ray, and $\beta$ spectroscopy, for example in neutrinoless double-beta decay searches. Due to its small

bandgap, germanium excels in energy resolution but needs to be operated at very low temperatures (liq. $N_2$) to reduce thermally generated reverse bias current. Besides Ge, also GaAs, CdTe and CdZnTe (CZT) feature high atomic numbers and hence much higher stopping power and shorter absorption length which is important, especially for X-ray detection. Diamond, fabricated by chemical vapour deposition (CVD) and strictly speaking classified as an insulator with a large bandgap (Tab. 35.10), features low Z and large radiation length $X_0$. Thanks to its radiation hardness (Sec. 35.8.5) it is used for particle detection in dedicated applications, especially in zones with high particle flux (see e.g. [354] and references therein).

Materials R&D for radiation sensors extends to other bulk semiconductors as well, for example, ZnS, SiC, GaN or InP, and also to new material structures and metamaterials. Examples are semiconductor Quantum Dots, realised by nanometer-sized semiconductor “particles” embedded in a semiconductor bulk, or graphene, which – as a zero bandgap 2D material – features extraordinarily high conductivity (electrical and thermal) as well as outstanding photonic properties. A bandgap can be introduced by employing doped bi- or multi-layer graphene structures, thus rendering transistor and sensor realisations possible. The interested reader is referred to references [355] or [356], for example.

Operating usually without intrinsic amplification, semiconductor detectors crucially depend on low-noise electronics (see Sec. 35.9), so the detection sensitivity is determined largely by signal charge and input capacitance. Reviews of semiconductor detectors and electronics can be found for example in Refs. [1, 8, 357–359]

### 35.8.1 Signal generation in semiconductors

#### 35.8.1.1 Creation of charges

Semiconductor detectors are solid-state ionisation chambers. Absorbed energy forms electron-hole (e-h) pairs, i.e. negative and positive charge carriers which—when moving in an applied electric field—generate a signal current on the electrodes by electrostatic induction (see Sec. 35.8.1.2). For the signal charge carriers to (freely) drift in the electric field and to become detectable, semiconductors must feature small intrinsic charge-carrier densities (as existing for example in diamond and to some extent also in GaAs and CZT) or they must be depleted by reverse-bias junction configurations (as for Si, Ge or CdTe). In addition, they should feature low intrinsic density of defects which can act as trapping or generation/recombination centers (see Sec. 35.8.5).

The minimum energy required to form an e-h pair is the bandgap energy (1.12 eV in Si, 0.66 eV in Ge, 5.5 eV in diamond). However, impinging radiation or particles also release energy to lattice vibrations (phonons) such that the average energy for e-h pair creation is higher. In an “indirect” semiconductor like Si, the valence-band maximum is not at the same position in k-space (crystal-momentum space) as the conduction-band minimum and additional momentum transfer is required for a band transition to occur. Since the “direct” bandgap energy without k-transfer in Si is 3.4 eV, fewer-eV photons must receive momentum from lattice phonons. Because phonons are Bose-Einstein distributed the needed momentum kick causes a steep rise in photon absorption probability between 1.12 eV and about 3.4 eV. For larger energy deposits the average energy $w_i$ needed to produce an e-h pair assumes a constant value of 3.65 eV at room temperature. For other semiconductors consult Tab. 35.10.

For minimum-ionising particles, the most probable charge deposition in a 300 µm thick silicon detector is about 3.7 fC (~23 000 electrons). In tracking detectors, a particle's energy loss and scattering in the detector material should be minimal (large $X_0$), whereas, for energy spectroscopy, e.g. of X-ray photons, the stopping power should be maximised by choosing high-Z semiconductors<fnref n="5" />. A smaller bandgap (in fact a smaller $w_i$) leads to a larger signal per deposited energy and improves the energy resolution, but also (exponentially) increases thermally excited carrier generation. To cope with excessive leakage currents at room temperature, Ge diodes are typically operated at liquid nitrogen temperature (77 K). In pure Si at 300 K, the intrinsic carrier concentration

<footnote n="4">⁴Characterised here by the peaking time of the signal pulse.</footnote>

<footnote n="5">⁵The cross section for photo effect scales as $Z^5$.</footnote>
<table><caption>**Table 35.10:** Properties of some detector-relevant semiconductors; temperature-dependent quantities given at 300 K (from [1] and references therein).</caption><thead><tr><th>Property</th><th>Si</th><th>Ge</th><th>GaAs</th><th>CdTe (CZT*)</th><th>Diamond</th></tr></thead><tbody><tr><td>atomic number (Z)</td><td>14</td><td>32</td><td>31/33</td><td>48/(30)/52</td><td>6</td></tr><tr><td>density ρ (g/cm³)</td><td>2.328</td><td>5.327</td><td>5.32</td><td>5.85</td><td>3.51</td></tr><tr><td>dielectric constant ε</td><td>11.9</td><td>16.0</td><td>13.1</td><td>10.2</td><td>5.7</td></tr><tr><td>semiconductor type</td><td>indirect</td><td>indirect</td><td>direct</td><td>direct</td><td>indirect</td></tr><tr><td>bandgap E<sub>G</sub> (eV)</td><td>1.12</td><td>0.66</td><td>1.424</td><td>1.44(1.44-2.2)</td><td>5.5</td></tr><tr><td>intr. carrier density (cm<sup>-3</sup>)</td><td>1.09×10<sup>10</sup></td><td>2.4×10<sup>13</sup></td><td>2.1×10<sup>6</sup></td><td>10<sup>7</sup></td><td>≈ 0</td></tr><tr><td>radiation length X<sub>0</sub> (cm)</td><td>9.36</td><td>2.30</td><td>2.29</td><td>1.52</td><td>12.15</td></tr><tr><td>average energy w<sub>i</sub> for (e/h) creation (eV)</td><td>3.65</td><td>2.96</td><td>4.35</td><td>4.43</td><td>13.1</td></tr><tr><td>mobility (cm<sup>2</sup>/Vs)</td><td></td><td></td><td></td><td></td><td></td></tr><tr><td>electrons μ<sub>n</sub></td><td>1450</td><td>3900</td><td>8500</td><td>1050</td><td>≈1800<sup>†</sup></td></tr><tr><td>holes μ<sub>h</sub></td><td>500</td><td>1800</td><td>400</td><td>90</td><td>≈2300<sup>†</sup></td></tr><tr><td>lifetime</td><td></td><td></td><td></td><td></td><td></td></tr><tr><td>electrons τ<sub>e</sub></td><td>>100 μs</td><td>~ms</td><td>1-10 ns</td><td>0.1-2 μs</td><td>≈100 ns</td></tr><tr><td>holes τ<sub>h</sub></td><td>>100 μs</td><td>~ms</td><td>20 ns</td><td>0.1-1 μs</td><td>≈50 ns</td></tr></tbody></table>

is $n_i \simeq 10^{10}$ cm$^{-3}$ (Tab. 35.10), corresponding to a resistivity in the order of $\rho \simeq (e\mu n)^{-1} \approx 400$ kΩ cm. In reality, crystal imperfections and minute impurity concentrations limit Si carrier concentrations to about $10^{11}$ cm$^{-3}$ at 300 K ($\rho \approx 40$ kΩ cm). In practice, wafer resistivities up to 20 kΩ cm are available, with mass production ranging from 1 to 10 kΩ cm.

The energy released in a semiconductor is absorbed by electronic excitations (e/h) and lattice excitations (phonons) in an anti-correlated way. Therefore, for a fixed released energy $E$ (for example of an X-ray photon), the variance in the number of charge carriers $N = E/w_i$ follows binomial statistics. Due to the energy constraint, this variance is reduced by the Fano factor $F$ relative to Poisson statistics ($F \approx 0.1$ in Si and Ge). Thus, $\sigma_N = \sqrt{FN}$ and the energy resolution is $\sigma_E/E = \sqrt{Fw_i/E}$. However, for semiconductors the measured fluctuations of a detected energy signal are usually dominated by electronic noise rather than by signal fluctuations. The electronic noise contribution depends much on the detector leakage current and the electrode capacitance as well as on pulse shaping (i.e. the shaping time) in the signal processing electronics (see Sec. 35.9).

For X-ray detection, a major effort is made to find high-Z materials with a bandgap that is sufficiently large to allow for room-temperature operation while still providing good energy resolution. Compound semiconductors, e.g. CZT, can allow this but typically suffer from charge collection problems, which are characterised by the product $\mu\tau$ of mobility and carrier lifetime; this is the depth per field strength that generated carriers can drift before being trapped, i.e. in this context the distance over which photon absorption can still be detected.

In Si and Ge $\mu\tau$ is orders of magnitude larger than in compound semiconductors for both electrons and holes (see Tab. 35.10). Since for holes $\mu\tau$ is typically much smaller than for electrons, detector configurations where the electron contribution to the charge signal near the readout electrode dominates—e.g. strip or pixel structures with electron collection—usually provide better performance (see also next section).

### 35.8.1.2 Signal formation

The signal and its pulse shape depend on the instantaneous carrier velocity $\vec{v}(\vec{x}) = \mu\vec{E}(\vec{x})$, $\mu$ = mobility, and the electrode configuration and its geometry which determine the distribution of induced current according to the Shockley-Ramo theorem

$$i_S(t) = Ne \vec{E}_w(\vec{x}) \vec{v}(\vec{x}(t)), \quad (35.20)$$

where $Ne$ represents a drifting charge cloud of $N$ elementary charges, $\vec{E}_w(\vec{x})$ is the “weighting field”, which accounts for the

coupling of the charge to a specific electrode and depends on the electrode configuration; $\vec{v}$ is the drift velocity. Note the difference between the electric field $\vec{E}$ and the weighting field $\vec{E}_w$ and take account of the fact that the mobility is, in general, field-dependent, $\mu = \mu(E)$, with $v \approx \text{const}$ at high fields (velocity saturation, for electrons in Si approaching $10^7$ cm/s at $E > 10^4$ V/cm). Both, electron and hole movements contribute to an electrode’s signal. Hence, if the carrier mobility is very different for electrons and holes, like e.g. in CdTe where $\mu_h \ll \mu_e$ (Tab. 35.10), a photon signal, for example, becomes absorption-point dependent.

Integration of the induced current signal on an electrode yields the “collected charge”. The average time to collect the created charge decreases with increasing bias voltage (i.e. field strength) until velocity saturation occurs at field strengths of about $10^4$ V/cm in Si.

For a simple parallel-plate geometry with two electrodes the weighting field is constant, whereas for structured electrode geometries, like for example strips or pixels, $E_w$ is position dependent which for small electrodes (compared to the sensor dimensions) strongly enhances the contribution of the movement close to the electrode (“small-pixel effect”). More details and practical accounts of the Shockley-Ramo theorem and its usage can be found in [1,8] and references therein.

Position resolution is ultimately limited by transverse diffusion of the moving charge cloud (typically 3–5 µm for 200–300 µm thickness) and by the emission of $\delta$ electrons. The performance then depends on optimal usage of charge sharing between neighbouring electrodes and on noise. In magnetic fields, Lorentz drift deflects the electron and hole trajectories thus increasing the spatial spreading. The total spreading and hence charge sharing between electrodes can be tuned (increased or decreased) by tilting the detector relative to the incoming (average) particle direction. Overall spatial resolutions of 2–4 µm (rms) have been obtained.

### 35.8.2 Junction detectors

Si and Ge detector substrates must be—others like CdTe, GaAs should be and CZT can be—depleted from free charge carriers by operating them as reverse-bias junctions (p-n or Schottky) to be sensitive to the charge created by impinging particles. A typical cross-section of a junction detector is shown in Fig. 35.24, here with structured electrodes at the top. A p-n junction—even without external voltage—forms a sensitive space-charge region across itself depleted of mobile charges, hence also called “depletion region”. The space charge establishes an electric field corresponding to a “built-in” voltage $V_{bi}$. Additional reverse-bias voltage $V$ applied externally increases the space-charge region such that in-

<footnote>*CZT = CdZnTe with the bandgap depending on the Cd to Zn ratio.</footnote>

<footnote>†Approximate averages. Values quoted in the literature for the mobility in diamond vary strongly.</footnote>
jected charge liberated by radiation is swept to the electrodes by the existing field. Detectors typically use an asymmetric structure, for example, a thin and highly doped $p^+$ electrode⁶ region and a lightly doped $n^-$ substrate region – or vice versa, so that the depletion region extends predominantly into the more lightly doped bulk volume.

In such planar Si (or Ge) devices, the thickness of the depleted region is

$$ 
\begin{align} 
d &= \sqrt{2\epsilon(V + V_{bi})/N\epsilon} = \sqrt{2\rho\mu\epsilon(V + V_{bi})} \tag{35.21} \\ 
&\approx \frac{0.5}{\mu\text{m}} \times \sqrt{\frac{\rho V}{\Omega \text{cm} \cdot \text{V}}} \quad \text{for } n\text{-type Si bulk} \nonumber \\ 
&\approx \frac{0.3}{\mu\text{m}} \times \sqrt{\frac{\rho V}{\Omega \text{cm} \cdot \text{V}}} \quad \text{for } p\text{-type Si bulk} \nonumber 
\end{align} 
$$

with (values for Si)

$V$ = external bias voltage

$V_{bi}$ = “built-in” voltage (≈ 0.5 V for typ. used resistivities)

$N$ = doping concentration

$e$ = elementary charge

$\epsilon$ = dielectric constant = 11.9 $\epsilon_0$ ≈ 1 pF/cm

$\rho$ = resistivity (typically 1–10 kΩ cm)

$\mu$ = charge carrier mobility

(~ 1450 cm²/Vs (electrons), ~ 500 cm²/Vs (holes) [360])

The conductive $p$ and $n$ regions together with the depleted volume form a capacitor with capacitance per unit area

$$ C' = \frac{\epsilon}{d} \approx \frac{1 \text{ pF/cm}}{d} \quad \text{in Si.} \tag{35.22} $$

The depletion depth $d$ becomes as large as the sensor’s thickness at the “full depletion voltage” $V = V_{fd}$ for which the sensor capacitance reaches a minimum (about 35 pF/cm² for $d = 300$ µm). In strip and pixel detectors (see next section) the capacitance is dominated by the fringing capacitance to neighbouring electrodes as the electrode pitch is typically much smaller than the sensor thickness. For example, the strip-to-strip Si fringing capacitance is about 1–1.5 pF per cm of strip length at a strip pitch of 25–50 µm.

<figure><img src="image_221.png" /><figcaption>**Figure 35.24:** Reverse-biased junction detector with planar readout electrodes at the top in a standard sensor configuration (p$^+$-in-n): lightly doped $n$-type substrate with $p^+$ implants forming the diode junction at the top side. Electrons move to the bottom electrode and holes to the top. Both movements contribute to the (induced) signal at the top electrodes with amounts as specified by the weighting field (see Sec. 35.8.1.2). The $n^+$ implant at the bottom side forms an ohmic contact between bulk and metal.</figcaption></figure>

The electric field strength and shape inside the semiconductor bulk are important for efficient signal charge collection. Governed not only by the applied external voltage but also by the space charge inside the semiconductor bulk, leads (for constant space charge) to a linear field-strength decrease from its maximum at the junction’s boundary into the depleted semiconductor bulk. Space charge can also occur from ionised lattice defects

<figure><img src="image_222.png" /><figcaption>**Figure 35.25:** Junction detector with vertical electrodes, $n$ (red) and $p$ (blue), and “active” edges called 3D-Si. Dimensions given are typical. This electrode configuration provides short drift paths for moving charges while keeping a large sensing thickness. An active edge minimises dead areas at the sensor boundaries. This geometry has also been employed in CVD diamond substrates.</figcaption></figure>

which either naturally exist for example in GaAs or can be created by irradiation in any semiconductor material. This can lead to low-field regions as well as to changes in the field’s shape (deviating from linear), both of which usually deteriorate the charge collection properties of a detector.

In partial depletion (i.e. $V < V_{fd}$) the field decreases to zero at the end of the depletion zone. Overbias ($V > V_{fd}$) adds a constant electric field component which avoids a vanishing field region and also provides faster charge collection as long as the carrier drift velocity has not yet saturated. At an average field of $E = 10^4$ V/cm, the onset of velocity saturation, where $\mu_e \approx 7 \times 10^3$ cm²/Vs, $\mu_h \approx 3.5 \times 10^3$ cm²/Vs, the collection times for Si are about 15 ps/µm for electrons and 30 ps/µm for holes. In typical fully-depleted detectors, 100–300 µm thick, electrons are collected within less than about 5 ns, and holes within less than about 10 ns.

Large volume (~$10^2 – 10^3$ cm³) germanium detectors, especially for γ-ray detection, are commonly configured in cylindrical or hexagonal rod shapes, for example, a 10 cm long cylindrical $n$-type crystal with 5–10 cm diameter with an inner 5–10 mm diameter $n^+$ electrode and an outer $p^+$ layer forming the diode junction. Germanium can be grown with fairly low impurity levels, $10^9–10^{10}$ cm⁻³ (HPGe, high-purity germanium), so these large volumes can be depleted with several kilovolts.

Diamond, featuring free charge carrier densities close to zero, needs no further depletion and is operated as a parallel plate capacitor with an insulator dielectric inside. Still, a substantial bias voltage is required to overcome charge trapping.

### 35.8.3 Detectors with structured electrodes

#### 35.8.3.1 Microstrip-, Si-drift- and hybrid-pixel detectors

In HEP experiments semiconductor detectors usually aim at good position resolution achieved with electrodes patterned in “strips” or “pixels” with typical dimension scales (electrode pitch) of 50–100 µm, or in “pads” (mm²-cm²) if coarser granularity is affordable for the benefit of fewer channels.

Electrodes are usually placed “planar”, i.e., at the surface of the sensing Si bulk (Fig. 35.24). In an alternative, but more elaborate way, electrodes can be shaped as columns or trenches running orthogonally to the surface and hence parallel to the average direction of impinging particles (Fig. 35.25) [361]. This geometry (termed “3D-Si”, but is also exercised with diamond) enhances the radiation tolerance due to shorter drift distances of charges at the same thickness (see also Sec. 35.8.5).

In strip detectors, the strip ends are connected to dedicated readout ICs where the signals are amplified and processed. Two-dimensional readout is realised either by angled double-layers of strip detectors or by “double-sided” strip sensors structured by readout strips on either side (see e.g. [1, 358] and references therein). For the latter, one electrode surface features $n^+n$ or $p^+p$ strip junctions (rather than $pn$), respectively, which require dedicated measures, e.g. intermediate strips or special doping profiles to break the electron accumulation layer that occurs at an

<footnote>$^6n^{+/+}$, $p^{+/+}$ as well as $n^-$, $p^-$ qualitatively denote relative doping-concentration levels.</footnote>
$n^+n$ interface. For $p^+p$, intermediate $n$ strips or similar are not necessary, thanks to positive oxide charges residing at the Si/SiO$_2$ interface.

As for gaseous driftchambers, silicon driftchambers provide the position of a hit orthogonal to a strip's coordinate from a measurement of the charge cloud's drift time, drifting over centimeter-long paths inside the silicon bulk. For this to work the bulk is "sideways depleted" by using junction strip implants of the same polarity on both sides of the sensor ($p$ in $n$ or $n$ in $p$, respectively), different to standard double-sided strip detectors (see e.g. [1]). This way a quadratic potential minimum for electrons is created, confining them amid the sensor bulk, on which a linear electric potential in drift direction is superimposed; holes, instead, drift to the sides. In comparison to strip detectors, much fewer electrodes are necessary for the same active area coverage. Low rate capability is a drawback caused by the long drift path of created charges. Arranged with only one small electrode with low capacitance in the center of a cylindrical disk, so-called silicon-drift diodes (SDD) are low-noise single-channel devices with large area coverage, used for example in X-ray fluorescence analysis and electron microscopy.

Very small pixel structures for particle detection were first realised with CCDs [362]. Most current applications, particularly those for high-rate applications such as at the LHC, employ the "hybrid-pixel" concept in which both, the sensing diode structure as well as the readout IC, have equal electrode structures and patterns (pixels). The connection is made by two-dimensional arrays of solder or indium bumps that mate the two parts by employing flip-chip technology (Fig. 35.26, usually chip-to-chip or chip-to-sensor; more details e.g. in [358, 363]). Further advancements in bonding techniques especially address chip-to-wafer or even wafer-to-wafer placements, such as SLID (Solid Liquid Interface Diffusion) employing a very thin liquid metal layer (Sn) in between metals with higher melting points (e.g. Cu). Also, Cu-Cu, Cu-Sn, or oxide-oxide (SiO$_2$) diffusion bonding without solder is feasible.

<figure><img src="image_223.png" /><figcaption>**Figure 35.26:** Cross-sectional view of a hybrid-pixel detector cell (one pixel) consisting of sensor (bottom) and readout chip (top) with CMOS circuitry illustrated by three transistors (red and blue implant areas plus metal lines). The sensor is a depleted diode structure (here $p$ substrate with $n$ electrode). Both parts are mated by a microbond (bump bond), typically solder or indium.</figcaption></figure>

The hybrid approach excels in optimally utilising chip and sensor technologies for radiation hardness and rate capability (see e.g. [364]). It is, therefore, the technology of choice for pixel trackers in high-rate, high-radiation environments optimised for the purpose by employing very thin modules ($\mathcal{O}(300 \text{ µm})$ total). A disadvantage is the complex and cost-intensive module assembly.

### 35.8.3.2 Monolithic pixels

Partially or fully monolithic pixel detectors directly connect the generated signal charge with an embedded active electronics device

(i.e. one or several transistors). An example of a partially monolithic device is the DEPFET (depleted $p$-channel FET) pixel sensor (see e.g. [365]), employed, for example, in the pixel tracker of the Belle-II detector. A single transistor (pMOSFET) is implemented in every pixel (Fig. 35.27 (a)). The sensor substrate is "sideways depleted", as similarly done in Si driftchambers (Sec. 35.8.3.1), by means of the backside $p$-contact and several $p$ regions near the transistor. Depletion is provided by the $n^+$ clear contact (see below) plus a bulk $n$-implant outside the matrix. In addition, the structure features a deep $n$-implant located a few micrometers underneath the transistor channel on floating potential. This implant becomes the most positive point of the structure, hence being an electron accumulation point, which acts as an "internal gate" of the transistor. The gate voltage changes by an amount $\alpha q_S/C$, with $q_S$ = signal charge, gate-oxide capacitance $C = C_{ox}WL$ with $W \times L$ = gate area, and $\alpha \lesssim 1$ accounting for stray capacitance. The gate voltage changes due to an accumulated signal charge $q_S$ and leads to a detectable change in drain current (in saturation):

$$I_D = \frac{W}{2L} \mu C_{ox} \left( V_G + \frac{\alpha q_S}{C} - V_{th} \right)^2 . \quad (35.23)$$

Other elements in Fig. 35.27 (a) like the external FET gate and the clear implant provide control and reset of the structure. The device gain $g_q = dI_D/dq_S \approx \alpha g_m/C$, with transconductance $g_m = \mu C_{ox} \frac{W}{L} (V_G - V_{th})$, is of order $500 \text{ pA/e}^{-}$. The small capacitance of the internal gate (few fF) enables very low noise operation ($\lesssim 2e^{-}$) when operating with long shaping times (see Sec. 35.9).

Fully monolithic pixels combine the sensing task and the complete readout circuitry, i.e. the two separated parts of hybrid pixels, in one chip. Both functions must be properly shielded against each other, which is achieved by exploiting multi-well IC technology. Realised as MAPS (monolithic active pixel sensors) standard CMOS wafers can be used for pixel detector fabrication. CMOS wafers often feature a Si epitaxial layer where the transistors are implanted. It is possible to use this epi-layer as a sensitive detection volume for particle detection. However, in such detectors, only the immediate regions near the collection nodes (see Fig. 35.27(b)) are depleted. Outside these confined regions there is no or little electric field and charge collection is dominantly governed by slow and non-directional charge diffusion, a problem for high-rate applications. Depleted MAPS (DMAPS) exploit non-standard high resistivity wafers or high-ohmic epi-layers, typically with resistivities $>1 \text{ k}\Omega \text{cm}$, as well as high bias voltage (up to $\gtrsim 300 \text{ V}$) resulting in charge collection by directed drift motion. Charge collecting electrodes are deep wells ($n$-wells in Fig. 35.27(b)), either formed as large (typ. $50 \times 100 \text{ µm}^2$) structures to fully contain the CMOS circuitry (Fig. 35.27(b), top) or as small nodes, set aside the electronics (Fig. 35.27(b), bottom), with obvious pros and cons resulting from shorter average drift distance (large electrode) versus much smaller capacitance (small electrode). The former benefits radiation tolerance, whereas the latter allows for smaller pixels and benefits from small sensor capacitance in noise and rise time. However, in both approaches radiation tolerance levels of $\gtrsim 10^{15} \text{ neq/cm}^2$ (fluence) and 1 MGy (ion. dose) have been demonstrated.

A variant of the monolithic approach are SOI (silicon on insulator) pixels providing high spatial resolution, but less radiation tolerance: A high-resistivity supporting wafer, employed for particle sensing, connects through an embedded insulator (a buried oxide layer, BOX) to the CMOS readout circuitry. The BOX separates the sensing volume from the electronics layer and provides shielding. Because the area between BOX and the transistor layer acts as a "back-gate", transistor operation can be affected requiring dedicated cures (see e.g. [366]).

### 35.8.4 Precise timing with silicon detectors

Typical time scales for silicon detector output signals are several tens of nanoseconds (e.g. rise times, shaping times, etc.) accompanied by a typical time mark precision in the order of nanoseconds. Essential for achieving precision timing ($< 100 \text{ ps}$) are steep signal rise and low noise according to eq. (35.38) in Sec. 35.9. This,
<figure><img src="image_224.png" /><figcaption>(a) DEPFET pixel structure.</figcaption></figure>

<figure><img src="image_225.png" /></figure>

<figure><img src="image_226.png" /><figcaption>(b) DMAPS pixel structures.</figcaption></figure>

**Figure 35.27:** Partially (a) and fully (b) monolithic pixels: (a) DEPFET pixel structure with an embedded pMOSFET transistor; source and drain are *p* implants (red), gate metal (black), oxide (grey), deep-*n* internal gate (light green), other metal (blue), clear contact (green), and deep *p*-well (brown). (b) Depleted MAPS with joint sensing and electronics volumes: (top) charge collection by a large-electrode housing the electronics in a deep-*n* well; (bottom) charge collection by a small electrode set aside from the electronics area.

in turn, requires fast charge collection, e.g. in thin planar or 3D-Si detectors with high E-fields, as well as fast and strong amplification, both together yielding large slew rates (dS/dt, with S="signal", e.g. an induced current or a voltage). Important parameters that influence timing precision are the detector capacitance (including stray contributions) and the amplification gain.

Detectors that have achieved precise time measurement are so-called LGADs (low gain avalanche diodes, Fig. 35.28). They have planar electrode geometry that includes an implanted low-gain (*g* = 10–50) amplification structure on either the top or bottom electrode side to maximise the slew rate. Electrons created from ionising traversing particles are accelerated towards a very high field created by an amplification layer near the (here) top electrode, where impact ionisation creates a multitude of *e*-*h* pairs. They induce a very fast and large signal rise, mainly governed

by the holes' movement away from the amplification layer. Sufficiently low amplification gain minimises excess noise contributions and avoids the creation of hole-induced avalanches moving in the opposite direction to electron avalanches.

<figure><img src="image_227.png" /><figcaption>**Figure 35.28:** LGAD amplification structure for precision timing. Amplification (typ. × 20–50) occurs at the structure boundary in a highly doped *p*-*n* junction near the (here: top) surface (JTE = Junction Termination Extension, deep *n*-well).</figcaption></figure>

The achievable time resolution has several contributions:

$$ \sigma_t^2 \approx \left( \frac{\sigma_S^{thr}}{dS/dt} \right)^2 + \left( \frac{\sigma_n}{dS/dt} \right)^2 + \sigma_{\text{arrival}}^2 + \sigma_{\text{dist}}^2 + \sigma_{\text{TDC}}^2. \quad (35.24) $$

The first term represents “time walk” coming from Landau fluctuations, defined as originating from number and energy-transfer fluctuations in the energy-loss process with $\sigma_S^{thr}$ being the signal-height variations at discriminator threshold. This term can be kept minimal, e.g., by employing constant fraction discrimination or by applying corrections using amplitude information. The second term is the contribution of noise to the time jitter, which can be approximated by $(t_r/SNR)^2$ with $t_r$ = rise time, SNR = signal-to-noise ratio. Both first terms are kept small by large signal slew rates. An irreducible contribution comes from fluctuations in non-uniform depositions of charge along the particle path (including fluctuations in the amplification process) which causes an intrinsic jitter in the arrival time (third term)$^7$. The thinner the detector the less disturbing is this effect. The fourth contribution is signal distortion due to non-uniform weighting field regions and variations in (non-saturated) drift velocities. “Junction Termination Extensions” (JTE, see Fig. 35.28) serve to lower this term. The final term denotes time fluctuations due to uncertainties in digitization, which can, however, be made negligible with GHz TDCs. Timing precision of order 30 ps has been reached. Structural and operational variants to improve charge collection and position resolution are “resistive AC-coupled LGADs”, “deep junction LGADs”, “trench-isolated LGADs” and others. More details on LGADs are given in [367] and [368].

Columnar electrode geometries such as shown in Fig. 35.25 with small diameter and small pitch are by design a very suitable choice for precision timing due to short collection paths and strong fields. Employing an optimised trench-like column shape field- and weighting-field distortions are minimised and time resolutions in the order of 10 ps have been achieved in prototypes [369].

High precision timing can also be approached by exploiting the benefits of SiGe BiCMOS technology: strain mismatch of Si-Ge alloy layers results in a smaller bandgap as well as higher mobility and hence larger transconductance. This benefits the speed in hetero-junction bipolar transistors (HBTs) which also feature less 1/f noise than MOS transistors. Resolutions of order 20 ps have been reached in a monolithic pixel matrix using this technology [370]. More elaborate discussions on timing in silicon detectors can be found e.g. in [371].

### 35.8.5 Radiation damage in silicon detectors

High channel density and response times in the nanosecond range render micro-patterned semiconductor detectors particularly suited for high particle rates. This is usually accompanied by high radiation causing damage of sensors and front-end electronics. We restrict ourselves here to damage in Si detectors and their electronics. Radiation damage occurs in semiconductor detectors through two basic mechanisms:

<footnote>$^7$In LGAD literature these fluctuations are sometimes also referred to as “Landau fluctuations”.</footnote>
1. Bulk damage due to displacement of atoms from their lattice sites resulting in defect energy levels inside the band gap. This leads to increased leakage current, carrier trapping, and build-up of space charge that changes the required operating voltage. Displacement damage results from non-ionizing energy loss (NIEL). The energy imparted to the recoil atoms can initiate a chain of subsequent displacements including “dam- age clusters”. For a description, it is critical to consider both, particle type and energy. Conventionally, for silicon, the re- ceived NIEL is normalised to the damage level caused by 1 MeV neutrons and specified in units of neutron-equivalent fluence $\Phi_{eq}$ with units $n_{eq}/cm^2$.

2. Surface damage due to charge build-up in surface layers and formation of interface traps at e.g. Si-SiO₂ boundaries. These influence transistor currents and thresholds or cause thin charge carrier layers at a silicon surface, leading to in- creased surface leakage currents in sensors or circuits and affecting electrodes' isolation and/or transistor characteris- tics. The effects of charge build-up are strongly dependent on the device structure and on fabrication details. The dam- age is dominantly due to ionising energy loss (IEL) and hence proportional to the absorbed total ionisation dose (TID) mea- sured in Gy (or rad), independent of the particle type.

The increase in reverse bias generation current (leakage current) due to bulk damage is $\Delta I_L = \alpha \Phi_{eq} V$, where V is the volume under an electrode and $\alpha \simeq 4 \times 10^{-17}$ A/cm, a universal constant when normalised to temperature (conventionally 20°C), and measured after annealing for 80 min at 60°C. Note that for devices with intrinsic amplification, generation current is amplified accordingly. Reverse bias leakage current depends strongly on temperature

$$I_L(T) \propto T^2 \exp \left( -\frac{E_a}{2kT} \right), \quad (35.25)$$

where $E_a \approx 1.2$ eV (activation energy), so rather modest cooling can reduce the current substantially (~6-fold reduction in cooling from room temperature to 0°C).

For bulk damage in silicon, the NIEL hypothesis is a good first-order description of the observed damage (especially regarding $I_L$). It states that all lattice radiation damage in silicon linearly scales with NIEL and can be traced back to the abundance of primary defects (point defects and clusters), irrespective of their initial distribution over energy and space, that is, regardless of the damage's topology and origin. Under the NIEL hypothesis, the observed differences in damage caused by neutrons, protons, pions and electrons are hence usually scaled to each other.

<figure><img src="image_228.png" /><figcaption>Figure 35.29: Characteristic locations of energy levels caused by bulk radiation damage and their main action effects.</figcaption></figure>

Three main bulk displacement damage effects are sketched in Fig. 35.29: (a) defects acting as charged donors/acceptors, (b) deep defects (near the middle of the bandgap) causing increased leakage current, and (c) carrier trapping centers. Damage effects are not constant with time; for example in n-type bulk, radiation-induced negative space charge anneals at first, reaching a stable damage minimum (beneficial annealing), but later electrically active defects (negative space charge from acceptor-type damage) build up over the time scale of months (reverse annealing).

Acceptor- (usually dominant) or donor-like (deep) defect states, when ionised, build up space charge (dominantly negative), which in turn normally requires an increase in the applied voltage to

sweep signal charge through the detector thickness. For n-type bulk starting material, however, donor (e.g. phosphorous) re- moval by forming electrically inactive vacancy-phosphor (V-P) complexes induced by the incipient radiation, leads to a faster decrease in depletion voltage (faster than only by the compensa- tion of donors through generated deep acceptors) until positive and negative space charges effectively balance; only little bias voltage then is required for operation (point of effective space charge inversion, also called “type inversion” (Fig. 35.30)). At larger fluences the negative space charge dominates, and the re- quired operating voltage increases proportional to the increasing effective space charge density $N_{eff}$, also called effective doping concentration. Also the space-charge distribution can no longer be regarded as constant and homogeneous over the detector vol- ume, thus destroying the linear slope of the electric field from the junction to the backside as was introduced in Sec. 35.8.2; instead “double-junction”-like field distributions typically occur.

Today, p-type silicon with $n^+$ electrode-implants is the pre- ferred choice for sensors operating in high-radiation environments for reasons of cost-effectiveness in production as well as high ra- diation hardness due to fast (high mobility) electron collection in high electric- and high weighting-field regions at small segmented $n^+$ electrodes. Space charge inversion is usually not observed since the initial space charge is negative already (Fig. 35.30).

Removal of acceptors (i.e. substitution of electrically active boron at its lattice site) in p material is found to be dominantly caused by interstitial silicon replacing boron on its lattice po- sition. The highly active interstitial boron then can cause the formation of other defects such as interstitial oxygen-boron com- plexes ($B_iO_i$, subscript i for “interstitial”) which absorb boron atoms into electrically active defects, dominantly as donors, that means that space charge is changed twice.

Various techniques have been explored to neutralise radiation damage effects by defect engineering. For n-type material, oxygen enrichment with a larger abundance than phosphorus, introduced during the Si growth, successfully reduces the formation of multi- vacancy, acceptor-like complexes which are dominantly produced by charged hadrons, because it enhances instead the formation of electrically neutral V-O complexes. Carbon enrichment, on the other hand, reduces acceptor removal in p-type Si by building stable $C_i-S_i$ complexes, thus trapping the Si interstitials which could else remove boron as described above.

<figure><img src="image_229.png" /><figcaption>Figure 35.30: Effective space charge concentration $N_{eff}$ (left axis) and required voltage for full depletion $V_{dep}$ (right axis) of approximately 300 µm thick Si sensors as a function of neutron-equivalent radiation fluence for typical n-type (blue) and p-type (black) silicon float zone (FZ) material before annealing. While initial n-type Si inverts to effectively p-type Si, this is not observed for initial p-type silicon. Bias voltage supply for large systems is usually limited to less than 600–1000 V. Figure adapted using [372, 373] and [374].</figcaption></figure>

Dopant removal plays a particularly important role in more
complex sensor structures such as, for example, DMAPS or LGADs. In particular, the high p-doping concentration in LGAD's gain layer is vulnerable to acceptor removal causing a drastic gain decrease with increasing fluence.

Deep level defects (Fig. 35.29(b)) typically are the origin of leakage current increase, whereas trapping centers (Fig. 35.29(c)) dominate carrier lifetime and signal loss at high fluences beyond $10^{15}$ n$_{eq}$/cm$^2$. The safe limit on the operating voltage ultimately limits the detector lifetime. Strip and pixel detectors, specifically designed for high voltages, have been extensively operated at bias voltages of 500–600 V. Sensors with columnar electrodes normal to the surface (3D-Si, Fig. 35.25) need significantly lower voltages for full depletion and are prime contenders for radiation-hard sensors.

Synthetic CVD diamond has proven strong against radiation damage due to its large bandgap (see Tab. 35.10) and strong lattice binding with a roughly twice higher threshold (~40 eV) for a displacement of lattice atoms than silicon (see e.g. [354]). Even for polycrystalline material, charge collection distances (CCD = sum of the mean free paths $\mu\tau E$ of e and h over which charges reach the electrodes) of 300–400 µm have been achieved. Diamond wafer cost, however, is comparatively very high. Smaller scale trackers acting e.g. as beam monitors have been built.

GaAs has received interest as a potentially radiation-hard material in the 1990ies. For particle detection it must still be depleted from charge carriers despite comparatively small intrinsic carrier densities (see Tab. 35.10). It suffers from a strong loss in charge collection efficiency when irradiated, dominantly because of electron trapping. Other semiconductor materials such as SiC or GaN feature wider bandgaps (3.26 eV and 3.39 eV, respectively) than Si (but smaller than diamond) and possess other suitable properties relevant to high-fluence operation, such as density, $e-h$ ionisation energy, and displacement energies. They have regained attention after the material quality has improved much due to an industrial push coming from power devices and LEDs. So far, charge collection degrades faster with radiation fluence than for Si and diamond, however.

Strip and pixel detectors have remained functional in large detectors even at particle fluences well beyond $10^{15}$ n$_{eq}$/cm$^2$ where charge loss due to recombination and trapping becomes significant. Thin planar (~100 µm) and 3D silicon detectors have been successfully operated at fluences of 2 and $3 \times 10^{16}$ n$_{eq}$/cm$^2$, respectively, and measurements indicate that operation above $10^{17}$ n$_{eq}$/cm$^2$ does not seem impossible. The large SNR obtainable with low capacitance pixel structures extends the detector lifetime. The higher mobility of electrons makes them less sensitive to carrier lifetime than holes, so detector configurations that emphasize the electron contribution to the charge signal are advantageous, i.e. $n^+$ strips or pixels on p- or n-substrates. The occupancy of the defect charge states is strongly temperature dependent; competing processes can increase or decrease space charge and the required operating voltage. It is critical to choose the operating temperature judiciously (–30 °C to 0 °C in typical collider detectors) and to limit warm-up periods during maintenance. Detailed discussions of radiation damage and its effects on semiconductor detectors can be found e.g. in [373, 375]; an introduction to the subject can be found in [1].

Tolerance against surface damage, especially in SiO$_2$ layers as in gates and in oxide trenches of CMOS transistor structures, largely depends on the feature size of a technology and on appropriately designed circuitry. Deep submicron technology nodes of 130 nm and 65 nm sustain total ionisation doses of up to 5 MGy (500 Mrad), corresponding to fluences of up to $5 \times 10^{15}$ cm$^{-2}$ of minimum ionising pions or protons.

## 35.9 Low-noise detector readout

Revised November 2021 by N. Wermes (Bonn U.), revised November 2013 by H. Spieler (LBNL).

Many detectors rely critically on low-noise readout electronics, for best energy resolution or to allow low thresholds for high detection efficiencies. A typical detector front-end is shown in Fig. 35.31.

<figure><img src="image_230.png" /><figcaption>**Figure 35.31:** Typical detector front-end. The dashed circuit part is a charge-sensitive amplifier (CSA) realization.</figcaption></figure>

In a model relevant to most readout applications, the detector, represented by a capacitance $C_d$, delivers a delta-function-shaped current signal, represented by a current source in parallel. Bias voltage is applied through resistor $R_b$ and the signal is (often) coupled to the amplifier through a blocking capacitor $C_c$. The series resistance $R_s$ represents the sum of all resistances present in the input signal path, e.g. the electrode resistance, any input protection network, and parasitic resistances in the input transistor. The amplification stage contains the preamplifier providing gain and a pulse shaper (characterized by a shaping time $\tau$), which tailors the overall frequency response to optimize the signal-to-noise ratio (SNR) while limiting the duration of the signal pulse to accommodate the signal-pulse rate. For a given input signal, purely random noise leads to a gaussiany smeared signal distribution behind the shaper. Even if not explicitly stated, all amplifiers provide some form of pulse shaping due to their limited frequency response.

It is useful to distinguish between noise inherent to a detector's signal amplification and processing circuitry (per channel) on the one hand, depending on the detector specifics and the experimental environment (e.g. the data rate), and — on the other hand — external noise sources introduced e.g. by systems external to a specific readout circuit often resulting in “common-mode” noise, i.e. common to all channels. External noise can be introduced from power supplies, digital signal switching, RF pick-up, or from effects due to “common grounding” allowing noise to couple to the current loop connecting the detector to the preamplifier. These noise sources differ from setup to setup and must be dealt with — and should at best be eliminated — individually. In the following, therefore, only the noise inherent to typical detector signal processing is discussed.

### 35.9.1 Principal noise origins

As principal noise origins in circuit elements we mainly distinguish thermal noise resulting from velocity fluctuations of charge carriers (Brownian motion), as well as shot noise and $1/f$ noise, both resulting from charge carrier number fluctuations. Shot noise is due to emission statistics of electrons crossing a barrier, $1/f$ noise is often — especially in MOSFETs — due to charge trapping-detrapping processes leading to a spectral behaviour as $1/f^\alpha$ with $\alpha = 0.5...2$. Further noise nomenclature, like for example RTS (random telegraph signal) noise, also called “burst noise” or “popcorn noise”, is in its origin usually related to trapping/detrapping processes, like $1/f$ noise. The popping-up nature of individual RTS bursts eventually leads to the $1/f$ noise spectral density when the noise of several traps with (very) different trapping times are superimposed. RTS noise is omitted in the following for simplicity reasons.

### 35.9.2 Equivalent noise analysis

The equivalent circuit for noise analysis (Fig. 35.32) shows contributions of the mentioned noise sources at several circuit points. Originating from fluctuations, noise is expressed as the variance $\langle i^2 \rangle$ or $\langle v^2 \rangle$ of a noise current or voltage distribution. Shot noise, such as that produced in a semiconductor detector by leakage current fluctuations, is represented by a current noise generator in parallel with the detector and the amplifier input. The statistical fluctuations in a charge measurement will scale with the square root of the total number of recorded charges, so this noise contribution increases with the measurement (shaping) time. Thermal noise in resistors produces a white noise, i.e. a noise power density independent of frequency. Hence limiting the bandwidth by
a shaper, i.e. increasing the shaping time, will decrease the noise at the expense of slowing down the detector response. Usually, resistors shunting the input act as noise current sources and resistors in series with the amplifier input act as noise voltage sources, so that they are often referred to as “parallel” and “series” noise. Thermal fluctuations in the bias resistor result in fluctuations of the voltage at point P in Fig. 35.31 and its noise current source $\langle i_b^2 \rangle$ has the same effect as the shot noise current $\langle i_d^2 \rangle$ from the detector. Conversely, the series resistor $R_s$ acts as a voltage noise generator $\langle v_s^2 \rangle$ for the amplifier input.

Shot noise and thermal noise have a “white” frequency distribution, i.e. the (power) spectral densities ($f$ = frequency) $d\langle i^2 \rangle/df$ and $d\langle v^2 \rangle/df$ are constant with magnitudes

$$ 
\begin{aligned}
d\langle i_d^2 \rangle &= 2eI_d df, \\
d\langle i_b^2 \rangle &= \frac{4kT}{R_b} df, \\
d\langle v_s^2 \rangle &= 4kT R_s df,
\end{aligned} 
\quad (35.26) 
$$

where $e$ is the elementary charge, $I_d$ the detector leakage current, and $k, T$ Boltzmann constant and temperature. Hence, in a $1 \text{ k}\Omega$ resistor at room temperature, for example, one finds a current-independent thermal noise of $\sqrt{d\langle i^2 \rangle/df} = 4 \text{ pA}/\sqrt{\text{Hz}}$ or $\sqrt{d\langle v^2 \rangle/df} = 4 \text{ nV}/\sqrt{\text{Hz}}$, respectively. For $1/f$ noise, by contrast, the spectral power density is proportional to $1/f^\alpha$ ($\alpha = 0.5-2$) with a device specific proportionality constant. In what follows, we will assume $\alpha = 1$ for simplicity reasons, noting though that more complex models exist.

<figure><img src="image_231.png" /><figcaption>Figure 35.32: Equivalent circuit diagram for noise analysis.</figcaption></figure>

An important noise source is the preamplifier input stage, very often dominated by the first amplification transistor. In what follows we assume a MOSFET at this point, as the large majority of amplification circuits is based on MOS transistors. Noise sources in bipolar amplification is shortly mentioned at the end of this section. A MOS transistor’s channel features thermal noise with “resistance” $1/g_m$, where $g_m = \partial I/\partial V$ is the transconductance. Correction factor(s) for parametrization depend on the transistor operation point. Conventionally, a factor $\gamma$ is applied varying between 0.5 and 1 when temperature dependence is implicit, but is commonly taken as $\gamma \approx 2/3$ for operation in strong inversion and 0.5 in weak inversion.

In addition, especially in MOSFETs, there is $1/f$ noise originating from trapping/detrapping processes of charge carriers at the channel’s Si–SiO$_2$ interface. The channel’s current noise is equivalent to a voltage noise at the transistor input (the gate) via

$$ \langle i_{ch}^2 \rangle = \langle (g_m v_a)^2 \rangle \quad (35.27) $$

leading to respective noise power spectral densities of

$$ \frac{d\langle v_a^2 \rangle}{df} = 4kT\gamma \frac{1}{g_m} + K_f \frac{1}{C'_{ox}WL} \frac{1}{f}, \quad (35.28) $$

where the individual (uncorrelated) noise contributions add in quadrature. The “$1/f$ noise constant” $K_f$ has magnitude in the order of $10^{-24}–10^{-26}$ J depending on the device type and the technology$^8$. $C'_{ox}$ is the oxide capacitance per unit area, and $W, L$ are width and length of the gate.

Particle and radiation detectors typically convert the deposited energy into charge (usually e-ion or e-h pairs) which can be measured by integrating the current induced on an electrode by the drifting charge pairs (see e.g. Sec. 35.8) employing a “charge-sensitive (pre)amplifier” (CSA), realized by feeding back the preamp’s output to the input through capacitor $C_f$ (dashed circuit part of Fig. 35.31). The feedback capacitance must be discharged for the next pulse to appear, for example by a resistor, a (constant) current source or a switch. Noise contributions from such components can usually be kept small in comparison.

For a CSA the noise output voltage after the preamplifier $\langle v_{pa}^2 \rangle$ relates to the input (current or voltage) noise sources via the transimpedance $1/\omega C_f$ as

$$ \langle v_{pa}^2 \rangle = \langle i_{in}^2 \rangle \left( \frac{1}{\omega C_f} \right)^2 \quad \text{or} \quad \langle v_{pa}^2 \rangle = \langle v_{in}^2 \rangle \left( \frac{\omega C_D}{\omega C_f} \right)^2 \quad (35.29) $$

with $\omega = 2\pi f$. The capacitance $C_D$ includes the detector capacitance $C_d$ plus all capacitances shunting the input. Considering only the (usually) dominant noise sources, being the detector shot noise as well as the transistor channel noise, and neglecting the resistor parallel and serial noise sources of (35.26) by choosing $R_b$ large and keeping $R_s$ small, one obtains

$$ \frac{d \langle v_{pa}^2 \rangle}{d\omega} = \sum_{k=-2}^{0} c_k \omega^k \quad (35.30) $$

with

$$ c_{-2} = \frac{e}{\pi} I_d \frac{1}{C_f^2}, \quad c_{-1} = K_f \frac{1}{C'_{ox}WL} \frac{C_D^2}{C_f^2}, \quad c_0 = \frac{2\gamma}{\pi} kT \frac{1}{g_m} \frac{C_D^2}{C_f^2}, \quad (35.31) $$

which correspond to the discussed main noise components, $c_{-2}$: shot noise, $c_{-1}$: $1/f$ noise, $c_0$: thermal noise, respectively.

The filtering effect of the shaper limits the bandwidth of the system and is generally described by a transfer function $H(\omega)$ for $\langle v_{pa}^2 \rangle$ integrated over the full bandwidth

$$ \langle v_{sh}^2 \rangle = \int_0^\infty \frac{d\langle v_{pa}^2 \rangle}{d\omega} |H(\omega)|^2 d\omega \quad (35.32) $$

with $\langle v_{sh}^2 \rangle$ being the shaper output voltage noise. $H(\omega)$ depends on the filter circuit, i.e. the shape of the pulse formed by the shaper, and is characterized by a characteristic time $\tau$, the shaping time, which can, for example, be the peaking time of a semi-gaussian pulse or the sampling interval in a correlated double-sampler (see e.g. [8]). The peaking time is an important quantity for readout systems since for peaking times large compared to the duration of the input signal the amplitude at peaking time corresponds to the total input charge.

For a simple time-invariant CR–RC filter, consisting of a high-pass followed by a low-pass filter with equal time constants, the transfer function is

$$ |H(\omega)|^2 = A^2 \left( \frac{\omega\tau}{1 + \omega^2\tau^2} \right)^2 \quad (35.33) $$

and (35.30) becomes

$$ \langle v_{sh}^2 \rangle = \frac{\pi}{4} A^2 \left( c_{-2} \tau + \frac{2}{\pi} c_{-1} + c_0 \frac{1}{\tau} \right) \quad (35.34) $$

with $A$ being an overall gain factor.

The system noise for a charge measurement is conveniently expressed as an equivalent noise charge (ENC) referring the noise to the equivalent signal of one electron at the input:

$$ \mathrm{ENC}^2 = \frac{\langle v_{\mathrm{sh}}^2 \rangle}{v_{\mathrm{signal}}^2 (1e^{-})} \quad (35.35) $$

While being dimensionless by definition, ENC is commonly expressed as an equivalent number of electrons ($e^{-}$) at the input or alternatively as equivalent Coulombs (C) when multiplying by the

<footnote n="8">⁸Other $1/f$ noise parametrizations also exist with correspondingly different numerical values for $K_f$.</footnote>
elementary charge or as equivalent deposited energy (eV), where $1e^{-}$ equivalent charge corresponds to 3.65 eV of equivalent deposited energy in Si (see Tab. 35.10 in Sec. 35.8).

A charge of one electron at the input yields a voltage signal (peak) behind the shaper of $v_{\text{signal}} = \frac{A}{2.71} \frac{e}{C_f}$. Hence with (35.35) and (35.34) one obtains:

$$ \begin{aligned} \text{ENC}^2 (e^{-2}) &= \frac{(2.71)^2}{4e^2} \left( eI_d\tau + 2C_D^2 K_f \frac{1}{C'_{ox}WL} + \gamma \frac{2kT}{g_m} \frac{C_D^2}{\tau} \right) \\ &= a_{\text{shot}}\tau + a_{1/f}C_D^2 + a_{\text{therm}}\frac{C_D^2}{\tau}, \end{aligned} \quad (35.36) $$

where 2.71 is Euler's number resulting from the amplitude peak value behind the (CR-RC) shaper (for $t = \tau$), while $e$ is the elementary charge and $(e^{-})$ explicitly denotes the equivalent number of electrons at the input.

Equation (35.36) accentuates the main noise dependencies of the system. The contribution from parallel current shot noise (i.e. detector leakage) increases with shaping time, i.e. with pulse duration, whereas thermal noise (here serial voltage noise at the gate, originating from transistor channel noise) decreases with increasing shaping time, i.e. with reduced bandwidth. Noise with a $1/f$ spectrum depends only on the ratio of upper to lower cutoff frequencies (low pass to high pass time constants) and is hence independent of $\tau$ for a given shaper topology. Furthermore, the contribution of serial noise voltage ($1/f$ and thermal noise) to ENC increases with input (detector) capacitance $C_D$. Pulse shapers can be designed to optimize the noise performance of a system or to mitigate operation variations with time (e.g. due to radiation damage) deteriorating the performance, at the expense of loss of simplicity. For example, a shaper with one high-pass filter followed by four cascaded low-pass filters, increases the pulse symmetry and tends to decrease the current noise, but increases the voltage noise contributions for the circuit of Fig. 35.32, if the same peaking time is assured by adjusting the individual time constants of the filter stages. If instead the low-pass filters cause the peaking time of the pulse to increase, the opposite can be true. More details about shaping filters can be found e.g. in [8, 376-378].

Figure 35.33 shows a typical example of eq. (35.36) displaying ENC as a function of shaping time $\tau$ for a system. At short shaping times, thermal noise dominates, whereas for longer shaping times the shot noise contribution takes over. The total noise has a minimum at a shaping time where shot and thermal noise contributions are equal. The minimum is flattened by the presence of $1/f$ noise. Increasing the detector capacitance increases the thermal and $1/f$ contributions and shifts the noise minimum to larger shaping times. One can hence exploit the shaping time dependence of the total noise to determine the individual noise contributions of a system.

For quick estimates (35.36) is cast into a useful equation for a typical CSA-shaper system [1]:

$$ \text{ENC}^2 (e^{-2}) = 11 \frac{I_d}{\text{nA}} \frac{\tau}{\text{ns}} + 740 \frac{1}{WL/(\mu\text{m}^2)} \frac{C_D^2}{(100 \text{ fF})^2} + 4000 \frac{1}{g_m/\text{mS}} \frac{C_D^2/(100 \text{ fF})^2}{\tau/\text{ns}}, \quad (35.37) $$

where $\gamma = 2/3$, $K_f = 33 \times 10^{-25} \text{J}$, and $C'_{ox} = 6 \text{ fF}/\mu\text{m}^2$ has been used, the latter being typical for CMOS technologies as employed for example for the detector readout chips of the LHC experiments and their upgrades. Generally, noise performance is improved by reducing the detector capacitance and the leakage current, judiciously selecting all resistances in the input circuit, and by choosing the optimum shaping time constant $\tau$. Another noise contribution to consider is the fact that noise can cross-couple from the neighbouring front-ends in detectors with structured electrodes through the inter-electrode capacitance.

As mentioned, the noise parameters of the amplifier depend primarily on the input device (input transistor(s)). In field-effect transistors, the input noise current contribution of the transistor itself is very small. Hence, if the experiment's time structure permits, reducing the detector leakage current and increasing the

<figure><img src="image_232.png" /><figcaption>Figure 35.33: Equivalent noise charge vs shaping time. Changing the detector capacitance $C_D$ affects thermal and $1/f$ noise contributions and moves the noise minimum as indicated by the dotted and dash-dotted lines. The $1/f$ noise contribution is small for the chosen parameters ($C_D = 20 \text{ pF}$, $I_d = 10 \text{ nA}$, $g_m = 1 \text{ mS}$) in this example.</figcaption></figure>

bias resistance will allow long shaping times with correspondingly lower noise. For amplifiers employing bipolar transistors, shot noise sources originate from both, base-emitter and collector-base, junctions and from the series resistance at the base which is especially important for small structures optimized for radiation hardness. For the shot-noise contribution, the input base current $I_B$ sets a lower bound on the noise current, such that these devices are best at short shaping times.

**35.9.2.0.1 Examples** Using (35.37) one finds for a typical pixel detector (before heavy irradiation) with $C_D = 200 \text{ fF}$, $I_d = 1 \text{ nA}$, $\tau = 50 \text{ ns}$, $W = 20 \text{ µm}$, $L = 0.5 \text{ µm}$, $g_m = 0.5 \text{ mS}$:

$$ \text{ENC}^2 \approx (24 e^{-})^2 \Big|_{\text{shot}} + (17 e^{-})^2 \Big|_{1/f} + (25 e^{-})^2 \Big|_{\text{therm}} \approx (40 e^{-})^2. $$

For a typical silicon microstrip detector after radiation damage (fluence $\gtrsim 10^{14} \text{n}_{\text{eq}}/\text{cm}^2$, assuming no degradation of the front-end electronics due to radiation) one obtains for $C_D = 20 \text{ pF}$, $I_d = 1 \text{ µA}$, $\tau = 50 \text{ ns}$, $W = 2000 \text{ µm}$, $L = 0.4 \text{ µm}$, $g_m = 5 \text{ mS}$:

$$ \text{ENC}^2 \approx (750 e^{-})^2 \Big|_{\text{shot}} + (200 e^{-})^2 \Big|_{1/f} + (800 e^{-})^2 \Big|_{\text{therm}} \approx (1100 e^{-})^2. $$

Apart from the larger leakage current, the larger capacitance of strips compared to pixels leads to a much worse noise performance which can only be partially compensated by allowing more power in the amplification transistor, i.e. by increasing $g_m$.

A liquid argon calorimeter cell is a suitable example of a detector with a large electrode capacitance with typical parameters (similar to the ATLAS central electromagnetic calorimeter, see Sec. 35.10). Using $C_D = 1.5 \text{ nF}$, $I_d = < 2 \text{ µA}$, $\tau = 50 \text{ ns}$, $W = 3000 \text{ µm}$, $L = 0.25 \text{ µm}$, $g_m = 100 \text{ mS}$, one obtains:

$$ \begin{aligned} \text{ENC}^2 &\approx (1000 e^{-})^2 \Big|_{\text{shot}} + (15000 e^{-})^2 \Big|_{1/f} + (13500 e^{-})^2 \Big|_{\text{therm}} \\ &\approx (20200 e^{-})^2. \end{aligned} $$

Here only a small (negligible) parallel shot noise (leakage current) contribution is assumed, which is typical for liquid argon calorimeters.

Practical noise levels range from $\sim 1e^{-}$ for CCD's at long shaping times to $\sim 10^4 e^{-}$ in high-capacitance liquid argon calorimeters. Gaseous micropattern detectors like GEMs or MicroMegas (see Sec. 35.6.4) typically feature noise levels between $1000 e^{-}$ and $1500 e^{-}$, depending on the ability to correct for common-mode noise. Silicon strip detectors typically operate at levels of $\sim 10^3 e^{-}$, whereas pixel detectors with fast readout typically have noise levels below about 100 electrons.
### 35.9.3 Timing measurements

In timing measurements, the slope-to-noise ratio must be optimized, rather than the signal-to-noise ratio alone, so the rise time $t_r$ of the pulse is important. The “jitter” $\sigma_t$ of the timing distribution is

$$ \sigma_t = \frac{\sigma_n}{(dS/dt)_{\text{trig}}} \approx \frac{t_r}{SNR}, \qquad (35.38) $$

where $\sigma_n$ is the rms noise, SNR the signal-to-noise ratio and the “slew rate” $dS/dt$, i.e. the derivative of the signal, is evaluated at the trigger level. The rise-time of a CSA again depends on the detector and feedback capacitances and on the amplifier transconductance

$$ t_r \propto \frac{C_D}{C_f \cdot g_m}. \qquad (35.39) $$

To increase $dS/dt$ without incurring excessive noise, the amplifier bandwidth should match the rise time of the detector signal. The 10% to 90% rise time of an amplifier with bandwidth $f_U$ is 0.35/f$_U$. For example, an oscilloscope with 350 MHz bandwidth has a 1 ns rise time. For cascaded amplifiers, the individual rise times add in quadrature to first order.

As increasing SNR also improves the time resolution, minimizing the total input capacitance is extremely important for timing measurements (see also Sec. 35.8). At high signal-to-noise ratios, the time jitter can be much smaller than the rise time. The time-mark distribution of pulses may shift with the signal level (“time walk”), but this can be corrected by various means, either in hardware or in software.

For applications aiming at extreme time resolution charge-sensitive (integrating) amplification can be less optimal. Transimpedance or voltage amplifiers converting current/voltage to voltage are usually preferred. Also, when aiming for picosecond timing using voltage amplifiers, e.g., with LGAD detectors having $C_D$ of O(pF) (see Sec. 35.8), not only the preamplifier rise time $t_r$ but also the signal pulse duration $t_d$ at the preamplifier input is important, i.e., the preamplifier must react draining the charge from $C_D$ before the input voltage $v_{in} = Q_S/C_D$ ($Q_S$ = signal charge) reaches its peak. Both times must reasonably match, $t_r \approx t_d$, resulting in a time jitter of [379]

$$ \sigma_t^2 \approx \frac{C_D^2}{Q_S^2} \frac{d\langle v_n^2 \rangle}{df} t_d, \qquad (35.40) $$

where $d\langle v_n^2 \rangle/df$ is the voltage noise spectral density.

### 35.9.4 Digital signal processing

The filtering principles apply to both analog and digital signal processing. In digital signal processing the pulse shaper shown in Fig. 35.31 is replaced by an analog to digital converter (ADC) followed by a digital processor that determines the pulse shape. Digital signal processing allows great flexibility in implementing filtering functions. The software can be changed readily to adapt to a wide variety of operating conditions and it is possible to implement filters that are impractical or even impossible using analog circuitry. However, this comes at the expense of increased circuit complexity and increased demands on the ADC compared to analog shaping.

If the sampling rate of the ADC is too low, high frequency components will be transferred to lower frequencies (“aliasing”). The sampling rate of the ADC must be high enough to capture the maximum frequency component of the input signal. Apart from missing information on the fast components of the pulse, undersampling introduces spurious artifacts. If the frequency range of the input signal is much larger than the sampling rate, the noise at the higher frequencies will be transferred to lower frequencies and will increase the noise level in the frequency range of pulses formed in the subsequent digital shaper. The Nyquist criterion states that the sampling frequency must be at least twice the maximum relevant input frequency. This requires that the bandwidth of the circuitry preceding the ADC must be limited. The most reliable technique is to insert a low-pass filter.

The digitization process also introduces inherent noise, since the voltage range $\Delta V$ corresponding to a minimum bit step introduces

quasi-random fluctuations relative to the exact amplitude

$$ \sigma_n = \frac{\Delta V}{\sqrt{12}}. \qquad (35.41) $$

When the Nyquist condition is fulfilled the noise bandwidth $\Delta f_n$ is spread nearly uniformly and extends to 1/2 the sampling frequency $f_S$, so the spectral noise density is

$$ \frac{\sigma_n}{\sqrt{\Delta f_n}} = \frac{\Delta V}{\sqrt{12}} \cdot \frac{1}{\sqrt{f_S/2}} = \frac{\Delta V}{\sqrt{6f_S}}. \qquad (35.42) $$

Sampling at a higher frequency spreads the total noise over a larger frequency range, so oversampling can be used to increase the effective resolution. In practice, this quantization noise is increased by the ADC’s differential non-linearity (DNL). Furthermore, the equivalent input noise of ADCs is often rather high, so the overall gain of the stages preceding the ADC must be sufficiently large for the preamplifier to override the ADC input noise.

### 35.9.5 What to use when?

When implemented properly, digital signal processing provides significant advantages in systems where the shape of detector signal pulses changes greatly, for example in large semiconductor detectors for gamma rays or in gaseous detectors (e.g. TPCs) where the duration of the current pulse varies with drift time, which can range over orders of magnitude. Analog signal processing is best or most efficient in systems that require fast time response, but the high power requirements of high-speed ADCs are prohibitive. Systems that are not sensitive to pulse shape can use fixed shaping time constants and rather simple filters (like CR-RC), which can be either continuous or sampled. In high-density systems that require small circuit area and low power (e.g., in strip and pixel detectors), analog filtering often yields the required response and tends to be most efficient.

As stressed already in the introduction, it is important to consider that additional noise is often introduced externally. Recognizing additional noise sources and minimizing cross-coupling to the detector current loop is often essential to obtain the best overall noise performance. Understanding basic physics and its practical effects is important in forming a broad view of the detector system and recognizing potential problems (e.g. modified data), rather than merely following standard recipes.

More comprehensive treatments of low noise detector readout and signal processing can be found, for example, in [8, 376, 378] and in [1, 363].

## 35.10 Calorimeters

### 35.10.1 Introduction

Revised August 2023 by F. Sefkow (DESY, Hamburg) and F. Simon (KIT).

A calorimeter measures the energy and direction of particles by absorption in the detector material and registration of the energy deposited in an (ideally) contained electromagnetic (EM) or hadronic shower. Calorimeters are central components of modern high energy physics experiments, due to their ability to measure not only the energy of charged particles (with the exception of muons), but also of photons and neutral hadrons, thus enabling the reconstruction of $\pi^0$ and $\eta$ decays and of exclusive final states involving long-lived neutral kaons, or neutrons. They are indispensable for the measurement of particle jets and for the reconstruction of total event properties, which, via the measurement of missing energy (or missing transverse energy in hadron colliders), enable the detection of the presence of "invisible" particles such as neutrinos and hypothetical particles such as dark-matter candidates. Calorimeters are also important for the identification of particle species, using information on the longitudinal and transverse shape of the energy deposition to separate electrons, photons, hadrons and muons. While the performance of calorimeters is typically assessed by the quality of their energy measurements, position resolution, both for EM and hadronic showers, is also
highly relevant, for example for the reconstruction of effective jet masses. The capability to measure high-level observables that serve to classify events, such as particle and jet energies, missing energy and isolated leptons, makes calorimeters central components of the trigger systems in high-energy physics experiments.

In collider experiments, the importance of calorimeters tends to increase with increasing collision energies since the relative energy resolution improves with increasing particle energy while the depth required for full containment of the showers shows only logarithmic growth with energy. This is in contrast to the precision of track-based measurements, which is decreasing with increasing momentum. With recent advances in timing capabilities calorimeters are also contributing to the rejection of pile-up from multiple interactions within the same bunch crossings at colliders.

<figure><img src="image_233.png" /><figcaption>**Figure 35.34:** Nuclear interaction length $\lambda_I/\rho$ (circles) and radiation length $X_0/\rho$ (+'s) in cm for the chemical elements with $Z > 20$ and $\lambda_I < 50$ cm.</figcaption></figure>

The characteristic length scale for EM showers is the radiation length $X_0$, which ranges from 1.8 cm (13.8 g cm⁻²) in iron to 3.2 mm (6.0 g cm⁻²) in uranium for materials used to generate showers in calorimeters.<fnref n="9" /> Similarly, the characteristic nuclear interaction length $\lambda_I$ varies from 16.8 cm (132.1 g cm⁻²) (Fe) to 11.0 cm (209 g cm⁻²) (U).<fnref n="10" /> There is a premium on small $\lambda_I/\rho$ and $X_0/\rho$ (both with units of length). These quantities are shown for elements with $Z > 20$ in Fig. 35.34. The minima for both $X_0$ and $\lambda_I$ correspond to elements between W and Au. Some of these elements are very difficult to work with (e.g. W) or expensive (e.g. Au), so, depending on the application (size of the required calorimeter) other materials, such as Fe, Cu, Pb, and different alloys like brass and CuW are often chosen. For EM calorimeters high $Z$ is preferred; here Pb is a popular choice, while W provides even higher density and is generally affordable due to the limited volume of EM systems.

Most existing calorimeters are subdivided into a front EM section (ECAL) and a hadronic part (HCAL) behind; electrons and photons are measured in the ECAL, while hadrons and jets are measured in the combined ECAL and HCAL system. The detailed design depends on energy range and performance requirements as well as on size and cost constraints for the entire system. EM calorimeters tend to be 15–30 $X_0$ deep, while hadronic calorimeters are usually optimised for cost and performance at 5–8 $\lambda_I$. The depth of the ECAL typically corresponds to approximately 1 $\lambda_I$, with the exact value depending on the material. This means that approximately 70% of all hadronic showers will already begin in the electromagnetic calorimeter, making its response to a hadronic cascade highly relevant for the overall performance of the system. The choice of the calorimeter technology for the ECAL is thus a result of simultaneous optimisation for EM and hadronic performance of the overall system.

Sampling calorimeters consist of a high-density, normally metallic absorber sandwiched (or threaded) with an active ma-

terial which generates a signal in response to shower particles. The active medium may be a scintillator, a noble liquid, a gas, silicon, or a Cherenkov radiator. These active media all have a relatively low $Z$, a significantly lower density, and larger $X_0$ and $\lambda_I$ values than typical absorber materials. The average radiation and interaction lengths in the full detector are thus larger than those of the absorber alone.

There are also homogeneous calorimeters, in which the entire volume contributes to the signal. Homogeneous calorimeters may be built with inorganic heavy scintillating crystals or non-scintillating Cherenkov radiators such as lead glass and lead fluoride. Nuclear interaction lengths in inorganic crystals range from 17.8 cm (LuAlO₃) to 42.2 cm (NaI). Materials with low $X_0$ used in large systems are for example BGO with $\lambda_I = 22.3$ cm and $X_0 = 1.12$ cm, and PbWO₄ (20.3 cm and 0.89 cm). Properties of these and other commonly used inorganic crystal scintillators can be found in Table 35.4. Cryogenic noble liquids, where scintillation light and/or inonization can be detected, are also suitable materials for homogeneous detectors.

Homogeneous calorimeters at colliders are usually only used for the EM section. For the use of homogeneous calorimeters for hadron energy measurement, the large differences in the response to EM and hadronic parts of the shower are a significant challenge, as is the three-dimensional segmentation. This is still requiring substantial R&D, including the search for affordable materials. In non-accelerator physics experiments or at neutrino beams, homogeneous calorimeters, where the sensitive medium can be water or ice, scintillator, a noble liquid or the atmosphere itself, are used to detect both EM and hadronic showers.

Comprehensive tables of particle-physics calorimeters are given as Appendix C in Ref. [380].

### 35.10.1.1 Energy Resolution and System Performance

The energy resolution of calorimeters is a complex observable, due to the variety of contributing processes with different energy dependencies, and response functions often not perfectly Gaussian. Nevertheless, a simplified picture is useful in practice, and in particular for EM calorimeters also numerically accurate. For hadronic calorimeters additional complications must be taken into account. In such a simplified picture, due to the stochastic nature of shower evolution, the intrinsic calorimeter energy resolution, $\sigma$, is proportional to $\sqrt{E}$, as the number of charged particles, or the total ionising track length in a shower are on average proportional to the incident particle energy $E$. The relative resolution $\sigma/E$ therefore improves with $A/\sqrt{E}$, where $A$ denotes the so-called stochastic term. The readout system of the active medium will contribute noise to the resolution, $\sigma_N = B$, which in general is not energy-dependent. Effects that are proportional to the total deposited energy result in a constant term, $\sigma_C = C \cdot E$. Different sources contribute to this term, depending on the type of calorimeter. For both EM and hadronic calorimeters, imperfections of the detector, inhomogeneities such as density variations or those introduced by the detector mechanics, instabilities in time, imperfections of the readout or incorrect calibration of channels contribute. Shower leakage, which depends on particle energy, also contributes to the resolution with approximately linear dependence on energy. In non-compensating hadronic calorimeters, fluctuations of the EM fraction $f_{em}$ from shower to shower, together with the energy dependence of the average $f_{em}$ value, can lead to a significant constant term that often dominates over the instrumental effects. Adding up all contributions in quadrature yields the standard parameterisation of the relative energy resolution of a calorimeter:

$$ \frac{\sigma}{E} = \frac{A}{\sqrt{E}} \oplus \frac{B}{E} \oplus C. \qquad (35.43) $$

In particular the effects specific to hadronic showers give rise to non-Gaussian distributions of the energy response. Therefore care must be used in performance comparisons, as different parameterisations of the line shape and different definitions of the resolution are in use. In some cases, a linear rather than a quadratic addition of the stochastic and constant term may provide a better description of the energy resolution as a function of energy. It should be noted that the individual terms then lose the simplified

<footnote n="9">⁹$X_0 = 120 \text{ g cm}^{-2} Z^{-2/3}$ to better than 5% for $Z > 23$.</footnote>

<footnote n="10">¹⁰$\lambda_I = 37.8 \text{ g cm}^{-2} A^{0.312}$ to within 0.8% for $Z > 15$.</footnote>

<footnote>See https://pdg.lbl.gov/current/AtomicNuclearProperties for actual values.</footnote>
interpretation discussed above, and care has to be taken when comparing performance based on fits to the energy dependence of the resolution.

Typically, primarily the stochastic term — which is determined by the calorimeter design in terms of material and geometry — is considered in order to describe the overall properties of a calorimeter. For the calorimeters of the multi-purpose experiments at the LHC, the stochastic terms are 3 – 10% for electromagnetic and 50 – 80% for hadronic calorimeters. In practice, the energy resolution of a calorimeter at high energies is limited by the constant term *C*, which in the EM case mainly reflects the precision and stability of the mechanical construction, electronic readout system and calibration. Typical constant terms are a few per-mil for EM and a few percent for hadronic calorimeters. For concrete examples and references see Tables 35.11 and 35.13.

For sampling calorimeters, the stochastic term depends on the sampling fraction $f_{\text{samp}}$, i.e. the ratio of energies deposited in the active and passive material. Also the sampling frequency enters, which is determined by the number *N* of different sampling elements present in the region in which the shower develops. The stochastic term *A* scales approximately with $1/\sqrt{f_{\text{samp}}}$, and for given $f_{\text{samp}}$ and total depth, with $1/\sqrt{N}$.

While the energy resolution for single hadrons (most commonly pions) is often used as the key performance criterion for a hadronic calorimeter, it has to be noted that this value is only of limited relevance in high-energy physics experiments. In most experiments, the calorimetric measurement of hadrons is based on the combined system consisting of ECAL and HCAL. Moreover, for the physics capabilities of a detector at a high-energy particle collider, also the combined calorimetric resolution for single hadrons is not a sufficient criterion to fully characterise hadronic performance, but rather the jet energy resolution, the resolution for missing (transverse) energy, and the capability to cope with high background and pile-up levels. These quantities cannot be measured directly with prototypes in beam tests, and strongly depend on overall system aspects and reconstruction tools. However, the performance in these observables can be reliably inferred from system simulations once the simulated response to single particles and the simulated topology of showers has been validated in detail by beam tests.

Besides energy resolution, response linearity is an important factor in the design of calorimeters. While a non-linear response for single particles can be corrected for if appropriate calibration measurements exist, such corrections deteriorate the energy resolution, in particular in the case of superposition of several show- ers, as it often occurs in jets. Sources for non-linearities can be intrinsic to the design, for example due to saturation effects in the active medium with increasing energy density, due to leakage, or connected to shower physics as discussed in section 35.10.3.

The energy resolution for hadrons is intrinsically limited by large event-to-event fluctuations of the shower evolution and of "invisible" components not contributing to the detector signal. It remains the limiting factor for single particles and for the high-level performance for jets and total event properties. This has motivated intense research in the past decades. One direction aims at improving the hadronic resolution by extracting additional signals to disentangle the shower composition, e.g. with so-called dual read-out methods, and is explained in the hadron calorimeter section 35.10.3. Another, so-called "particle flow" approach described below optimises the combination of measurements of individual particles in different detector components. Ideally, both methods can be combined.

### 35.10.1.2 Role of Simulations

Simulations have become indispensable for the design of detectors and the development of reconstruction algorithms. Since event-to-event fluctuations drive calorimeter performance, Monte Carlo techniques that accurately model the evolution of particle cascades in material are required. By far the most common computer code in use today is the GEANT4 toolkit [381, 382], which provides a step-based simulation of the passage of particles through matter. Thanks to the relative simplicity of EM cascades, simulations of EM showers are typically highly accurate.

The modelling of hadronic showers is more complex, and suffers from larger uncertainties. Significant improvement has been achieved in this area, moving from simpler parameterised models to physics-driven interaction models. From early on, detailed codes and data describing nuclear break-up and neutron transport like CALOR [383] or FLUKA [384, 385] contributed crucially to the understanding of hadron calorimetry. The simulations reproduce the general features of the substructure of hadronic showers, characterised by dense shower activity and sparser ionising track segments, and their accuracy is adequate for most purposes of quantitative design optimisations. Details of the implementation of physics models in GEANT4 are discussed in [386].

### 35.10.1.3 Particle flow approach

In a typical collider experiment, the EM and hadronic calorimeter system surrounds a charged-particle tracking volume devised for momentum measurement in a magnetic field. Matching tracks to calorimetric energy deposits provides corrections for the magnetic deflection of the charged particles, necessary for the reconstruction of invariant masses in multi-jet final states, or of the total momentum imbalance. Furthermore, for charged particles the track-based measurements are far more precise for particles in jets with energies of up to several hundred GeV, so using these instead of calorimeter energies may optimise the jet energy resolution. This so-called "particle flow" approach aims at reconstructing each particle individually, using a combination of the best measurements from the detector.

About 60% of the energy in a typical jet is carried by charged particles, predominantly hadrons, 30% by photons and only 10% by long-lived neutral hadrons ($K_L^0$ and $n$), for which hadronic calorimetry is unavoidable. Assuming, as motivated by detector designs proposed for a future Higgs factory, $15\%/\sqrt{E(\text{GeV})}$ for photons and $55\%/\sqrt{E(\text{GeV})}$ for hadrons, then, in the ideal case, where each particle is resolved, a jet energy resolution of $19\%/\sqrt{E(\text{GeV})}$ could be obtained. Here the dominant part ($18\%/\sqrt{E(\text{GeV})}$) is still due to the calorimeter resolution for neutral hadrons.

The particle flow method places high demands on the imaging capabilities of the calorimeters, and on the pattern recognition performance of the reconstruction algorithms. Only energy deposits not associated with charged particles and not identified as photons, will be interpreted as neutral hadrons. In practice, this cannot always be done unambiguously, and mis-assignments give rise to an additional measurement uncertainty, which is called con-fusion. For simulated detectors at proposed future Higgs factories jet energy resolutions of 3-4% have been demonstrated, signifi-cantly larger than in the ideal case, but sufficient for the required efficient separation of W and Z hadronic final states.

Particle flow-like techniques were first applied in the ALEPH detector [387], which achieved a jet energy resolution of $60\%/\sqrt{E}$, or 6.2 GeV for hadronic Z decays. More recently, particle flow techniques are successfully used in the CMS experiment [388], for example improving the missing energy resolution by one third over a wide range.

The Pandora particle flow algorithm (PFA) [389] is the most developed and best performing today in the context of future lep-ton colliders. The algorithms make use of topological information, including the sub-structure of showers, as well as the compatibil-ity of calorimetric and track-based measurements. In this way the purely calorimetric performance for the jet is either retained or im-proved. In the framework of studies for CLIC [390], it was shown that the required jet energy resolution of 3.5% can be achieved with the PF technique for jet energies up to 1500 GeV.

For the use of energy-momentum matching in the assignment of energy depositions, and for energy flow treatment of dense jets, particle flow calorimeters with their emphasis on imaging must still feature a good energy resolution. Furthermore, the neutral hadron energy uncertainty is the dominant contribution to the jet resolution for low energy jets, where particles are well separated, while at higher energies the confusion effects take over.

High granularity in all three space dimensions comparable to or smaller than the length scales of particle showers given by $X_0$
and $\rho_M$ for both ECAL and HCAL brings additional advantages; for example it offers ideal conditions for the application of software compensation methods, which improves the intrinsic resolution and also reduces "confusion" [391]. A particular strength is the possibility to use topological information such as the reconstructed starting point of the shower for the estimation of leakage. Moreover, the combination of fine-grained topological reconstruction and cuts on cluster-wise timing with a precision of few tens of pico-seconds allows for powerful pile-up rejection. This extends the application range of particle flow methods towards collider environments with less benign background conditions, like multi-TeV $e^+e^-$ collisions, and it is an asset on its own for high-intensity hadron colliders, even if particle flow methods are difficult to apply.

The performance of highly granular calorimeters depends, among others, on the particle separation and pattern recognition capabilities and on the single particle energy and timing resolution. They are thus not only optimised for energy measurement, but also for multi-dimensional shower reconstruction and particle separation in space and time. The potential of such calorimeters can be further enhanced by combining a fine spatial segmentation with maximised information on energy, time, and possibly the nature of the energy deposition, for example by using dual-readout methods. This may be achieved, for example, by combining multiple optical materials sensitive to different components in the shower in finely segmented sandwich calorimeters, or by using timing information to extract the position of the energy deposition along a fibre. For future collider experiments, different approaches based on silicon, scintillator, noble liquid, fibre and crystal readout are being pursued [392]. They all foresee, with different level of emphasis, high granularity to enable the application of particle flow methods, and in some cases to further enhance their performance by additional information not exploited in current algorithms. One such example is the proposal [393] to benefit from the superior electromagnetic energy performance of crystals to pair photons from $\pi^0$ decays and thereby resolve jet finding ambiguities to improve the jet energy resolution.

### 35.10.2 Electromagnetic calorimeters

Revised August 2023 by C.L. Woody (BNL) and R.-Y. Zhu (HEP California Inst. of Technology).

The development of electromagnetic showers is discussed in the section on "Passage of Particles Through Matter" (Sec. 34 of this Review). Formulae are given which approximately describe average showers, but since the physics of electromagnetic showers is well understood, a detailed and reliable Monte Carlo simulation is possible. EGS4 [394] and GEANT [381] have emerged as the standards.

Electromagnetic calorimeters are devices that are designed to measure the total energy of electrons and photons by total absorption. They come in two general categories: homogeneous and sampling. In a homogeneous calorimeter, all of the particle's energy is deposited in the active detector volume and is used to produce a measurable signal (either scintillation light, Cherenkov light or charge). Homogeneous electromagnetic calorimeters are typically constructed using high density, high Z inorganic scintillating crystals such as BaF₂, BGO, CsI, CsI(Tl), LYSO, NaI(Tl) and PWO, non-scintillating Cherenkov radiators such as lead glass and lead fluoride (PbF₂), or ionizing noble liquids such as liquid argon, liquid krypton or liquid xenon. The properties of some commonly used inorganic crystal scintillators can be found in Table 35.4. Total absorption homogeneous calorimeters such as those built with heavy crystal scintillators provide the best energy resolution for measuring electromagnetic showers and are generally used when the best possible performance is required, particularly at lower energies. Attention, however, needs to be paid to radiation damage in inorganic scintillators when exposed to the typical environment of high luminosity proton colliders, as discussed below and in Section 35.4.

A sampling calorimeter consists of an active medium which generates a signal and a passive medium which functions as an absorber. In this case, most of the particle's energy is deposited in

the absorber and only a fraction of the energy is detected in the active medium. The ratio of energy in the sampling medium to the total energy in calorimeter is called the sampling fraction. The active medium may be a scintillator, an ionizing noble liquid, a semiconductor, or a gas ionization detector. The absorber is typically a heavy metal with a high Z such as lead, tungsten, iron, copper, or depleted uranium. The active material is interspersed with the passive absorber in a variety of ways, e.g. by using alternating plates of active material and absorber or embedding the active material, such as scintillating fibers, into the absorber. The main difficulty in this approach is extracting the signal from the active material. One possibility is a simple stack of alternating absorber and active material, which, however, usually leads to gaps for services and hence non-uniformities. This can be improved by using a so-called "spaghetti" design, where scintillating fibers are brought to the front or back of the detector and read out. This can also be done with either wavelength shifting plates or fibers, such as in a so-called "shashlik" design where wavelength shifting fibers run through the stack of alternating scintillator and absorber plates and are read out at end, or embedding wavelength shifting fibers in the scintillating plates which are then brought out to the edges or back of the detector and read out. For ionization detectors, there is also an "accordion" design which avoids all gaps for services and where the absorber plates are folded into an accordion shape along with interspersed electrodes to collect the ionization charge [395]. While these readout schemes are generally more complicated than those for homogeneous calorimeters, the sampling calorimeter design allows the construction of large calorimeters at much lower cost than homogeneous calorimeters.

The energy resolution $\sigma_E/E$ of a calorimeter can be parameterized as $a/\sqrt{E} \oplus b \oplus c/E$, where $\oplus$ represents addition in quadrature and $E$ and $\sigma_E$ are in GeV. The stochastic term $a$ represents statistics-related fluctuations such as intrinsic shower fluctuations, photoelectron statistics, dead material at the front of the calorimeter, and sampling fluctuations for minimum ionizing particles. For a fixed number of radiation lengths, the stochastic term $a$ for a sampling calorimeter is expected to be proportional to $\sqrt{t/f}$, where $t$ is plate thickness and $f$ is sampling fraction [396–398]. The stochastic term $a$ is typically of the order of a few percent for a homogeneous calorimeter, and is generally in the range of 8 to 20% for sampling calorimeters, depending on the sampling fraction.

The main contributions to the systematic, or constant, term $b$ are detector non-uniformity and calibration uncertainties. In the case of hadronic cascades discussed below, non-compensation also contributes deviations from $\sqrt{E}$ scaling. Another important contribution to the energy resolution of calorimeters that are used in high radiation environments such as high luminosity colliders is radiation damage of the active medium. Radiation damage can induce optical absorption in scintillating materials which reduces the measured light output and produces non-uniformities in light collection. This can be mitigated by developing radiation-hard active media [399], by reducing the signal path length [400] and by frequent in situ calibration and monitoring [75, 398]. With effort, the constant term $b$ can be reduced to below one percent. The term $c$ is due mainly to electronic noise summed over the readout channels required to measure the shower energy (typically a few Molière radii).

The position resolution depends on the effective Molière radius and the transverse granularity of the calorimeter. Like the energy resolution, it can be factored as $a/\sqrt{E} \oplus b$, where $a$ is the stochastic term, typically of the order of a few mm to 20 mm, and $b$ can be as small as a fraction of mm for a dense calorimeter with fine granularity. Fine granularity also helps particle flow analysis discussed in the hadron calorimeters section below.

Electromagnetic calorimeters may also provide angular measurements for electrons and photons. This is particularly important for photon-related physics to identify the correct primary vertex, since photons are not detected by the tracking system of the overall experiment. The typical photon angular resolution is about $45 \text{ mrad}/\sqrt{E}$, which can be achieved by implementing longitudinal segmentation [395] for a sampling calorimeter or by adding a preshower detector [401] for a homogeneous calorimeter
<table><caption>Table 35.11: Resolution of typical electromagnetic calorimeters. *E* is in GeV.</caption><thead><tr><th>Technology (Experiment)</th><th>Depth</th><th>Energy resolution</th><th>Date</th></tr></thead><tbody><tr><td>NaI(Tl) (Crystal Ball)</td><td>20X<sub>0</sub></td><td>2.7%/E<sup>1/4</sup></td><td>1983</td></tr><tr><td>Bi<sub>4</sub>Ge<sub>3</sub>O<sub>12</sub> (BGO) (L3)</td><td>22X<sub>0</sub></td><td>2%/√E ⊕ 0.7%</td><td>1993</td></tr><tr><td>CsI (KTeV)</td><td>27X<sub>0</sub></td><td>2%/√E ⊕ 0.45%</td><td>1996</td></tr><tr><td>CsI(Tl) (BaBar)</td><td>16–18X<sub>0</sub></td><td>2.3%/E<sup>1/4</sup> ⊕ 1.4%</td><td>1999</td></tr><tr><td>CsI(Tl) (BELLE)</td><td>16X<sub>0</sub></td><td>1.7% for E<sub>γ</sub> > 3.5 GeV</td><td>1998</td></tr><tr><td>CsI(Tl) (BES III)</td><td>15X<sub>0</sub></td><td>2.5% for E<sub>γ</sub> = 1 GeV</td><td>2010</td></tr><tr><td>PbWO<sub>4</sub> (PWO) (CMS)</td><td>25X<sub>0</sub></td><td>3%/√E ⊕ 0.5% ⊕ 0.2/E</td><td>1997</td></tr><tr><td>PbWO<sub>4</sub> (PWO) (ALICE)</td><td>19X<sub>0</sub></td><td>3.6%/√E ⊕ 1.2%</td><td>2008</td></tr><tr><td>Lead glass (OPAL)</td><td>20.5X<sub>0</sub></td><td>5%/√E</td><td>1990</td></tr><tr><td>Liquid Kr (NA48)</td><td>27X<sub>0</sub></td><td>3.2%/√E ⊕ 0.42% ⊕ 0.09/E</td><td>1998</td></tr><tr><td>Scintillator/depleted U (ZEUS)</td><td>20–30X<sub>0</sub></td><td>18%/√E</td><td>1988</td></tr><tr><td>Scintillator/Pb (CDF)</td><td>18X<sub>0</sub></td><td>13.5%/√E</td><td>1988</td></tr><tr><td>SciFi/Pb spaghetti (KLOE)</td><td>15X<sub>0</sub></td><td>5.7%/√E ⊕ 0.6%</td><td>1995</td></tr><tr><td>SciFi/W (sPHENIX)</td><td>20X<sub>0</sub></td><td>13%/√E</td><td>2022</td></tr><tr><td>Liquid Ar/Pb (NA31)</td><td>27X<sub>0</sub></td><td>7.5%/√E ⊕ 0.5% ⊕ 0.1/E</td><td>1988</td></tr><tr><td>Liquid Ar/Pb (SLD)</td><td>21X<sub>0</sub></td><td>8%/√E</td><td>1993</td></tr><tr><td>Liquid Ar/Pb (H1)</td><td>20–30X<sub>0</sub></td><td>12%/√E ⊕ 1%</td><td>1998</td></tr><tr><td>Liquid Ar/depl. U (DØ)</td><td>20.5X<sub>0</sub></td><td>16%/√E ⊕ 0.3% ⊕ 0.3/E</td><td>1993</td></tr><tr><td>Liquid Ar/Pb accordion (ATLAS)</td><td>25X<sub>0</sub></td><td>10%/√E ⊕ 0.4% ⊕ 0.3/E</td><td>1996</td></tr></tbody></table>

without longitudinal segmentation.

There have been many electromagnetic calorimeters built and used in particle physics experiments for a variety of applications. Table 35.11 provides a short list of the major ones used in some of the larger experiments. Also listed are calorimeter depths in radiation lengths ($X_0$) and the achieved energy resolution. Whenever possible, the performance of the calorimeters in situ is quoted, which is usually in good agreement with prototype test beam results as well as EGS or GEANT simulations, provided that all systematic effects are properly included. Details about detector design and performance can be found in Appendix C of reference [398] and Proceedings of the International Conference series on Calorimetry in High Energy Physics.

### 35.10.3 Hadronic calorimeters

Revised August 2021 by F. Sefkow (DESY, Hamburg) and F. Simon (KIT).

Hadronic calorimetry [380, 402, 403] is considerably more complex than electromagnetic (EM) calorimetry due to the wider range and different nature of physical processes contributing to shower development and energy deposition, which in turn are characterised by different length and time scales. Hadronic showers are initiated by inelastic strong interactions of highly-energetic charged and neutral hadrons with atomic nuclei. These interactions result in the production of secondary particles, which drive the development of the shower. Among these are energetic hadrons, as well as lower-energy nucleons, photons and nuclear fragments. Energy transferred to nuclear break-up, excitation or recoil does not, in general, produce a signal, but remains invisible, and event-to-event fluctuations of this invisible energy deposit ultimately limit the resolution of hadronic calorimeters (HCALs).

The length scale of the interaction of relativistic hadrons is given by the nuclear interaction length $\lambda_I$. As discussed in Section 35.10.1, $\lambda_I$ is a factor 10 to 30 larger than $X_0$ for common materials used in the construction of calorimeters. HCALs thus require a significantly larger geometrical depth for full containment than electromagnetic calorimeters (ECALs), albeit not by the factor suggested by the ratio of $\lambda_I/X_0$ due to the different nature of the showers. A key role is played by the production of $\pi^0$s and their subsequent decay into two photons. These result in the formation of electromagnetic sub-showers which evolve on the scale given by the radiation length $X_0$, and thus require sufficiently fine sampling of the shower activity to capture also this electromagnetic component. The two different length scales occurring in hadronic

<figure><img src="image_234.png" /><figcaption>Figure 35.35: Longitudinal profile of hadronic showers induced by 45 GeV negative pions measured in the CALICE highly-granular steel-scintillator sampling calorimeter [404]. Left: Reconstructed position of the first inelastic interaction, compared to simulations (GEANT4 9.4p03, FTFP_BERT), and an exponential fit yielding a slope consistent with the expected pion interaction length. Right: Longitudinal shower profile measured from the front face of the calorimeter (dots, without corrections for dead cells) and relative to the position of the shower start given by the first inelastic interaction (filled histogram). The visible energy is given in units of the most probable energy loss of a minimum-ionizing particle (MIP). The integrals of the distributions are normalized to unity.</figcaption></figure>

showers, and the large fluctuations of hadronic, electromagnetic and invisible activity, result in significant event-to-event variations of the energy response and of the shower topology. This topology is characterized by a lumpy structure, with compact regions of high local energy density originating from electromagnetic sub-showers, and sparser hadronic activity with minimum-ionizing hadrons.

Figure 35.35 (left) shows the distribution of the longitudinal position of the first inelastic interaction measured for pion-induced hadronic showers with the highly-granular scintillator-steel HCAL of the CALICE collaboration [404]. The figure illustrates the exponential distribution, with a slope consistent with the pion interaction length expected from the geometry and material composition of the calorimeter. This distribution is well reproduced by simulations using GEANT4 [381, 386]. Figure 35.35 (right) shows the mean longitudinal shower profile, given by the mean energy deposition in each calorimeter layer, both relative to the front face of the calorimeter (dots, without corrections for dead
cells) and relative to the measured shower starting point given by the first inelastic interaction (filled histogram). The latter is much more compact than the former, which is a convolution of the latter with the distribution of shower starting point shown in the left panel of the figure. This shows that the detector depth required for adequate shower containment is significantly influenced by the fluctuations of the position of the first hadronic interaction. Detection of this position allows for an estimate of leakage from a finite calorimeter volume, and it enables stringent tests of shower evolution models. In the past years, motivated by precision needs at the next generation of $e^{+}e^{-}$ colliders, the CALICE collaboration has constructed a number of prototypes with high 3-dimensional spatial granularity and recorded large sets of data at test beams, allowing for studies of shower evolution processes in unprecedented detail. The refined simulations support the trend to proceed from research focused on understanding the global intrinsic properties of showers, e.g. for the purpose of compensation, towards the study of detailed information, in space, time, and energy deposition type, through many and multiple readout channels.

In an inelastic hadronic collision a significant fraction $f_{\text{fem}}$ of the energy is removed from further hadronic interaction by the production of secondary $\pi^{0}/\eta$'s, whose decay photons generate high-energy electromagnetic showers. Charged secondaries ($\pi^{\pm}$, p, ... ) deposit energy via ionization and excitation, but also interact with nuclei, producing evaporation neutrons, spallation protons and neutrons, and heavier spallation fragments. The charged collision products produce detectable ionization, as do the showering $\gamma$-rays from the prompt de-excitation of highly excited nuclei. The recoiling nuclei generate little or no detectable signal, as mentioned previously. The neutrons lose kinetic energy in elastic collisions which generate ionization signals via recoiling protons, thermalize on a time scale of several $\mu$s, and are finally captured, with the production of more $\gamma$-rays—usually outside the acceptance gate of the electronics. Between endothermic spallation losses, nuclear recoils, and late neutron capture, a significant fraction of the hadronic energy (20%–40%, depending on the absorber and energy of the incident particle) is used to overcome nuclear binding energies and is therefore lost or “invisible.”

In a hadron-nucleus collision a large fraction of the incident energy is carried by a “leading particle” with the same quark content as the incident hadron. If the projectile is a charged pion, the leading particle is usually a pion, which can be neutral and hence contributes to the EM sector. This is not true for incident protons. The result is an increased mean hadronic fraction for incident protons.

The complexity of hadronic showers also has a significant impact on the energy measurement. In contrast to EM showers, hadronic cascade processes are characterised by the production of relatively few high-energy particles. The number multiplicity of these particles produced in hadronic interactions increases only logarithmically with energy. The lost energy and $f_{\text{fem}}$ are highly variable from event to event, and on average increase with increasing energy [405]. Electromagnetic sub-showers typically result in a higher response than the hadronic parts of the cascade, where undetectable energy loss due to nuclear dissociation, the long time scales and the material dependence of neutron signals, as well as other effects reduce the measured signal. This difference in response is often expressed by the $\langle h/e \rangle$ ratio, a calorimeter-dependent quantity which is smaller than unity for many, but not all, HCALs. The increase of the electromagnetic fraction with energy thus introduces a non-linear contribution to the response. Combined with the significant event-by-event fluctuations between electromagnetic and hadronic fractions of the showers and between different hadronic processes the non-equality of h and e deteriorates the energy resolution of HCALs. Different strategies to address this exist, as discussed further below.

Most large HCALs are parts of complex $4\pi$ detectors at colliding beam facilities. To date, all these HCALs are sampling calorimeters. This choice is imposed by the physics of hadronic showers, both by the required depth for containment which favours high-density materials with short $\lambda_{I}$, and by the differences in response to electromagnetic and hadronic parts of the cascade, which are

<figure><img src="image_235.png" /><figcaption>Figure 35.36: Two examples of geometrical structures of scintillator-based HCALs. (a) A wedge of the ATLAS central tile calorimeter consisting of scintillator tiles in iron, read out via wavelength-shifting fibers and PMTs [406]. The coordinate system is that of the ATLAS calorimeter within the experiment, with the z axis along the beam direction, r pointing radially outward, and $\phi$ being the azimuthal angle. (b) An illustration of the “SiPM-on-tile” structure used in the CALICE analogue HCAL prototype, and in the CMS High-Granularity Calorimeter (HG-CAL), highly granular calorimeters with steel absorbers and small scintillator tiles directly read out via SiPMs with embedded electronics [403, 407–409].</figcaption></figure>

particularly severe for homogeneous calorimeters. Common absorber materials are Fe, Cu, Pb, and U, with W also used occasionally. A large variety of different active materials are used, depending on application and optimisation, from plastic scintillators (plates, tiles, bars, fibers), crystals and Cherenkov media, silicon, liquid argon (LAr), to gaseous detectors. The energy loss of particles in the active medium is either detected directly by collecting charge, or via scintillation or Cherenkov light observed with conventional photomultipliers (PMTs), photodiodes or silicon photomultipliers (SiPMs). The choice of both active and passive materials is driven by different, sometimes conflicting, constraints, including performance requirements, space and other mechanical boundary conditions, radiation tolerance, and cost considerations.

A wide range of different geometries of absorbers and sensors is used, with design choices depending on the chosen priorities of addressing these constraints, also considering the need to bring the signals to the outside of the detector while achieving a hermetic coverage and other constraints. In this context it is important to note that a classic sandwich structure with absorber plates and active elements approximately perpendicular to the particle incidence is not required, and arbitrary orientations are viable for good calorimetric measurements, as long as channelling, meaning the extended passage of primary particles through low-density active regions, is excluded. Figure 35.36 shows two examples of plastic scintillator-based calorimeters to illustrate differences in design between coarsely-segmented and highly-granular calorimeters. The ATLAS tile calorimeter [406] uses scintillator tiles coupled to wavelength-shifting fibers which collect the light from the tiles and guide them to PMTs outside of the active region of the calorimeter. The calorimeter is segmented in $\phi$ (azimuthal angle) and $\eta$ (pseudorapidity, defined as $\eta = -\ln \tan(\theta/2)$, where $\theta$ is the angle relative to the beam axis), with coarse longitudinal segmentation. The technological prototype of the CALICE analog HCAL, a highly granular SiPM-on-tile calorimeter is based on scintillator tiles directly coupled to SiPMs, which, together with
the front-end electronics, are embedded inside of the active volume of the calorimeter. The CMS HGCAL [409] uses the same concept in part of the detector, and in addition embeds elements for digital data concentration and power distribution in the active volume. Other detector solutions include scintillating fibres threading an absorber [410], liquid-argon-filled tubes [411] and the "accordion" LAr detector [412]. The latter has zig-zag absorber plates to minimize channeling effects; the calorimeter is hermetic (no cracks), and plates are oriented so that cascades cross the same plate repeatedly.

In particular, but not exclusively, the combination of heavy absorber materials (Pb, U) with plastic scintillators allows the construction of HCALs that have a near-equal response to electromagnetic and hadronic parts of the cascade, so-called compensating calorimeters [413]. In this first study, it was recognized that nuclear fission can amplify the hadronic signal when using uranium absorbers. However, the key drivers of compensation are the reduction of the electromagnetic response with high-Z absorbers and an increased sensitivity to neutrons, which are strongly correlated to otherwise invisible energy loss due to nuclear dissociation, with a hydrogeneous active medium [414–418]. Since the electromagnetic cross section increases, and the critical energy decreases with Z, and since most of the energy of an electromagnetic shower is deposited by low-energy, short-range electrons, a disproportionate fraction of the total electromagnetic energy is deposited in the absorber in the high-Z case. Hydrogenous active media, such as organic scintillators, have a high sensitivity to spallation neutrons via elastic $n-p$ scattering. The number of produced neutrons is highly correlated with the invisible energy of the hadronic cascade. Increasing the sensitivity to these particles can thus boost the visible hadronic signal. Achieving compensation requires carefully-chosen sampling fractions and frequencies, with the response to the hadronic parts of the shower also sensitive to the integration time of the electronics due to the time structure of the neutron component of the signal.

Since Cherenkov light, for example in quartz plates or fibers, or in crystals, is produced only by relativistic particles in the cascade and thus predominantly by the electromagnetic component of the shower, such media are less common in hadronic than in ECALs. Notable exceptions are applications that require high radiation tolerance, and dual-readout calorimeters which specifically exploit this feature, as discussed below.

Silicon offers high compactness, high granularity, high radiation tolerance, long-term stability and fast charge collection, and is thus an interesting active material for sampling calorimeters. A thorough overview of the development and of the features of this technology can be found in a recent review paper [419]. The first silicon-based calorimeter in a collider experiment was a HCAL, the H1 PLUG calorimeter [420] covering the very forward region of the H1 experiment at HERA. The SICAPO collaboration has demonstrated the conceptual possibility of constructing compensating HCALs using silicon sensors [421]. Silicon is currently the technology of choice for several ECALs for future Higgs factories [422–424]. It is also being used extensively in the CMS HG-CAL [409] in both electromagnetic and hadronic sections, complemented by scintillator tiles with on-tile SiPM readout where the radiation levels allow.

More generally, high-granularity calorimeters play an increasingly important role, in particular motivated by the use of particle-flow algorithms for global event reconstruction (see Section 35.10.1). The associated technologies for both electromagnetic and HCALs have been pioneered by the CALICE collaboration, which has built and tested an increasingly sophisticated series of “imaging” calorimeters with a highly granular read-out [425]. In the area of HCALs, this includes the scintillator-based analog HCAL [426] with the latest SiPM-on-tile technological prototype with fully-integrated electronics having approximately 22,000 channels [407, 408], as well as digital [427] and semidigital [428] calorimeters using gas detectors, such as RPCs (Sec. 35.6.7) and micropattern gas detectors (Sec. 35.6.4), with channel counts of up to 500,000. The large numbers of channels of high-granularity calorimeters presents a significant integration challenge for full detector systems, and requires the full integra-

tion of the front-end electronics inside of the active volume of the detector, as well as very compact data concentration and interface units. The first such detector in construction for a collider experiment is the CMS HGCAL [409]. Beam tests with a combined prototype using both silicon and scintillator-SiPM instrumented active layers have confirmed simulation-based expectations [429]. The total silicon area of the full HGCAL amounts to about 600 m², and about 240,000 SiPMs are foreseen. The calibration of such calorimeters requires the monitoring of a large number of cells, which is achieved in-situ using reconstructed track segments within hadronic showers [409, 430] or externally identified muons. For this method, the capability to detect the most probable energy loss of a minimum-ionizing particle in a single cell is essential. This is required over the full lifetime of the detector, also after the active elements have received significant radiation damage, resulting in increased noise and reduced charge or signal collection efficiency. Due to the large number of cells contributing to the measurement of one shower, the requirements on the precision of the calibration of individual cells is relaxed relative to the global energy calibration of the calorimeter.

The energy resolution of HCALs is severely affected by fluctuations between different components of the cascade, exacerbated by differences in response to purely hadronic and to electromagnetic sub-showers. In many detectors, fluctuations in the electromagnetic energy fraction, $f_{em}$, and the related, consequential variations in nuclear energy dissociation losses, represent the biggest single contribution to the hadron energy resolution. One strategy to address this problem is the construction of intrinsically-compensating calorimeters, which imposes stringent constraints on materials and geometries as discussed above. Compensating calorimeters are not used in current large collider experiments, and are at the moment not considered for future collider detectors. Two different strategies are presently followed to improve the energy resolution in non-compensating calorimeters: Offline weighting or software compensation in longitudinally-segmented or in highly-granular calorimeters; and dual-readout calorimetry.

Software compensation techniques exploit the fact that electromagnetic sub-showers typically have a higher spatial density than the purely hadronic parts of the cascade. Amplitude (or energy-density) dependent weights are applied in the reconstruction to reduce the effects of shower-to-shower fluctuations. These techniques were pioneered by the CDHS collaboration for a longitudinally segmented steel-plastic scintillator calorimeter [431], where an improvement of the energy resolution of 10% (at 10 GeV) to 30% (at 140 GeV) for charged pion showers was achieved. Similar techniques were successfully applied in the H1 [432] liquid argon calorimeter system resulting in a stochastic term of $51\%/\sqrt{E}$, and in the ATLAS [433] endcap calorimeters, also based on liquid argon, with an energy resolution of $84\%/\sqrt{E}$. Inspired by these approaches, a software compensation technique using the detailed spatial information provided by highly-granular calorimeters of the CALICE collaboration has been developed, achieving up to 25% improvement of the energy resolution compared to the resolution without software compensation, resulting in a stochastic term of $45\%/\sqrt{E}$ [434] in a scintillator tile calorimeter with steel absorbers. This technique has also been successfully transferred to particle-flow reconstruction [435], resulting in an improvement of the jet-energy resolution in simulated events by 8% - 15%, depending on jet energy. Highly-granular calorimeters with software compensation and particle-flow reconstruction are currently studied as the baseline configuration for several Higgs-factory detectors.

The dual-readout method, originally proposed by Mockett in 1983 [436], measures $f_{em}$ event by event in parallel to the total deposited energy. It uses the fact that most of the relativistic particles in the shower originate from the electromagnetic part, and that only those produce Cherenkov light, while the signal of the hadronic part is mostly due to non-relativistic protons. In practice either two different active media, e.g. scintillator and quartz, are used to register scintillation and Cherenkov light, respectively, or the optical signals from the two processes occurring in heavy crystals are disentangled, using their different spectral, directional or timing properties.
The Cherenkov and scintillation signals, normalised to the response for electrons, are given by

$$C = [f_{em} + (h/e)_C(1 - f_{em})]E, \quad (35.44)$$

$$S = [f_{em} + (h/e)_S(1 - f_{em})]E, \quad (35.45)$$

respectively, which can be solved for the fraction $f_{em}$ and the energy

$$E = (\xi S - C)/(\xi - 1), \quad (35.46)$$

where $\xi = [1 - (h/e)_C]/[1 - (h/e)_S]$ and $(h/e)_{C,S}$ denote the average ratios of hadronic to electromagnetic response in the Cherenkov and scintillator parts, respectively. This is illustrated in Figure 35.37, which shows their correlation for a set of simulated negative pion events [437] using FLUKA [385].

<figure><img src="image_236.png" /><figcaption>Figure 35.37: Scatter plot of Monte Carlo $C/E$ (Cherenkov) vs $S/E$ (scintillator) signals for individual events in a dual-readout calorimeter for 100 GeV negative pions and photons. Hadronic events are shown in blue, and scatter about the indicated event locus. Electromagnetic events cluster about $(C/E, S/E) = (1,1)$. In this case worse resolution (fewer p.e.'s) was assumed for the Cherenkov events, leading to the “elliptical” distribution.</figcaption></figure>

It was noted that the method demands a steep slope $\xi$, which implies that the scintillator read-out should be as compensating as possible, which however reduces the room for improvement by adding Cherenkov information.

The method was tested by the DREAM/RD52 collaboration [438,439], using a 1 ton copper matrix with embedded quartz and scintillating fibers. The value of $\xi$ was about 3 in this detector. With this detector, a resolution of $70\% / \sqrt{E(\text{GeV})}$ was obtained for single hadrons [440]. Due to the small size of the module, this includes contributions from transverse leakage which prevent the full exploitation of key features of dual readout, and thus underestimates the potential of the method.

The separate Cherenkov read-out evidently provides excellent pion-electron separation for particle identification. In another RD52 prototype, each fiber is read out individually by SiPMs, giving also a superior transverse granularity. A fiber-based calorimeter with full solid-angle coverage requires a pointing geometry due to the limited or missing longitudinal segmentation. The resulting challenges for a mechanical design are studied in the framework of the IDEA detector concept [441].

Table 35.12 shows selected examples of the energy resolution of HCALs for single charged pions achieved in beam tests. The examples are selected to illustrate the performance achieved with different designs ranging from intrinsic compensation to software compensation and dual readout, with a focus on results by R&D

projects. It should be noted that the exact values of the different resolution terms depend on the functional form used in the fit, here the addition in quadrature is used for the cases where more than just the stochastic term is quoted. The results shown in the table illustrate that (close to) compensating calorimeters with optimized sampling fraction and frequency, such as HELIOS (U-plastic scintillator) and SPACAL (Pb-plastic scintillator), achieve a very good energy resolution. The comparison with Bernardi et al., which has the same Pb/scintillator ratio by volume as SPACAL, but coarser sampling in a sandwich structure, illustrates the importance of the geometrical details. Beyond the examples shown in the table, liquid argon has also been explored as an active medium, for example in the context of the SLD detector with different absorber options [446]. Due to the reduced sensitivity to neutrons in the shower, not the same resolution as for plastic-scintillator-based systems is achieved. This technology has also been used in the D0 experiment, as discussed below. The dual-readout method has the potential to reach or surpass this performance, but would require a prototype sufficiently large for full longitudinal and transverse shower containment for an experimental demonstration. The two CALICE calorimeters shown, which use the same active elements (5 mm thick scintillator tiles) but different absorbers (21.4 mm Fe vs 10 mm W + 4 mm Fe) per layer, illustrate the impact of the absorber choice on energy resolution and reconstruction possibilities. While the tungsten-based W-AHCAL setup is very close to compensating, the steel-based AHCAL achieves a better energy resolution when software compensation is applied, profiting from the finer sampling of the electromagnetic parts of the cascade and the correction for shower-to-shower fluctuations of the electromagnetic fraction in the reconstruction. In the case of tungsten, software compensation does not significantly improve the energy resolution, as expected. The comparison of the CALICE AHCAL performance with the one of CDHS illustrates the benefits of higher granularity for software compensation techniques, but it should be noted that the absorber thickness of the latter is 25 mm, with the same scintillator thickness as in the case of CALICE.

As explained in the introduction, in most high-energy physics experiments, the HCAL follows after an ECAL, making the response of the latter to hadronic cascades highly relevant for the overall performance of the combined ECAL HCAL system. For scenarios where the electromagnetic and the HCAL have very different $\langle h/e \rangle$, as is typically the case for crystal-based ECALs, the fluctuations of the fraction of the hadronic shower contained within the ECAL result in a significant deterioration of the energy resolution for hadrons. A deterioration of the hadronic performance also results from larger amounts of not-instrumented material, e.g. supports and services, between electromagnetic and hadronic sections. In particle-flow calorimeters, a large value of the $\lambda_I/X_0$ ratio of the absorber material, like in tungsten, maximises the longitudinal separation of electromagnetic and hadronic showers. This is reflected in the design of particle-flow-based detector concepts for future Higgs Factories.

Table 35.13 summarizes the single hadron energy resolution obtained from test beams of the combined ECAL and HCAL systems of the large multi-purpose experiments at HERA, the Tevatron and at the LHC. These systems are examples of different optimization strategies. D0 and ZEUS are near-compensating systems with the same technology in ECAL and HCAL emphasizing hadronic performance. ATLAS and H1 use sampling ECALs with good electromagnetic resolution and weighting techniques exploiting longitudinal and transverse shower information for hadronic energy reconstruction. CMS, with a crystal ECAL and a scintillator-brass HCAL, prioritizes electromagnetic performance, with very different $\langle h/e \rangle$ in the electromagnetic and hadronic system. Of the detectors shown in the table, CMS has the best electromagnetic resolution by a comfortable margin, but consequently the weakest hadronic resolution. The best hadronic performance is achieved with the compensating calorimeter of ZEUS, which however has a weaker electromagnetic performance than the other calorimeter systems shown here. It should be noted that an excellent single-hadron resolution in general does not fully propagate into the jet-energy performance. Inactive material in
<table><caption>**Table 35.12:** Energy resolution of selected hadron calorimeters for single charged hadrons obtained in beam tests.</caption><thead><tr><th>Calorimeter</th><th>Passive</th><th>Active</th><th>Resolution</th><th>Ref.</th></tr></thead><tbody><tr><td>Bernardi et al.</td><td>Pb</td><td>Scintillator layers</td><td>44.2%/$\sqrt{E}$<sup>§</sup></td><td>[442]</td></tr><tr><td>CALICE AHCAL</td><td>Fe</td><td>Scintillator tiles</td><td>44.3%/$\sqrt{E}$ ⊕ 1.8%<sup>†</sup></td><td>[434]</td></tr><tr><td>CALICE W-AHCAL</td><td>W</td><td>Scintillator tiles</td><td>57.9%/$\sqrt{E}$ ⊕ 4.6% ⊕ 0.065/$E$<sup>§</sup></td><td>[443]</td></tr><tr><td>CDHS</td><td>Fe</td><td>Scintillator layers</td><td>58%/$\sqrt{E}$<sup>‡</sup></td><td>[431]</td></tr><tr><td>DREAM/RD52</td><td>Pb</td><td>Scint.+ Quartz fibers</td><td>70%/$\sqrt{E}$<sup>*</sup></td><td>[440]</td></tr><tr><td>HELIOS</td><td>U</td><td>Scintillator layers</td><td>34%/$\sqrt{E}$<sup>§</sup></td><td>[444]</td></tr><tr><td>SPACAL</td><td>Pb</td><td>Scintillating fibers</td><td>33.3%/$\sqrt{E}$ ⊕ 2.2%<sup>§</sup></td><td>[445]</td></tr></tbody></table>

§ Bernardi et al., CALICE W-AHCAL, HELIOS, SPACAL: (near-)compensating calorimeters.

† CALICE AHCAL: Local software compensation exploiting the high granularity of the calorimeter.

‡ CDHS: Offline weighting using longitudinal information.

* DREAM/RD52: Due to the relatively small transverse size of the detector lateral leakage was significant, deteriorating the energy resolution with respect to the full potential of the dual readout method.

<table><caption>**Table 35.13:** Energy resolution of selected combined electromagnetic and hadronic calorimeter systems in past and present high-energy collider experiments for single hadrons. The results are taken from beam tests of prototypes with the electromagnetic calorimeter upstream of the hadronic calorimeter.</caption><thead><tr><th>Experiment</th><th>technology (ECAL, HCAL)</th><th>Combined hadronic resolution</th><th>Reference</th></tr></thead><tbody><tr><td>H1</td><td>Pb/LAr, Steel / LAr</td><td>46%/$\sqrt{E}$ ⊕ 2.6% ⊕ 0.73/$E$</td><td>[447]</td></tr><tr><td>ZEUS</td><td>depleted U / plastic scintillator</td><td>35%/$\sqrt{E}$</td><td>[448]</td></tr><tr><td>CDF</td><td>Pb/plastic scint., Steel/plastic scint.</td><td>68%/$\sqrt{E}$ ⊕ 4.1%</td><td>[449]</td></tr><tr><td>D0</td><td>depleted U / LAr</td><td>44.6%/$\sqrt{E}$ ⊕ 3.9%</td><td>[450]</td></tr><tr><td>ATLAS</td><td>Pb/LAr, Steel/plastic scintillator</td><td>52%/$\sqrt{E}$ ⊕ 3.0% ⊕ 1.6/$E$</td><td>[451]</td></tr><tr><td>CMS</td><td>PbWO<sub>4</sub>, brass/plastic scintillator</td><td>84.7%/$\sqrt{E}$ ⊕ 7.4%</td><td>[452]</td></tr></tbody></table>

front of the calorimeter can significantly worsen the energy resolution for jets, with an impact in particular on lower-energy particles. For example, the core of the invariant mass distribution of hadronically-decaying $Z^0$ bosons measured in ZEUS, which had a superconducting coil in front of the calorimeter, is well described by a Gaussian with a $\sigma$ of 6 GeV [453], approximately 40% wider than would be expected for a jet-energy resolution that is identical to the single-hadron performance.

### 35.10.4 Free electron drift velocities in liquid ionization chambers

Revised August 2009 by W. Walkowiak (Siegen U.).

Drift velocities of free electrons in LAr [259] are given as a function of electric field strength for different temperatures of the medium in Fig. 35.38. The drift velocities in LAr have been measured using a double-gridded drift chamber with electrons produced by a laser pulse on a gold-plated cathode. The average temperature gradient of the drift velocity of the free electrons in LAr is described [259] by

$$ \frac{\Delta v_d}{\Delta T v_d} = (-1.72 \pm 0.08) \text{ %/K.} \qquad (35.47) $$

Previous measurements [454, 455, 457, 458] range from 13% higher [455] to 18% lower [457] than these measurements. They used different techniques and show drift velocities for free electrons which cannot be explained by the temperature dependence mentioned above.

Drift velocities of free electrons in LXe [454] as a function of electric field strength are also displayed in Fig. 35.38. The drift velocity saturates for $|E| > 3$ kV/cm, and decreases with increasing temperature for LXe as well as measured e.g. by [459].

The addition of small concentrations of other molecules like $N_2$, $H_2$ and $CH_4$ in solution to the liquid typically increases the drift velocities of free electrons above the saturation value [454, 457], see example for $CH_4$ admixture to LAr in Fig. 35.38. Therefore, actual drift velocities are critically dependent on even small additions or contaminations.

<figure><img src="image_237.png" /><figcaption>**Figure 35.38:** Drift velocity of free electrons as a function of electric field strength for LAr [259], LAr + 0.5% $CH_4$ [454] and LXe [455]. The average temperatures of the liquids are indicated. Results of a fit to an empirical function [456] are superimposed. In case of LAr at 91 K the error band for the global fit [259] including statistical and systematic errors as well as correlations of the data points is given. Only statistical errors are shown for the individual LAr data points.</figcaption></figure>

## 35.11 Accelerator-based neutrino detectors

Written by M.O. Wascko (Imperial Coll. London). Minor revision in August 2023 by A. Fava (FNAL) and F. Pietropaolo (CERN;
INFN, Padova).

### 35.11.1 Introduction

Accelerator-based neutrino experiments span many orders of magnitude in neutrino energy, from a few MeV to hundreds of GeV. This wide range of neutrino energy is driven by the many physics applications of accelerator-based neutrino beams. Foremost among them is neutrino oscillation, which varies as the ratio $L/E_ν$, where L is the neutrino baseline (distance traveled), and $E_ν$ is the neutrino energy. But accelerator-based neutrino beams have also been used to study the nature of the weak interaction, to probe nucleon form factors and structure functions, and to study nuclear structure.

The first accelerator-based neutrino experiment used neutrinos from the decays of high energy pions in flight to show that the neutrinos emitted from pion decay are different from the neutrinos emitted by beta decay [460]. The field of accelerator-based neutrino experiments would likely not have expanded beyond this without Simon van der Meer's invention of the magnetic focusing horn [461], which significantly increased the flux of neutrinos aimed toward the detector. In this mini-review, we focus on experiments employing decay-in-flight beams—pions, kaons, charmed mesons, and taus—producing fluxes of neutrinos and antineutrinos from ~10 MeV to ~100 GeV.

Neutrino interactions with matter proceed only through the weak interaction, making the cross section extremely small and requiring high fluxes of neutrinos and large detector masses in order to achieve satisfactory event rates. Therefore, neutrino detector design is a balancing act taking into account sufficient numbers of nuclear targets (often achieved with inactive detector materials), adequate sampling/segmentation to ensure accurate reconstruction of the tracks and showers produced by neutrino-interaction secondary particles, and practical readout systems to allow timely analysis of data.

### 35.11.2 Signals and Backgrounds

The neutrino interaction processes available increase with increasing neutrino energy as interaction thresholds are crossed; in general neutrino-interaction cross sections grow with energy; for a detailed discussion of neutrino interactions see [462]. The multiplicity of secondary particles from each interaction process grows in complexity with neutrino energy, while the forward-boost due to increasing $E_ν$ compresses the occupied phase space in the lab frame, impacting detector designs. Because decay-in-fight beams produce neutrinos at well-defined times, leading to very small duty factors, the predominant backgrounds usually stem from unwanted beam-induced neutrino interactions, i.e. neutrinos interacting via other processes than the one being studied. A noteworthy exception is time projection chambers, wherein the long drift times can admit substantially more cosmic backgrounds than most other detection methods. Cosmic backgrounds are more rare at higher energies because the secondary particles produced by neutrino interactions yield detector signals that resemble cosmic backgrounds less and less.

Below, we describe a few of the dominant neutrino interaction processes, with a focus on the final state particle content and topologies.

#### 35.11.2.1 Charged-Current Quasi-Elastic Scattering and Pion Production

Below ~2 GeV neutrino energy, the dominant neutrino-nucleus interaction process is quasi-elastic (QE) scattering. In the charged current (CC) mode, the CCQE base neutrino reaction is $v_\ell n \to \ell^- p$, where $\ell = e, \mu, \tau$, and similarly for antineutrinos, $\bar{v}_\ell p \to \ell^+ n$. The final state particles are a charged lepton, and perhaps a recoiling nucleon if it is given enough energy to escape the nucleus. Detectors designed to observe this process should have good single-particle track resolution for muon neutrino interactions, but should have good $\mu/e$ separation for electron neutrino interactions. Because the interaction cross section falls sharply with $Q^2$, the lepton typically carries away more of the neutrino's kinetic energy than the recoiling nucleon. The fraction of backward-scattered leptons is large, however, so detectors with $4\pi$ coverage are desirable. The dominant backgrounds in this channel tend to

come from single pion production events in which the pion is not detected.

Near 1 GeV, the quasi-elastic cross section is eclipsed by pion production processes. A typical single pion production ($CC1\pi$) reaction is $v_\ell n \to \ell^- \pi^+ n$, but many more final state particle combinations are possible. Single pion production proceeds through the coherent channel and many incoherent processes, dominated by resonance production. With increasing neutrino energy, higher-order resonances can be excited, leading to multiple pions in the final state. Separating these processes from quasi-elastic scattering, and indeed from each other, requires tagging, and ideally reconstructing, the pions. Since these processes can produce neutral pions, electromagnetic (EM) shower reconstruction is more important here than it is for the quasi-elastic channel. The predominant backgrounds for pion production change with increasing neutrino energy. Detection of pion processes is also complicated because near threshold the quasi-elastic channel creates pion backgrounds through final state interactions of the recoiling nucleon, and at higher energies backgrounds come from migration of multiple pion events in which one or more pions is not detected.

#### 35.11.2.2 Deep Inelastic Scattering

Beyond a few GeV, the neutrino has enough energy to probe the nucleon at the parton scale, leading to deep inelastic scattering (DIS). In the charged-current channel, the DIS neutrino reaction is $v_\ell N \to \ell^- X$, where N is a nucleon and X encompasses the entire recoiling hadronic system. The final state particle reconstruction revolves around accurate reconstruction of the lepton momentum and containment and reconstruction of the hadronic shower energy. Because of the high neutrino energies involved, DIS events are very forward boosted, and can have extremely long particle tracks. For this reason, detectors measuring DIS interactions must be large to contain the hadronic showers in the detector volume.

#### 35.11.2.3 Neutral Currents

Neutrino interactions proceeding through the neutral current (NC) channel are identified by the lack of a charged lepton in the final state. For example, the NC elastic reaction is $v_l N \to v_l N$, and the NC DIS reaction is $v_l N \to v_l X$. NC interactions are suppressed relative to CC interactions by a factor involving the weak mixing angle; the primary backgrounds for NC interactions come from CC interactions in which the charged lepton is misidentified.

### 35.11.3 Instances of Neutrino Detector Technology

Below we describe many of the actual detectors that have been built and operated for use in accelerator-based neutrino beams.

#### 35.11.3.1 Spark Chambers

In the first accelerator-based neutrino beam experiment, Lederman, Schwartz, and Steinberger [460] used an internally-triggered spark chamber detector, filled with 10 tons of Al planes and surrounded by external scintillator veto planes, to distinguish muon tracks from electron showers, and hence muon neutrinos from electron neutrinos. The inactive Al planes served as the neutrino interaction target and as radiators for EM shower development. The detector successfully showed the presence of muon tracks from neutrino interactions. It was also sensitive to the hadronic showers induced by NC interactions, which were unknown at the time. In 1963, CERN also built and ran a large (20 ton) Al plane spark chamber in a wideband beam based on the PS accelerator [463]. More than a decade later, the Aachen-Padova [464] experiment at CERN employed a 40 ton Al spark chamber in the PS-WBB.

#### 35.11.3.2 Bubble Chambers

Several large bubble chamber detectors were employed as accelerator neutrino detectors in the 1970s and 80s, performing many of the first studies of the properties of the weak interaction. Bubble chambers provide exquisite granularity in the reconstruction of secondary particles, allowing very accurate separation of interaction processes. However, the extremely slow and labor-intensive acquisition and analysis of the data from photographic film led to them being phased out in favor of electronically read out detectors.
<table><caption>**Table 35.14:** Properties of detectors for accelerator-based neutrino beams. Revised in August 2023 by A. Fava (FNAL) and F. Pietropaolo (CERN; INFN, Padova).</caption><thead><tr><th>Name</th><th>Type</th><th>Target</th><th>Total Mass (fiducial) [t]</th><th>⟨E<sub>ν</sub>⟩ [GeV]</th><th>Location</th><th>Dates</th></tr></thead><tbody><tr><td>Lederman et al.</td><td>Spark</td><td>Al</td><td>10</td><td>[0.2-3]</td><td>BNL</td><td>1962</td></tr><tr><td>CERN-spark.</td><td>Spark</td><td>Al</td><td>20</td><td>1.5</td><td>CERN</td><td>1963</td></tr><tr><td>Serpukhov</td><td>Spark</td><td>Al</td><td>20</td><td>[3-30]</td><td>IHEP</td><td>1974-82</td></tr><tr><td>Aachen-Padova</td><td>Spark</td><td>Al</td><td>40(20)</td><td>1.4</td><td>CERN</td><td>1976-77</td></tr><tr><td>Gargamelle</td><td>Bubble</td><td>Freon</td><td>12</td><td>[1-10]</td><td>CERN</td><td>1970-79</td></tr><tr><td>BEBC</td><td>Bubble</td><td>H,D,Ne-H</td><td>2-42</td><td>[50-150],20</td><td>CERN</td><td>1977-84</td></tr><tr><td>SKAT</td><td>Bubble</td><td>Freon</td><td>8</td><td>7</td><td>IHEP</td><td>1976-1987</td></tr><tr><td>ANL-12ft</td><td>Bubble</td><td>H,D</td><td>1-2</td><td>0.5</td><td>ANL</td><td>1970</td></tr><tr><td>BNL-7ft</td><td>Bubble</td><td>HD</td><td>0.4-0.9</td><td>1.3,3</td><td>BNL</td><td>1976-82</td></tr><tr><td>Fermilab-15ft</td><td>Bubble</td><td>D, Ne</td><td>1,20</td><td>[50-180],[25-100]</td><td>FNAL</td><td>1973-92</td></tr><tr><td>CITF</td><td>Iron</td><td>Fe</td><td>92</td><td>[50-180]</td><td>FNAL</td><td>1974-83</td></tr><tr><td>CDHS</td><td>Iron</td><td>Fe</td><td>1250(520)</td><td>10-200</td><td>CERN</td><td>1976-84</td></tr><tr><td>MINOS</td><td>Iron</td><td>Fe</td><td>980(23.7), 5.5k (4.2k)</td><td>3</td><td>FNAL, SUL</td><td>2005-2012</td></tr><tr><td>MINOS+</td><td>Iron</td><td>Fe</td><td>980, 5.5k</td><td>4-10</td><td>FNAL, SUL</td><td>2013-2016</td></tr><tr><td>INGRID</td><td>Iron</td><td>Fe</td><td>160</td><td>0.6-3</td><td>J-PARC</td><td>2010-</td></tr><tr><td>SuperKamiokande</td><td>Cherenkov</td><td>H<sub>2</sub>O</td><td>50k (22.5k)</td><td>0.6</td><td>Kamioka</td><td>1996-2018</td></tr><tr><td>SuperK-Gd</td><td>Cherenkov</td><td>H<sub>2</sub>O doped with Gd</td><td>50k (22.5k)</td><td>0.6</td><td>Kamioka</td><td>2020-</td></tr><tr><td>HyperK</td><td>Cherenkov</td><td>H<sub>2</sub>O</td><td>260k(190k)</td><td>0.6</td><td>Kamioka</td><td>2027-</td></tr><tr><td>K2K-1kt</td><td>Cherenkov</td><td>H<sub>2</sub>O</td><td>25</td><td>0.8</td><td>KEK</td><td>1998-2004</td></tr><tr><td>MiniBooNE</td><td>Cherenkov</td><td>CH<sub>2</sub></td><td>818(440)</td><td>0.5</td><td>FNAL</td><td>2002-19</td></tr><tr><td>ANNIE</td><td>Cherenkov</td><td>H<sub>2</sub>O with GD sulfate</td><td>26</td><td>0.8</td><td>FNAL</td><td>2021-</td></tr><tr><td>HWPF</td><td>Scintillation</td><td>CH<sub>2</sub>, Fe</td><td>160</td><td>[50-180]</td><td>FNAL</td><td>1974-78</td></tr><tr><td>LSND</td><td>Scintillation</td><td>CH<sub>2</sub></td><td>167</td><td>0.003-0.06</td><td>LANL</td><td>1993-1998</td></tr><tr><td>NOvA</td><td>Scintillation</td><td>CH<sub>2</sub></td><td>300, 14k</td><td>2</td><td>FNAL,Ash River</td><td>2014-</td></tr><tr><td>SciBar</td><td>Scintillation</td><td>CH</td><td>15(9.5)</td><td>0.6</td><td>KEK</td><td>2003-2004</td></tr><tr><td>SciBooNE, Captain Mills</td><td>Scintillation</td><td>CH</td><td>15(9.5)</td><td>0.8</td><td>FNAL</td><td>2007-08</td></tr><tr><td>ICARUS</td><td>LArTPC</td><td>Ar</td><td>10</td><td>0.003-0.06</td><td>LANL</td><td>2022-2024</td></tr><tr><td>ICARUS</td><td>LArTPC</td><td>Ar</td><td>760(476)</td><td>17</td><td>LNGS</td><td>2006-12</td></tr><tr><td>Argoneut</td><td>LArTPC</td><td>Ar</td><td>760(476)</td><td>0.8</td><td>FNAL</td><td>2020-</td></tr><tr><td>MicroBooNE</td><td>LArTPC</td><td>Ar</td><td>0.025</td><td>3</td><td>FNAL</td><td>2009-10</td></tr><tr><td>SBND</td><td>LArTPC</td><td>Ar</td><td>170(85)</td><td>0.8</td><td>FNAL</td><td>2014-21</td></tr><tr><td>DUNE</td><td>LArTPC</td><td>Ar</td><td>220(112)</td><td>0.8</td><td>FNAL</td><td>2024-</td></tr><tr><td>FNAL-E-531</td><td>Emulsion</td><td>Ag, Br</td><td>0.009</td><td>3</td><td>FNAL,SURF</td><td>2027-</td></tr><tr><td>CHORUS</td><td>Emulsion</td><td>Ag, Br</td><td>0.8</td><td>22</td><td>FNAL</td><td>1984</td></tr><tr><td>DONuT</td><td>Emulsion</td><td>Fe</td><td>0.26</td><td>27</td><td>CERN</td><td>1994-97</td></tr><tr><td>OPERA</td><td>Emulsion</td><td>Pb</td><td>1.25k</td><td>53</td><td>FNAL</td><td>1997</td></tr><tr><td>NINJA</td><td>Emulsion</td><td>Fe</td><td>0.002(0.001)</td><td>17</td><td>LNGS</td><td>2008-12</td></tr><tr><td>FASERν</td><td>Emulsion</td><td>W</td><td>1.2</td><td>0.6</td><td>J-PARC</td><td>2015-</td></tr><tr><td>CHARM</td><td>Hybrid</td><td>CaCO<sub>3</sub></td><td>(27),156(122)</td><td>[360-6300]</td><td>CERN (LHC)</td><td>2022-</td></tr><tr><td>CHARM-II</td><td>Hybrid</td><td>glass</td><td>692</td><td>20</td><td>CERN</td><td>1978-84</td></tr><tr><td>BNL-E-734</td><td>Hybrid</td><td>CH<sub>2</sub></td><td>172</td><td>20</td><td>CERN</td><td>1984-91</td></tr><tr><td>BNL-E-776</td><td>Hybrid</td><td>concrete</td><td>240</td><td>1.3</td><td>BNL</td><td>1981-86</td></tr><tr><td>NOMAD</td><td>Hybrid</td><td>CH</td><td>2.9(2.7)</td><td>1.4</td><td>BNL</td><td>1986</td></tr><tr><td>CCFR</td><td>Hybrid</td><td>Fe</td><td>690</td><td>27</td><td>CERN</td><td>1995-98</td></tr><tr><td>NuTeV</td><td>Hybrid</td><td>Fe</td><td>690</td><td>[30-260]</td><td>FNAL</td><td>1985-88</td></tr><tr><td>MINERvA</td><td>Hybrid</td><td>CH,H<sub>2</sub>O,Fe,Pb,C,He</td><td>8</td><td>[70-180]</td><td>FNAL</td><td>1996-97</td></tr><tr><td>T2K-ND280</td><td>Hybrid</td><td>CH,H<sub>2</sub>O</td><td>2</td><td>3.8</td><td>FNAL</td><td>2010-19</td></tr><tr><td>COHERENT</td><td>Hybrid</td><td>several</td><td>between 0.01 and 1.3</td><td>0.6</td><td>J-PARC</td><td>2010-</td></tr><tr><td>SND@LHC</td><td>Hybrid</td><td>Emulsions & SciFi</td><td>0.81</td><td>[100-1000]</td><td>CERN (LHC)</td><td>2022-</td></tr></tbody></table>

The Gargamelle [465] detector at CERN used Freon and propane gas targets to make the first observation of neutrino-induced NC interactions and more. The BEBC [466] detector at CERN was a bubble chamber that was alternately filled with liquid hydrogen, deuterium, and a neon-hydrogen mixture; BEBC was also outfitted with a track-sensitive detector to improve event tagging, and sometimes used with a small emulsion chamber. The SKAT [467] Freon bubble chamber was exposed to wideband neutrino and antineutrino beams at the Serpukhov laboratory in the former Soviet Union. A series of American bubble chambers in the 1970's and 1980's made measurements on free nucleons that are still crucial inputs for neutrino-nucleus scattering predictions. The 12-foot bubble chamber at ANL [468] in the USA used both deuterium and hydrogen targets, as did the 7-foot bubble cham-

ber at BNL [469]. Fermilab's 15 foot bubble chamber [470] used deuterium and neon targets.

### 35.11.3.3 Iron Tracking Calorimeters

Because of the forward boost of high energy interactions, long detectors made of magnetized iron interspersed with active detector layers have been very successfully employed. The long magnetized detectors allow measurements of the momentum of penetrating muons. The iron planes also act as shower-inducing layers, allowing separation of EM and hadronic showers; the large number of iron planes provide enough mass for high statistics and/or shower containment. Magnetized iron spectrometers have been used for studies of the weak interaction, measurements of structure functions, and searches for neutrino oscillation. Non-
magnetized iron detectors have also been successfully employed as neutrino monitors for oscillation experiments and also for neutrino-nucleus interaction studies.

The Caltech-Fermilab counter (CITF) [468] combined a 92 ton iron-scintillator target-calorimeter detector with a downstream toroidal magnet to perform early studies of weak interactions—including observations of neutral currents. The CDHS [471] detector used layers of magnetized iron modules interspersed with wire drift chambers, with a fiducial mass of 1250 t, to detector neutrinos in the range 10–200 GeV. Within each iron module, 5 cm (or 15 cm) iron plates were interspersed with scintillation counters. The MINOS [472] detectors, a near detector of 980 t at FNAL and a far detector of 5500 t in the Soudan Underground Laboratory (SUL), were functionally identical magnetized iron calorimeters, comprised of iron plates interleaved with layers of 4 cm wide plastic scintillator strips in alternating orientations. The T2K [473] on-axis detector, INGRID, consists of 16 non-magnetized iron scintillator sandwich detectors, each with nine 6.5 cm iron plane (7.1 t total) interspersed between layers of 5 cm wide plastic scintillator strips readout out by multi-pixel photon counters (MPPCs) coupled to WLS fibers. Fourteen of the IN-GRID modules are arranged in a cross-hair configuration centered on the neutrino beam axis.

### 35.11.3.4 Cherenkov Detectors

Open volume water Cherenkov detectors were originally built to search for proton decay. Large volumes of ultra-pure water were lined with photomultipliers to collect Cherenkov light emitted by the passage of relativistic charged particles. See Sec. 36.3.1 for a detailed discussion of deep liquid detectors for rare processes. The Cherenkov light, which has significant production in the visible range, appears on the walls of the detectors in distinctive ring patterns, and topological characteristics of the rings are employed to separate muon-induced rings from electron-induced with very high accuracy. As neutrino detectors, Cherenkov detectors optimize the design balance since the entire neutrino target is also active detector medium.

When used to detect ~ GeV neutrinos, the detector medium acts as a natural filter for final state particles below the Cherenkov threshold; this feature has been exploited successfully by the K2K, MiniBooNE (using mineral oil instead of water), and T2K neutrino oscillation experiments. This makes event reconstruction simple and robust since electrons and muons have very different signatures, but does require making assumptions when inferring neutrino energy since not all final state particles are observed. At higher energies Cherenkov detectors become less accurate because the overlapping rings from many final state particles become increasingly difficult to resolve.

The second-generation Cherenkov detector in Japan, Super-Kamiokande [86] (Super-K), comprises 22.5 kt of water viewed by 50 cm photomultiplier tubes with 40% photocathode coverage; it is surrounded by an outer detector region viewed by 20 cm photomultipliers. Super-K is the far detector for K2K and T2K, and is described in greater detail elsewhere in this review. The K2K experiment also employed a 1 kt water Cherenkov detector in the suite of near detectors [474], with 40% photocathode coverage. The MiniBooNE detector at FNAL was a 0.8 kt [475] mineral oil Cherenkov detector, with 20 cm photomultipliers giving 10% photocathode coverage, surrounded by a veto detector also with 20 cm photomultipliers. The ANNIE detector also at FNAL consists of a 26-ton water Cherenkov detector loaded with gadolinium sulfate (Gd2O12S3), complemented by a muon range detector and a veto wall. In addition to traditional PMTs, it exploits novel Large Area Picosecond Photodetectors (LAPPDs) for the detection of the Cherenkov light.

### 35.11.3.5 Scintillation Detectors

Liquid and solid scintillator detectors also employ fully (or nearly fully) active detector media. Typically organic scintillators, which emit into the ultraviolet range, are dissolved in mineral oil or plastic and read out by photomultipliers coupled to wavelength shifters (WLS). Open volume scintillation detectors lined with photomultipliers are conceptually similar to Cherenkov detectors, although energy reconstruction is calorimetric in nature

as opposed to kinematic (see also Sec. 36.3.1). For higher energies and higher particle multiplicities, it becomes beneficial to use segmented detectors to help distinguish particle tracks and showers from each other.

The HWPF collaboration [476] employed a 2 t liquid scintillator total-absorption hadron calorimeter followed by a magnetic spectrometer to observe neutral current events in the early days of Fermilab. The LSND [477] detector at LANL was a 130 t open volume liquid scintillator detector employed to detect relatively low energy (<300 MeV) neutrinos. The NOvA [478] detectors use segmented volumes of liquid scintillator in which the scintillation light is collected by WLS fibers in the segments that are coupled to avalanche photodiodes (APDs) at the ends of the volumes. The NOvA far detector, located in Ash River, MN, is comprised of 896 layers of 15.6 m long extruded PVC scintillator cells for a total mass of 14 kt; the NOvA near detector is comprised of 214 layers of 4.1 m scintillator volumes for a total mass of 300 t. Both are placed in the NuMI beamline at 0.8° off-axis. The SciBar (Scintillation Bar) detector was originally built for K2K at KEK in Japan and then re-used for SciBooNE [479] at FNAL. SciBar used plastic scintillator strips with 1.5 cm×2.5 cm rectangular cross section, read out by multianode photomultipliers (MAPMTs) coupled to WLS fibers, arranged in alternating horizontal and vertical layers. Both SciBooNE and K2K employed an EM calorimeter downstream of SciBar and a muon range detector (MRD) downstream of that.

The Coherent Captain Mills (CCM) detector is an upright cylindrical cryostat 2.58 m in diameter and 2.25 m high that holds 10 tons of LAr and is designed to be movable for distances between 20 m to 40 m from the neutrino source. It is instrumented with 120 8-inch PMTs, out of which 96 are coated with Tetraphenyl Butadiene (TPB) wavelength shifter.

### 35.11.3.6 Liquid Argon Time Projection Chambers

Liquid argon time projection chambers (LAr-TPCs) were conceived in the 1970s as a way to achieve a fully active detector with sub-centimeter track reconstruction [480]. A massive volume of purified liquid argon is put under a strong electric field (hundreds of V/cm), so that the liberated electrons from the paths of ionizing particles can be drifted to the edge of the volume and read out, directly by collecting charge from wire planes or non-destructively through charge induction in the wire planes. Dual-phase readout methods have also been developed, in which the charge is drifted vertically and then passed through an amplification region inside a gas volume above the liquid volume; the bottom of the liquid volume is equipped with a PMT array for detecting scintillation photons form the liquid argon. The first large scale LAr-TPC was the ICARUS T-600 module [481], comprising 760 t of liquid argon with a charge drift length of 1.5 m read out by wires with 3 mm pitch, which operated in LNGS, both standalone and also exposed to the CNGS high energy neutrino beam. The ICARUS detector has been transported to Fermilab and installed and presently operated in an on-axis position in the Booster Neutrino Beamline at 600 m from target, where it is also exposed to off-axis neutrinos from the NuMI beamline. It will be complemented by the SBND detector (100 m from BNB target, 112 t fiducial mass), currently in the final phases of installation to search for possible anomalies in short baseline neutrino oscillation in the context of the SBN Program.

The ArgoNeuT [482] detector at FNAL, with fiducial mass 25 kg of argon read out with 4 mm pitch wires, was exposed to the NuMI neutrino and antineutrino beams. The MicroBooN [483] detector at FNAL comprises 170 t (85t active) of liquid Ar, read out with 3 mm wire pitch, which began collecting data in the Booster Neutrino Beam Oct 2015. A LAr-TPC has also been chosen as the multi-kton detector design for the future DUNE [484] neutrino oscillation experiment, from FNAL to Sanford Underground Research Facility. The design choices for the first two DUNE far detector modules (each with a total LAr mass of 17 kt) are being validated in the NP04 (horizontal drift) and the NP02 (vertical drift) experiments at the CERN Neutrino Platform facility [485,486].
35.11.3.7 Emulsion Detectors

Photographic film emulsions have been employed in particle
physics experiments since the 1940s [487]. Thanks to advances in
scanning technology and automation [488], they have been suc-
cessfully employed as neutrino detectors. Emulsions are used for
experiments observing CC tau neutrino interactions, where the
short lifetime of the tau, $\tau_{\tau} = 2.90 \times 10^{-13}$s, leading to the short
mean path length, $c \times \tau = 87\mu$m, requires extremely precise track
resolution. They are employed in hybrid detectors in which the
emulsion bricks are embedded inside fine-grained tracker detec-
tors. In the data analysis, the tracker data are used to select
events with characteristics typical of a tau decay in the final state,
such as missing energy and unbalanced transverse momentum.
The reconstructed tracks are projected back into an emulsion brick
and used as the search seed for a neutrino interaction vertex.

E531 [489] at Fermilab tested many of the emulsion-tracker hy-
brid techniques employed by later neutrino experiments, in a de-
tector with approximately 9 kg of emulsion target. The CHO-
RUS [488] experiment at CERN used 1,600 kg of emulsion, in
a hybrid detector with a fiber tracker, high resolution calorime-
ter, and muon spectrometer, to search for $\nu_{\mu} \rightarrow \nu_{\tau}$ oscillation.
The DONuT [490] experiment at FNAL used a hybrid detector,
with 260 kg of emulsion bricks interspersed with fiber trackers,
followed by a magnetic spectrometer, and calorimeter, to make
the first direct observation of tau neutrino CC interactions. The
OPERA [491] [492] [493] experiment used an automated hybrid
emulsion detector, with 1,25x0 t of emulsion, to make the first
direct observation of the appearance of $\nu_{\tau}$ in a $\nu_{\mu}$ beam. Re-
cently, the NINJA collaboration has developed an emulsion cloud
chamber detector to observe neutrinos in the J-PARC neutrino
beam [494].

FASER$\nu$ [495] (a sub-detector of the FASER [496] experiment) is designed to detect and study all neutrino flavors produced at the LHC in the far-forward region of the ATLAS interaction point at the TeV energies scale. It consists of emulsion films interleaved with tungsten plates with a total target mass of 1.2 tons, enabling the identification of the leptons in charged current $\nu$ interactions. Placed 480m downstream of ATLAS, FASER$\nu$ will measure neutrino cross sections at energies where they are currently unconstrained and will bound models of forward particle production.

35.11.3.8 Hybrid Detectors

In the previous neutrino detector examples, one can point to a
specific detection technology or configuration that defines a cate-
gory of detectors. In this section we look at detectors that combine
multiple elements or techniques, without one facet being specifi-
cally dominant or crucial; we call these detectors hybrids.

The CHARM detector [497] at CERN was built to study neutral-current interactions and search for muon neutrino oscillation. It was a fine-grained ionization calorimeter tracker with approximately 150 t of marble as neutrino target, surrounded by a magnetized iron muon system for tagging high angle muons, and followed downstream by a muon spectrometer. The CHARM II detector [498] at CERN comprised a target calorimeter followed by a downstream muon spectrometer. Each target calorimeter module consists of a 4.8 cm thick glass plate followed by a layer of plastic streamer tubes, with spacing 1 cm, instrumented with 2 cm wide pickup strips. Every fifth module is followed by a 3 cm thick scintillator layer. The total mass of the target calorimeter was 692 t.

The Brookhaven E-734 [499] detector was a tracking calorime-
ter made up of 172 t liquid scintillator modules interspersed with
proportional drift tubes, followed by a dense EM calorimeter and
a muon spectrometer downstream of that. The detector was ex-
posed to a wideband horn-focused beam with peak neutrino en-
ergy near 1 GeV. The Brookhaven E-776 [500] experiment com-
prised a finely segmented EM calorimeter, with 2.54 cm concrete
absorbers interspersed with planes of drift tubes and acrylic scin-
tillation counters, with total mass 240 t, followed by a muon spec-
trometer.

The FNAL Lab-E neutrino detector was used by the CCFR
[501] and NuTeV [502] collaborations to perform a series of ex-
periments in the Fermilab high energy neutrino beam (50 GeV<
$E_\nu$ < 300 GeV). The detector was comprised of six iron target

calorimeter modules, with 690 t total target mass, followed by
three muon spectrometer modules, followed by two drift cham-
bers. Each iron target calorimeter module comprised 5.2 cm thick
steel plates interspersed with liquid scintillation counters and drift
chambers.

The NOMAD [503] detector at CERN consisted of central tracker detector inside a 0.4 T dipole magnet (the magnet was originally used by the UA1 experiment at CERN) followed by a hadronic calorimeter and muon detectors downstream of the magnet. The main neutrino target is 3 t of drift chambers followed downstream by transition radiation detectors which are followed by an EM calorimeter. NOMAD was exposed to the same wide-band neutrino beam as was CHORUS.

MINERvA [504] is a hybrid detector based around a central plastic scintillator tracker: 8.3 t of plastic scintillator strips with triangular cross section read out by MAPMTs coupled to WLS fibers. The scintillator tracker is surrounded by electromagnetic and hadronic calorimetry, which is achieved by interleaving thin lead (steel) layers between the scintillator layers for the ECAL (HCAL). MINERvA is situated upstream of the MINOS near detector which acts as a muon spectrometer. Upstream of the scintillator tracker is a nuclear target region containing inactive layers of C (graphite), Pb, Fe (steel), and O (water). MINERvA's physics goals span a wide range of neutrino-nucleus interaction studies, from form factors to nuclear effects.

T2K [473] in Japan employs two near detectors at 280 m from the neutrino beam target, one centered on the axis of the horn-focused J-PARC neutrino beam and one placed 2.5° off-axis. The on-axis detector, INGRID, is described above. The 2.5° off-axis detector, ND280, employs the UA1 magnet (at 0.2 T) previously used by NOMAD. Inside the magnet volume are three separate detector systems: the trackers, the Pi0 Detector (P0D), and several ECAL modules. The tracker detectors comprise two fine-grained scintillator detectors (FGDs), read out by MPPCs coupled to WLS fibers, interleaved between three gas TPCs read out by micromegas planes. The downstream FGD contains inactive water layers in addition to the scintillators. Upstream of the tracker is the P0D, a sampling tracker calorimeter with active detector materials comprising plastic scintillator read out by MPPCs and WLS fibers, and inactive sheets of brass radiators and refillable water modules. Surrounding the tracker and P0D, but still inside the magnet, are lead-scintillator EM sampling calorimeters.

The COHERENT collaboration aims to measure CEvNS (Co-
herent Elastic Neutrino-Nucleus Scattering) using the high-
quality pion-decay-at-rest neutrino source at the Spallation Neu-
tron Source in Oak Ridge, Tennessee. For doing so, it is deploying
a suite of detector systems in a hallway in the basement of the SNS
target hall with a phased approach. This includes CsI[Na] crystal,
LAr scintillation detector, NaI[Tl], interwoven EJ200 scintillator
and gadolinium-doped Mylar, thorium-232 metal plates and Ger-
manium detectors.

The SND@LHC experiment [505] will perform measurements with high-energy neutrinos (100 GeV–1 TeV) produced at the LHC in the pseudo-rapidity region $7.2 < \eta < 8.6$. It allows the identification of all neutrino flavours with high efficiency. It is located 480 m downstream and off-axis with respect to the ATLAS interaction point and consists of a target region followed by a muon identification system. The target (810 kg) is instrumented with nuclear emulsions and Scintillating Fibre (SciFi) planes. The muon identification system is made of iron slabs interleaved with scintillating bars planes. SND@LHC is complementary to FASER$\nu$ as it intercepts different components of the neutrino flux from LHC collisions, mainly those produced in heavy flavour decays, mostly charm.

35.11.4 Outlook

Detectors for accelerator-based neutrino beams have been in
use, and constantly evolving, for six decades now. The rich pro-
gram of neutrino oscillation physics and attendant need for newer
and better neutrino-nucleus scattering measurements means that
more neutrino detectors with broader capabilities will be needed
in the coming decades.

One of the most intriguing prospects is a large volume,
high pressure gas time projection chamber (HPTPC). With the
prospect of megawatt power accelerator-based neutrino beams, it is entirely feasible to collect high statistics data sets with a gas target. The low momentum thresholds for particle detection, and excellent momentum resolution and particle identification capabilities, of an HPTPC would open a new window into the physics of neutrino-nucleus scattering. Moreover, the ability to change the gas mixtures in the HPTPC would allow measurements in the same detector on multiple nuclear targets, which would, in turn, allow unprecedentedly accurate constraints and tuning of neutrino-nucleus interaction models.

## 35.12 Superconducting magnets for collider detectors

Revised March 2024 by Y. Makida (KEK).

### 35.12.1 Solenoid Magnets

In all cases SI unit are assumed, so that the magnetic field, $B$, is in Tesla, the stored energy, $E$, is in joules, the dimensions are in meters, and vacuum permeability of $\mu_0 = 4\pi \times 10^{-7}$.

The magnetic field ($B$) in an simple solenoid with a flux return iron yoke, in which the magnetic field is lower than magnetic saturation of < 2 T, is given by

$$B = \frac{\mu_0 n I}{L} \qquad (35.48)$$

where $n$ is the number of turns, $I$ is the current and $L$ is the coil length.

In an air-core solenoid case, the central field is given by

$$B(0,0) = \mu_0 n I \frac{1}{\sqrt{L^2 + 4R^2}}, \qquad (35.49)$$

where $R$ is the coil radius.

In most cases, momentum analysis is made by measuring the circular trajectory of the passing particles according to $p = mv = qrB$, where $p$ is the momentum, $m$ the mass, $q$ the charge, $r$ the bending radius. The sagitta, $s$, of the trajectory is given by

$$s = qB\ell^2/8p, \qquad (35.50)$$

where $\ell$ is the path length in the magnetic field as shown in Fig. 35.39.

<figure><img src="image_238.png" /><figcaption>Figure 35.39: Geometric relation of sagitta between path length.</figcaption></figure>

In a practical momentum measurement in colliding beam detectors, it is more effective to increase the magnetic volume than the field strength, since

$$dp/p \propto p/B \ell^2, \qquad (35.51)$$

where $\ell$ corresponds to the solenoid coil radius $R$. The energy stored in the magnetic field of any magnet is calculated by integrating $B^2$ over all space:

$$E = \frac{1}{2\mu_0} \int B^2 dV \qquad (35.52)$$

If the coil thin and inside an iron return yoke, (which is the case if it is to superconducting coil), then

$$E \approx (B^2/2\mu_0)\pi R^2 L. \qquad (35.53)$$

For a detector in which the calorimetry is outside the aperture of the solenoid, the coil must be transparent in terms of radiation and absorption lengths. This usually means that the superconducting solenoid and its cryostat is of minimum real thickness and is made of a material with long radiation length. There are two major contributors to the thickness of a thin solenoid:

1. The conductor consisting of the current-carrying superconducting material (usually Nb-Ti/Cu) and the quench protecting stabilizer (usually aluminum) are wound on the inside of a structural support cylinder (usually aluminum alloy). The coil thickness scales as $B^2R$, so the thickness in radiation lengths ($X_0$) is

$$t_{\text{coil}}/X_0 = (R/\sigma_h X_0)(B^2/2\mu_0), \qquad (35.54)$$

where $t_{\text{coil}}$ is the physical thickness of the coil, $X_0$ the average radiation length of the coil/stabilizer material, and $\sigma_h$ is the hoop stress in the coil [506]. $B^2/2\mu_0$ is the magnetic pressure. In large detector solenoids, the aluminum stabilizer and support cylinders dominate the thickness; the superconductor (Nb-Ti/Cu) contributes a smaller fraction. The main coil and support cylinder components typically contribute about 2/3 of the total thickness in radiation lengths.

2. Another contribution to the material comes from the outer cylindrical shell of the vacuum vessel. Since this shell is susceptible to buckling collapse, its thickness is determined by the diameter, length and the modulus of the material of which it is fabricated. The outer vacuum shell represents about 1/3 of the total thickness in radiation length.

### 35.12.2 Properties of collider detector magnets

The physical dimensions, central field stored energy and thickness in radiation lengths normal to the beam line of the superconducting solenoids associated with the major collider are given in Table 35.15 [507]. Fig. 35.40 shows thickness in radiation lengths as a function of $B^2R$ in various collider detector solenoids.

<figure><img src="image_239.png" /><figcaption>Figure 35.40: Magnet wall thickness in radiation length as a function of $B^2R$ for various detector solenoids. Gray entries are for magnets no longer in use.</figcaption></figure>

The ratio of stored energy to cold mass ($E/M$) is a useful performance measure. It can also be expressed as the ratio of the stress, $\sigma_h$, to twice the equivalent density, $\rho$, in the coil [506]:

$$\frac{E}{M} = \frac{E}{\rho 2\pi t_{\text{coil}} RL} \approx \frac{\sigma_h}{2\rho} \qquad (35.55)$$

The $E/M$ ratio in the coil is approximately equivalent to $H, ¶$

<footnote>¶¶ The enthalpy, or heat content, is called $H$ in the thermodynamics literature. It is not to be confused with the magnetic field intensity $B/\mu$.</footnote>
<table><caption>Table 35.15: Progress of superconducting magnets for particle physics detectors.</caption><thead><tr><th rowspan="2">Experiment</th><th rowspan="2">Laboratory</th><th><i>B</i></th><th rowspan="2">Radius<br>[m]</th><th colspan="3">Length Energy <i>X</i>/<i>X</i><sub>0</sub></th><th rowspan="2"><i>E</i>/<i>M</i><br>[kJ/kg]</th></tr><tr><th>[T]</th><th>[m]</th><th>[MJ]</th></tr></thead><tbody><tr><td>TOPAZ*</td><td>KEK</td><td>1.2</td><td>1.45</td><td>5.4</td><td>20</td><td>0.70</td><td>4.3</td></tr><tr><td>CDF*</td><td>Tsukuba/Fermi</td><td>1.5</td><td>1.5</td><td>5.07</td><td>30</td><td>0.84</td><td>5.4</td></tr><tr><td>VENUS*</td><td>KEK</td><td>0.75</td><td>1.75</td><td>5.64</td><td>12</td><td>0.52</td><td>2.8</td></tr><tr><td>AMY*</td><td>KEK</td><td>3</td><td>1.29</td><td>3</td><td>40</td><td>†</td><td></td></tr><tr><td>CLEO-II*</td><td>Cornell</td><td>1.5</td><td>1.55</td><td>3.8</td><td>25</td><td>2.5</td><td>3.7</td></tr><tr><td>ALEPH*</td><td>Saclay/CERN</td><td>1.5</td><td>2.75</td><td>7.0</td><td>130</td><td>2.0</td><td>5.5</td></tr><tr><td>DELPHI*</td><td>RAL/CERN</td><td>1.2</td><td>2.8</td><td>7.4</td><td>109</td><td>1.7</td><td>4.2</td></tr><tr><td>ZEUS*</td><td>INFN/DESY</td><td>1.8</td><td>1.5</td><td>2.85</td><td>11</td><td>0.9</td><td>5.5</td></tr><tr><td>H1*</td><td>RAL/DESY</td><td>1.2</td><td>2.8</td><td>5.75</td><td>120</td><td>1.8</td><td>4.8</td></tr><tr><td>BaBar*</td><td>INFN/SLAC</td><td>1.5</td><td>1.5</td><td>3.46</td><td>27</td><td>†</td><td>3.6</td></tr><tr><td>D0*</td><td>Fermi</td><td>2.0</td><td>0.6</td><td>2.73</td><td>5.6</td><td>0.9</td><td>3.7</td></tr><tr><td>BELLE*</td><td>KEK</td><td>1.5</td><td>1.8</td><td>4</td><td>42</td><td>†</td><td>5.3</td></tr><tr><td>BES-III</td><td>IHEP</td><td>1.0</td><td>1.475</td><td>3.5</td><td>9.5</td><td>†</td><td>2.6</td></tr><tr><td>ATLAS-CS</td><td>ATLAS/CERN</td><td>2.0</td><td>1.25</td><td>5.3</td><td>38</td><td>0.66</td><td>7.0</td></tr><tr><td>ATLAS-BT</td><td>ATLAS/CERN</td><td>1</td><td>4.7–9.75</td><td>26</td><td>1080</td><td>(Toroid)<sup>†</sup></td><td></td></tr><tr><td>ATLAS-ET</td><td>ATLAS/CERN</td><td>1</td><td>0.825–5.35</td><td>5</td><td>2 × 250</td><td>(Toroid)<sup>†</sup></td><td></td></tr><tr><td>CMS</td><td>CMS/CERN</td><td>4</td><td>6</td><td>12.5</td><td>2600</td><td>†</td><td>12</td></tr><tr><td>SiD**</td><td>ILC</td><td>5</td><td>2.9</td><td>5.6</td><td>1560</td><td>†</td><td>12</td></tr><tr><td>ILD**</td><td>ILC</td><td>4</td><td>3.8</td><td>7.5</td><td>2300</td><td>†</td><td>13</td></tr><tr><td>SiD**</td><td>CLIC</td><td>5</td><td>2.8</td><td>6.2</td><td>2300</td><td>†</td><td>14</td></tr><tr><td>ILD**</td><td>CLIC</td><td>4</td><td>3.8</td><td>7.9</td><td>2300</td><td>†</td><td>13</td></tr><tr><td>FCC**</td><td></td><td>6</td><td>6</td><td>23</td><td>54000</td><td>†</td><td>12</td></tr></tbody></table>

* No longer in service

** Conceptual design in future

† EM calorimeter is inside solenoid, so small $X/X_0$ is not a goal

the enthalpy of the coil, and it determines the average coil temperature rise after energy absorption in a quench:

$$E/M = H(T_2) - H(T_1) \approx H(T_2) \quad (35.56)$$

where $T_2$ is the average coil temperature after the full energy absorption in a quench, and $T_1$ is the initial temperature. $E/M$ ratios of 5, 10, and 20 kJ/kg correspond to ~65, ~80, and ~100 K, respectively. The $E/M$ ratios of various detector magnets are shown in Fig. 35.41 as a function of total stored energy. One would like the cold mass to be as small as possible to minimize the thickness, but temperature rise during a quench must also be minimized. An $E/M$ ratio as large as 12 kJ/kg is designed into the CMS solenoid, with the possibility that about half of the stored energy can go to an external dump resistor. Thus the coil temperature can be kept below 80 K if the energy extraction system works well. The limit is set by the maximum temperature that the coil design can tolerate during a quench. This maximum local temperature should be <130 K (50 K + 80 K), so that thermal expansion effects, which are remarkable beyond 80 K, in the coil are manageable less than 50 K.

### 35.12.3 Toroidal magnets

Toroidal coils uniquely provide a closed magnetic field without the necessity of an iron flux-return yoke. Because no field exists at the collision point and along the beam line, there is, in principle, no effect on the beam. On the other hand, the field profile generally has $1/r$ dependence. The particle momentum may be determined by measurements of the deflection angle combined with the sagitta. The deflection (bending) power $BL$ is

$$BL \approx \int_{R_i}^{R_0} \frac{B_i R_i dR}{R \sin \theta} = \frac{B_i R_i}{\sin \theta} \ln(R_0/R_i), \quad (35.57)$$

where $R_i$ is the inner coil radius, $R_0$ is the outer coil radius, and $\theta$ is the angle between the particle trajectory and the beam line axis. The momentum resolution given by the deflection may be expressed as

$$\frac{\Delta p}{p} \propto \frac{p}{BL} \approx \frac{p \sin \theta}{B_i R_i \ln(R_0/R_i)} \quad (35.58)$$

The momentum resolution is better in the forward/backward (smaller $\theta$) direction. The geometry has been found to be optimal

<figure><img src="image_240.png" /><figcaption>Figure 35.41: Ratio of stored energy to cold mass for major detector solenoids. Gray indicates magnets no longer in operation.</figcaption></figure>

when $R_0/R_i \approx 3–4$. In practical designs, the coil is divided into 6–12 lumped coils in order to have reasonable acceptance and accessibility. This causes the coil design to be much more complex. The mechanical structure needs to sustain the decentering force between adjacent coils, and the peak field in the coil is 3–5 times higher than the useful magnetic field for the momentum analysis [508].

## 35.13 Measurement of particle momenta in a uniform magnetic field

The trajectory of a particle with momentum $p$ (in GeV/c) and charge $ze$ in a constant magnetic field $\vec{B}$ is a helix, with radius of curvature $R$ and pitch angle $\lambda$. The radius of curvature and momentum component perpendicular to $\vec{B}$ are related by

$$p \cos \lambda = 0.3 z B R, \quad (35.59)$$

where $B$ is in tesla and $R$ is in meters.

The distribution of measurements of the curvature $k \equiv 1/R$ is approximately Gaussian. The curvature error for a large number
of uniformly spaced measurements on the trajectory of a charged particle in a uniform magnetic field can be approximated by

$$ (\delta k)^2 = (\delta k_{\text{res}})^2 + (\delta k_{\text{ms}})^2, \qquad (35.60) $$

where $\delta k$ = curvature error

$\delta k_{\text{res}}$ = curvature error due to finite measurement resolution

$\delta k_{\text{ms}}$ = curvature error due to multiple scattering.

If many ($\ge$ 10) uniformly spaced position measurements are made along a trajectory in a uniform medium,

$$ \delta k_{\text{res}} = \frac{\epsilon}{L'^2} \sqrt{\frac{720}{N + 4}}, \qquad (35.61) $$

where $N$ = number of points measured along track

$L'$ = the projected length of the track onto the bending plane

$\epsilon$ = measurement error for each point, perpendicular to the trajectory.

If a vertex constraint is applied at the origin of the track, the coefficient under the radical becomes 320.

For arbitrary spacing of coordinates $s_i$ measured along the projected trajectory and with variable measurement errors $\epsilon_i$ the curvature error $\delta k_{\text{res}}$ is calculated from:

$$ (\delta k_{\text{res}})^2 = \frac{4}{w} \frac{V_{ss}}{V_{ss}V_{s^2s^2} - (V_{ss^2})^2}, \qquad (35.62) $$

where $V$ are covariances defined as $V_{s^ms^n} = \langle s^ms^n \rangle - s^ms^n$ with $\rangle s^m = w^{-1} \sum (s_i^m/\epsilon_i^2)$ and $w = \sum \epsilon_i^{-2}$.

The contribution due to multiple Coulomb scattering is approximately

$$ \delta k_{\text{ms}} \approx \frac{(0.016)(\text{GeV}/c)z}{Lp\beta \cos^2 \lambda} \sqrt{\frac{L}{X_0}}, \qquad (35.63) $$

where $p$ = momentum (GeV/c)

$z$ = charge of incident particle in units of $e$

$L$ = the total track length

$X_0$ = radiation length of the scattering medium (in units of length; the $X_0$ defined elsewhere must be multiplied by density)

$\beta$ = the kinematic variable $v/c$.

More accurate approximations for multiple scattering may be found in the section on Passage of Particles Through Matter (Sec. 34 of this Review). The contribution to the curvature error is given approximately by $\delta k_{\text{ms}} \approx 8s_{\text{plane}}^{\text{rms}}/L^2$, where $s_{\text{plane}}^{\text{rms}}$ is defined there.

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