dipole and quadrupole terms, which are treated as static in time, but s-dependent.

The solutions of the Hill's equations (31.4) for x and y with a restoring force periodic in s are those of quasi-harmonic oscillators:

$$x(s) = \sqrt{2J_x\beta_x} \cos \psi_x, \quad x'(s) = -\sqrt{\frac{2J_x}{\beta_x}} [\alpha_x \cos \psi_x + \sin \psi_x], \qquad (31.5)$$

where the action $J_x$ is a constant of integration, $\alpha_x = \alpha_x(s) \equiv -(1/2)d\beta_x(s)/ds$, and the envelope of oscillations is modulated by the beta-function $\beta_x(s)$. A solution of the same form describes the motion in y. The betatron oscillation phase advances according to $d\psi_x/ds = 1/\beta_x$; that is, $2\pi\beta_x$ also plays the role of a local wavelength of oscillations along the orbit. An extremely important parameter for circular machines is the tune, $Q_x$, which is the number of such oscillations per turn about the closed path:

$$Q_x = \frac{1}{2\pi} \oint d\psi_x = \frac{1}{2\pi} \oint \frac{ds}{\beta_x(s)}. \qquad (31.6)$$

While the integer part of the tune [$Q_{x,y}$] generally characterizes the extent of the focusing lattice, it is the fractional part of the tune {$Q_{x,y}$} that needs to be well defined and controlled by the machine operators in order to stay away from potentially detrimental resonances, which may occur under conditions of $kQ_x + lQ_y = m$, where k, l, and m are integers. For example, for the LHC a combination of horizontal and vertical tunes — also called the working point — equal to $(Q_x, Q_y)=(64.31, 59.32)$ has been selected, such that resonances up to the order of $|k|+|l| = 10$ or 12 are avoided. These resonances are driven by high order multipole components of the fields in the magnets, or by self-fields of the beam, or by the electromagnetic fields of the opposite bunch. Normally, the nonlinear components are very weak compared to linear ones, nevertheless, when the nonlinear resonance condition is encountered, the amplitudes of particle oscillations could grow over the beam lifetime, resulting in the escape of the particles to the machine aperture, in the increase of the average beam size, or in both; either of these is highly undesirable phenomena. Careful analysis of nonlinear beam dynamics is instrumental in determining and optimizing the dynamic aperture, which is defined as the maximum amplitude of a bounded particle motion.

Neglecting for now all nonlinear effects and usually small $x-y$ coupling, and considering only the linear dynamics, the beta-function is well defined and satisfies the following equation:

$$2\beta_x\beta_x'' - \beta_x'^2 + 4\beta_x^2 K_x = 4. \qquad (31.7)$$

In a region free of magnetic fields, such as in the neighborhood of a collider interaction point (IP), usually occupied by particle detectors, a symmetric solution of Eq. (31.7) is a parabola:

$$\beta_x(s) = \beta_x^* + \frac{s^2}{\beta_x^*}, \qquad (31.8)$$

where, in this case, s denotes the longitudinal distance from the IP. The location of the beam waist usually coincides with the IP and corresponds to the minimum value of the beta-function $\beta_x^*$; the asterisk is used to indicate IP parameters.

Note that individual quadrupole magnet focuses particles in one plane and defocuses in another, see Eq.(31.4), and a standard way to provide focusing in both planes is to employ an alternating gradient periodic focusing lattice, consisting of a sequence of equally-spaced quadrupoles with a magnetic field gradient equal in magnitude, but alternating in sign ("focusing quadrupole - drift space - defocusing quadrupole - drift space" - known as a FODO cell). Eq. (31.7) has stable periodic solutions $\beta_x(s), \beta_y(s)$ in both planes provided that the focal length of the quadrupoles is longer than half the focusing-lens spacing L, i.e., $f = p/(eB_2l) > L/2$ (where l is the length of a quadrupole magnet, here assumed to be short $l \ll L$, and $B_2 \equiv |\partial B_y/\partial x|$ the quadrupoles' field gradient). In that case, the beta-functions have maxima at the focusing quadrupoles and minima at the defocusing ones, equal to, for example, $\beta_{max,min} = (2 \pm \sqrt{2})L$ in the case of $f = L/\sqrt{2}$, which

corresponds to a betatron phase advance $\Delta\psi_{x,y} = 90^\circ$ per FODO cell.

Expressing the invariant $J_x$ in terms of $x, x'$ yields

$$J_x = \frac{1}{2} (\gamma_x x^2 + 2\alpha_x x x' + \beta_x x'^2) = \frac{x^2 + (\alpha_x x + \beta_x x')^2}{2\beta_x} \qquad (31.9)$$

with $\gamma_x = \gamma_x(s) \equiv (1 + \alpha_x^2(s))/\beta_x(s)$. In a periodic system, these Courant-Snyder parameters [6] (frequently referred to as Twiss parameters) $\alpha(s), \beta(s), \gamma(s)$ are usually defined by the focusing lattice; in a single pass system such as a linac, the parameters may be selected to match the $x-x'$ distribution of the input beam. For a given position $s$ in the ring, the transverse particle motion in {$x, x' \equiv dx/ds$} phase space describes an ellipse, the area of which is $2\pi J_x$, where the horizontal action $J_x$ is a constant of motion and independent of $s$. If the interior of that ellipse is populated by an ensemble of non-interacting and non-radiating particles, that area, given the name emittance, is constant over the trajectory as well and would only change with energy. In a typical case of the particle's energy change rate being much slower than betatron motion, and considering a Hamiltonian system (i.e., a hadron collider or a linear collider, either without significant synchrotron radiation), the adiabatic invariant $\int p_x dx$ is conserved, and given that for small angles $p_x = x' \cdot \beta\gamma mc^2$, it is common practice to consider an energy-independent normalized emittance that is equal to the product of the emittance and relativistic factor $\beta\gamma/\pi$ and denoted by $\epsilon_n$. For a beam with a Gaussian distribution in {$x, x'$}, average action value $\langle J_x \rangle$ and standard deviations $\sigma_x$, and $\sigma_{x'}$, the definition of the normalized rms emittance is

$$\epsilon_{nx} \equiv \beta\gamma\langle J_x \rangle = \beta\gamma \frac{\sigma_x^2(s)}{\beta_x(s)} = \beta\gamma \frac{\sigma_{x'}^2(s)}{\gamma_x(s)}, \qquad (31.10)$$

with a corresponding expression for the other transverse direction, $y$. The angular brackets denote an average over the beam distribution. For 1D Gaussian beam, 95% of the particles are contained within {$x, x'$} phase space area of $6\pi\epsilon_n/(\beta\gamma)$. Normalized beam emittances are conserved over the acceleration cycle in linear, static focusing lattices $K_{x,y}(s)$, and consequently, one would expect the same $\epsilon_n$ at the hadron (or linear) collider top energy as the one coming from the very initial low energy particle source. Unfortunately, that is rarely the case as many time-varying or nonlinear phenomena come into play. In an $e^+/e^+$ storage ring, the normalized emittance is not preserved during acceleration, but at each energy the beam's equilibrium emittance is determined by the effect of synchrotron radiation as a balance between radiation damping and quantum excitation [7]. In such a ring, for a given accelerator optics, the normalized equilibrium emittance increases with the third power of the beam energy [8].

As for the description of a particle's longitudinal motion, one takes the fractional momentum deviation $\Delta p/p$ from that of the reference particle as the variable conjugate to $z$. The factors $K_{x,y}$ and $\rho$ in Hill's equations (31.4) are dependent on momentum $p$, leading to a number of effects: first, the trajectory of off-momentum particles deviates by $\Delta x(s) = D_x(s)(\Delta p/p)$, where the dispersion function $D_x(s)$ is determined by the magnetic lattice and is usually positive, periodic, and of the order of $\sim \rho/Q_x^2$. Second, the radius of curvature and orbit path-length $C$ vary with the momentum and, to first order, are characterized by the momentum compaction factor $\alpha_c$,

$$\alpha_c \equiv \frac{\Delta C/C}{\Delta p/p} = \frac{1}{C} \oint \frac{D_x(s)}{\rho(s)} ds, \qquad (31.11)$$

which typically is of order $1/Q_x^2$. Energy deviations also result in changes of machine focusing lattice properties and variations of the particle tunes, characterized by the chromaticity $Q'_{x,y} \equiv \Delta Q_{x,y}/(\Delta p/p)$. The natural chromaticity due to momentum dependence of the quadrupole focusing is negative and large $\sim -Q_{x,y}$. Corresponding chromatic tune variations can, therefore, become unacceptably large even for relatively small energy deviations $(\Delta p/p) \sim (0.1 - 1) \cdot 10^{-3}$. To assure transverse particle stability, usually, the chromaticity is partially or fully com-
pensated by additional sextupole magnets placed at locations of non-zero dispersion.

Radiofrequency electric fields in s direction provide a longitudinal focusing effect, allowing a stable increase of particle energy. The frequency $f_s$ of such longitudinal synchrotron oscillations is (expressed in units of revolution frequency $f_0$, to become the synchrotron tune $Q_s$)

$$Q_s \equiv \frac{f_s}{f_0} = \sqrt{\frac{(\alpha_c - 1/\gamma^2)hZeV_{RF} \sin(\phi_s)}{2\pi\beta cp}}, \quad (31.12)$$

where $h = f_{RF}/f_0$ denotes the RF harmonic number, $V_{RF}$ the RF voltage, and $\phi_s = \cos^{-1}(\Delta E/ZeV_{RF})$ the synchronous phase, with $\Delta E$ the average energy loss per turn (e.g. due to synchrotron radiation and impedance). The synchrotron tune $Q_s$ determines the amplitude of longitudinal oscillations for a particle with an initial momentum offset, e.g., the rms bunch length $\sigma_z$ relates to the rms momentum spread $\delta p/p$ as:

$$\sigma_z = \frac{c(\alpha_c - 1/\gamma^2)}{2\pi Q_s f_0} \left(\frac{\delta p}{p}\right). \quad (31.13)$$

Similarly to the case of transverse oscillations, the area of the longitudinal phase space $\{\Delta E, \Delta t\}$, or $\{\gamma\beta\delta p/p = (1/\beta)\Delta\gamma, z = \beta c \Delta t\}$, encircled by a moving particle is an adiabatic invariant, and the corresponding normalized longitudinal emittance $\epsilon_{n,L} = \beta\gamma mc\sigma_z(\delta p/p)$ is a generally conserved quantity in hadron accelerators and also in linear accelerators. In the case of lepton storage rings, synchrotron radiation determines the equilibrium relative momentum spread, which grows linearly with beam energy [7, 8], and the corresponding bunch length follows from Eq. (31.13). In hadron synchrotrons, the longitudinal emittance sometimes is intentionally blown up during acceleration, so as to preserve longitudinal beam stability.

Longitudinal oscillations are the slowest of all the periodic processes which take place in the accelerators. For example, in the LHC, the frequency of synchrotron oscillations at the top energy of 7 TeV is about $f_s = 23$ Hz, the revolution frequency is $f_0 = 11.3$ kHz, the frequency of betatron oscillations is about $Q_{x,y}f_{rev} \approx 700$ kHz and the RF frequency is $f_{RF} = 400.8$ MHz ($h = 35640$). It should be noted that longitudinal motion is practically absent in linacs. In the absence of bending dipoles, dispersion $D_x(s)$ is zero and so are the momentum compaction factor $\alpha_c$ and the synchrotron tune $Q_s$. As a result, ultrarelativistic particles in a linac barely change their relative positions during acceleration despite significant energy spread.

Highest-energy circular colliders face a serious impediment in the form of synchrotron radiation (SR) that causes an energy loss per turn of

$$\Delta E_{SR} = \frac{1}{3\epsilon_0} \frac{Z^2 e^2 \beta^3 \gamma^4}{\rho}, \quad (31.14)$$

here, $\epsilon_0$ is the permittivity of vacuum. For electrons and positrons it is equal to $\Delta E_{SR} = 88.5$ [keV/turn] $E_b^4[GeV]/\rho[m]$ and requires correspondingly high total RF voltage per turn to replenish the loss. Above a few hundred GeV, the SR energy loss becomes comparable to beam energy $\Delta E_{SR} \sim E_b$, which makes circular $e^+e^-$ colliders impractical for c.m.e. above $\sim 500$ GeV.

Dynamics of the particle spin and sophisticated methods to maintain beam polarization along the acceleration chain, from the polarized sources to collisions, dedicated spin matching procedures to enable self polarization in $e^+/e^-$ storage rings and the resonant depolarization method of ultra-precise c.m.e. calibration are described in [9].

### 31.2.2 High Intensity Beams

Ultimate collider luminosity calls for high beam currents $I_b = Zef_0 n_b N$. Three related major difficulties include growing RF demands to compensate the synchrotron-radiation power loss $P = I_b \Delta E_{SR}$ in $e^+/e^-$ beams, the advent of so-called coherent (or collective) beam instabilities, and growing demands for mini- mization of radiation due to inevitable particle losses. Many types of single- and multi-bunch instabilities are caused by beam interactions with electromagnetic fields induced by the beam itself due

to the impedance of the vacuum chambers and RF cavities [4], or caused by unstable clouds of secondary particles, like electrons or ions, which are formed around the circulating beams [10]. These instabilities can develop as quickly as within tens to thousands of turns and need to be controlled. Mechanisms that are routinely employed to avoid coherent instabilities include the use of nonlinear magnets to generate sufficient spread of the tunes and therefore, provide Landau damping, fast beam-based transverse and longitudinal feedback systems, and electron/ion clearing (either by weak magnetic or electric fields or by modulation of the primary beam current profile rendering secondaries unstable, or by reducing the yield of secondary electrons via either a special coating or extensive beam scrubbing of the vacuum chamber walls).

High current beam operation is sensitive to even minuscule fractional intensity losses caused by particles' scattering at a large angle or with a large energy loss, sufficient for either the particle amplitudes $\sqrt{2J_x y \beta_{x,y}(s)}$, or their dispersive position deviations $\Delta x = D_x(s)(\delta p/p)$ to exceed the available transverse aperture, usually set by collimators (otherwise, by the vacuum chamber and magnet apertures). This can be due to residual vacuum molecules near the beam orbit or Compton scattering off thermal photons, due to Coulomb scattering off other particles within the same bunch (Touschek effect), or due to collisions with opposite beam particles and fields, such as inelastic interaction of protons, Bhabha scattering $e^+e^- \to e^+e^-$, or radiative Bhabha scattering $e^+e^- \to e^+e^-\gamma$ (see corresponding chapters in [11]).

Particles can also get lost on the aperture as a result of much slower mechanisms of diffusion caused either by the above processes with smaller scattering amplitudes, but stochastically repeated many times, such as intensity-dependent multiple Coulomb intrabeam scattering [12], by external noises such as ground motion or magnetic field fluctuations, or via chaotic mechanisms like Arnold diffusion, modulational diffusion, or resonance streaming in nonlinear fields, enhanced by minor tune modulations. Diffusion leads to a slow evolution of the beam distribution function and appearance of highly unwanted large-amplitude tails and beam emittance growth. The only way to counteract it is to arrange beam cooling (damping of particle oscillations). The cooling requires a reaction force opposite to particle momentum arranged such that, on average, the corresponding dissipative particle energy loss is compensated for by external power [13, 14].

In the case of electron or positron storage rings, such cooling occurs naturally due to synchrotron radiation and provides an automatic route to achieve small equilibrium emittances through a balance between radiation damping and excitation of oscillations by random radiation of individual photons. Fast radiation damping allows top-up injection of new particles without removing existing ones, a useful method to maximize the integrated luminosity of circular $e^+e^-$ colliders. Synchrotron radiation damping will also be an important cooling mechanism for future energy-frontier hadron colliders, like the proposed FCC-hh and SppC (see below). Four other methods of beam cooling have been developed and successfully employed to attain low emittances, namely electron cooling and stochastic cooling of heavy particles (ions and antiprotons), laser cooling of ion beams, and the ionization cooling of muons.

To avoid damage or excessive irradiation of accelerator components so that these remain accessible for maintenance in the tunnel, sophisticated collimation systems are utilized. These systems usually employ a series of targets or primary collimators which scatter the halo particles, and numerous absorbers (sometimes as many as a hundred, which intercept particles in dedicated locations) [15, Ch.9.7]. In the highest energy modern and future colliders, extreme total beam energies $n_b N E_b$ ranging from MJs to GJs and impacting surface energy densities reaching many GJ/mm² pose one of the biggest challenges for high efficiency and robust particle collimation.

### 31.2.3 High Luminosity Collisions

Eq. (31.3) for luminosity can be recast in terms of normalized transverse emittances Eq. (31.10) and the beta-functions $\beta^*$ at
the IP as:

$$ \mathcal{L} = f_0 \gamma n_b \frac{N^2}{4\pi \sqrt{\varepsilon_{nx} \beta_x^* \varepsilon_{ny} \beta_y^*}} \mathcal{F}. \qquad (31.15) $$

Here, equal bunch populations $N$ are assumed in two Gaussian beams with the same emittances. Naturally, to achieve a high luminosity, one has to maximize the total beam populations $n_b N$ within the lowest possible emittances, and collide the beams at high frequency at locations where the focusing beam optics provides the lowest possible values of the amplitude functions $\beta^*$, the so-called low-beta insertion. The latter requires sophisticated systems of strong focusing elements, sometimes occupying quite a significant fraction of the collider's total length. The lowest $\beta_{x,y}^*$ is determined by the maximum field gradients and apertures in the interaction region (IR) magnets and the effectiveness of compensation of chromatic and nonlinear aberrations.

The typical geometric reduction factor is $\mathcal{F} \approx 1$, and it rarely drops below 0.5 for the majority of colliders, unless this is specifically required by physics processes under study. The reduction due to the hourglass effect is caused by the increase in transverse beam sizes as one proceeds away from the IP, where $\beta(s)$ grows parabolically, as in Eq. (31.8). For long round bunches the effect scales as $\mathcal{F} \approx \sqrt{\pi} A \exp(A^2) \text{erfc}(A)$, where $A = \beta^*/\sigma_z$. Nonzero beam crossing angles $\theta_c$ in the horizontal plane and long bunches (rms bunch length $\sigma_z$) will reduce the luminosity, too, by a factor $\mathcal{F} \approx 1/(1+\Phi^2)^{1/2}$, where the parameter $\Phi = \sigma_z \tan(\theta_c/2)/\sigma_x^*$ is known as the Piwinski angle.

One of the most common limits to producing high luminosity arises from electric and magnetic forces of the opposite bunch at the IPs, characterized by a dimensionless beam-beam parameter:

$$ \xi_{x,y} = \frac{r_0 N \beta_{x,y}^*}{2\pi \gamma \sigma_{x,y}^* (\sigma_x^* + \sigma_y^*)}, \qquad (31.16) $$

where $r_0 = Z^2 e^2 / (4\pi\varepsilon_0 mc^2)$ is the classical radius of the colliding particle (with charge $Ze$ and mass $m$). From Eqs. (31.3) or (31.15) and (31.16), one can note that the path to higher luminosity via higher beam intensity and smaller beam sizes almost automatically calls for a higher beam-beam parameter as $\mathcal{L} \propto \xi$. Cited in the Tables, the beam-beam parameter is roughly equal to the betatron tune shift experienced by small-amplitude particles – positive in the case of opposite charge beams, like $e^+e^-$, and negative for same charge beams as in pp collisions. Beam-beam forces can lead to coherent effects, such as unstable beam oscillations or blow-up of one beam's size while the other beam remains small or even shrinks (flip-flop effect). The tune spread arising from $\xi$ and the nonlinear nature of beam-beam interactions results in strong diffusion along high-order transverse resonances $kQ_x + lQ_y = m$ and, ultimately, in beam size growth and beam losses. Operational experience indicates that the aforementioned effects are tolerable below certain beam-beam limit of $\xi_{x,y} \approx 0.003 - 0.012$ in hadron colliders [16], and – due to strong synchrotron radiation damping – an order of magnitude higher one in $e^+e^-$ colliders, with maximum $\xi_{x,y} \approx 0.03 - 0.12$ [17, 18]. The accessible beam-beam parameter range can also be restricted by coherent beam-beam instabilities. These various limits translate into a maximum allowed single bunch intensity $N$ and call for an increase of the number of bunches $n_b$ to achieve higher luminosities.

In linear colliders, where each bunch collides only once, with typically much smaller beam size and experiencing much stronger forces, the strength of the collision is measured by the ratio of the rms bunch length $\sigma_z$ to the beam-beam focal length. This ratio, called disruption parameter $D_y$, is related to $\xi_y$ via $D_y = 4\pi\sigma_z\xi_y/\beta_y^*$. Stronger beam-beam focusing and higher $D_y$ lead to effectively smaller beam sizes and a resulting luminosity enhancement; but it also makes the collisions more sensitive to small offsets, resulting in a kink instability. Additional beam-beam effects arising in the collisions at linear colliders are the emission of beamstrahlung (synchrotron radiation in the field of the opposing beam), along with $e^+e^-$ pair creation, and depolarization by various mechanisms.

Beamstrahlung is relevant for both linear colliders, where it may significantly degrade the luminosity spectrum, and for future

highest-energy circular colliders, where it may limit the beam lifetime, and also increases the energy spread and bunch length of the stored beam. For both types of colliders, the beamstrahlung is mitigated by making the colliding beams as flat as possible at the interaction point ($\sigma_x^* \gg \sigma_y^*$) to lower the beams' EM fields. The photon energy spectrum of the beamstrahlung is characterized by the parameter $\Upsilon = (2/3)\hbar\omega_c/E_b$ [19], with $\hbar\omega_c$ denoting the critical photon energy. The spectrum strongly deviates from the classical synchrotron radiation spectrum for $\Upsilon$ approaching or exceeding 1.

For hadron colliders, two fundamental luminosity limits are the beam lifetime, determined by burn-off in the collisions due to in-elastic pp interaction $dN/dt = -\mathcal{L}\sigma_{in}$, and the radiation from the collision debris, which may induce “quenches” (transitions to the normal-conducting state) of the superconducting final quadrupole magnets, and, in the long term, affect the equipment lifetime. Another limit on the achievable integrated luminosity in circular colliders is set by the minimum or average turnaround time (the time between the beam abort at the end of a physics fill and the start of the next physics collisions). Achieving practical filling times with many bunches in the collider requires either fast cycling injector machines and/or the top-up injection operation. The latter makes the average luminosity of circular electron-positron colliders approximately equal to the peak luminosity.

## 3.1.3 Recent High Energy Colliders

In this and the following section, elaboration is made on various issues associated with some of the recently operating colliders, particularly factors which impact integrated luminosity. Only general references are provided, where further information can be obtained. A more complete list of recent colliders and their parameters can be found in the High-Energy Collider Parameters tables.

### 3.1.3.1 Tevatron

The first superconducting synchrotron in history, the Tevatron [20] was converted into a proton-antiproton collider in 1985. Its 4.4 T dipole magnets employed Nb-Ti superconducting cable operating at 4.5 K, requiring what was then the world's largest cryogenic system. With $\sqrt{s}$ up to 1.96 TeV it was the highest energy collider for 25 years and delivered more than 12 fb$^{-1}$ of the integrated luminosity to each $p\bar{p}$ detector experiments (CDF and D0) before being shut off in 2011. The route to high integrated luminosity in the Tevatron was governed by the antiproton production rate, the turn-around time to produce another store, and the resulting optimization of store time [21]. The antiproton production complex consisted of three 8 GeV $p\bar{p}$ accelerators (the Accumulator, Debuncher, and Recycler – the latter was the first high energy accelerator built with permanent magnets), and employed 25 independent stochastic cooling systems and one pioneering high-energy electron cooling set-up to accumulate up to a record high value of $25 \cdot 10^{10} \bar{p}$ per hour. Despite severe parasitic long-range interactions of the two beams, each consisting of 36 bunches placed on helical orbits by two dozen $\pm 150$ kV high-voltage (HV) separators, a total beam-beam tune shift parameter of $n_{IP}\xi \approx 0.025-0.03$ was achieved, a record for hadron beams, with $n_{IP} = 2$ primary collision points. Other notable advances in the accelerator science and technology included advanced longitudinal beam manipulation techniques of slip-stacking and momentum mining and the first operational use of electron lenses for beam collimation and for compensation of long-range beam-beam effects [22]. The Tevatron ultimately achieved luminosities a factor of 430 higher than the original design specification.

### 3.1.3.2 HERA

The first lepton-proton collider, the 6.4 km long Hadron-Elektron-Ring-Anlage (HERA) at DESY in Germany [23], operated between 1992 and 2007 and delivered nearly 1 fb$^{-1}$ of integrated luminosity at $\sqrt{s}$ of about 320 GeV to the electron-proton collider experiments H1 and ZEUS [24, Ch.10.5]. It was the first facility to employ both applications of superconductivity: 5 T magnets in the 920 GeV proton ring and SRF accelerating structures to provide about 12 MW of RF power to compensate for synchrotron radiation losses of 30 GeV lepton beams (positrons or
electrons, in a conventional-magnet ring). With proper orbit and optics control, the HERA lepton beam would naturally become transversely polarized to about 60% (within about 40 minutes) thanks to the Sokolov-Ternov effect. Special magnets called spin rotators were implemented on either side of the collider IPs to produce 30–45% longitudinal polarization at the experiments.

### 31.3.3 LEP

Installed in a tunnel of 26.7 km circumference, LEP [25] was the largest circular $e^+e^-$ collider built so far. LEP was operated from 1989 to 2000 with beam energies ranging from 45.6 to 104.5 GeV. The synchrotron radiation loss per turn reached some 3% of beam energy and it was the total available RF voltage and power, respectively, that determined LEP maximum energy and luminosity of $10^{32} \text{ cm}^{-2}\text{s}^{-1}$. At a beam energy of 98 GeV, LEP operated with a beam-beam parameter $\xi \approx 0.083$, i.e., the total beam-beam tune shift for four interaction points was $n_{\text{IP}}\xi \approx 0.33$. In the last year of operation, 288 SRF cavities were powered by 36 klystrons with an average power of 0.6 MW each and provided $V_{\text{RF}} = 3.63$ GV. Up to about 60 GeV, LEP used resonant depolarization to measure the beam energy with 0.001% accuracy.

### 31.3.4 SLC

Based on an existing 3-km long 2.85 GHz warm RF linac, the SLC [26] was the first and only linear collider. It was operated from 1987 to 1998 with a constant beam energy of 45.6 GeV, up to about 80% electron-beam polarization, quasi-flat beams, a final-focus optics with local chromatic correction based on four interleaved sextupoles and $\beta_y^* \approx 1$ mm. In its last year, SLC achieved a peak luminosity of about $3 \times 10^{30} \text{ cm}^{-2}\text{s}^{-1}$, roughly half of the design value. The SLC had a high-efficiency positron source providing $5 \times 10^{12} \text{ e}^+$ per s for 120 Hz injection into the linac. It also employed the BNS damping to suppress the single-bunch beam break up instability, and also demonstrated an about 2-fold increase of luminosity from disruption enhancement due to the mutual focusing of the colliding electron and positron bunches at the interaction point,

## 31.4 Presently Operating Colliders

### 31.4.1 LHC

With a beam energy of 6.8 TeV (to be raised to the design value of 7 TeV through consolidation and magnet training), the superconducting Large Hadron Collider [27] presently is the world's highest energy collider. In the latest runs, peak luminosities of up to $2.1 \times 10^{34} \text{ cm}^{-2}\text{s}^{-1}$ have been achieved at 13 TeV cme - more than twice the design value (the current status is best checked at the Web site [28]). To meet its luminosity goals, the LHC operates with a high beam current of approximately 0.5 A, leading to stored energies of about 330 MJ per beam. Controlled energy deposition and component protection are given a high priority and a sophisticated highly efficient system of more than 100 beam collimators is employed [29]. At the energy of about 7 TeV per particle, synchrotron radiation poses a challenge, as the cryogenic system must remove roughly 7 kW due to synchrotron radiation, intercepted with a specially designed beamscreen inside the vacuum chamber, at a temperature of about 5–20 K, to be compared with a temperature of 1.9 K for the magnet cold bore. The elevated temperature allows for a more energy-efficient removal of beam-induced heat. The beamscreen also provides an effective cryo-pump for the vacuum system. When synchrotron-radiation photons hit the beamscreen, they can generate photoelectrons. These photoelectrons, and also any other electrons generated in the vacuum system, e.g. by residual-gas ionization, are accelerated in the electric field of the beam and may multiply via secondary-electron emission, with consequent electron cloud development. To mitigate this issue, the beamscreen is regularly subjected to beam-induced surface conditioning (scrubbing), thereby lowering the secondary emission yield. The two proton beams of 2556 bunches spaced by 25 ns are contained in separate pipes throughout most of the circumference and are brought together into a single 130 m long beam pipe at the interaction points. To avoid approximately 30 head-on collisions a small crossing angle of about 0.3 mrad is employed, which reduces the luminosity by about 15%. Still, the bunches moving in one direction experience multi-

ple long-range encounters with the counter-rotating bunches and the resulting perturbations of the particle motion substantially contribute to the beam lifetime reduction. The dominant source of approximately 8 hour characteristic luminosity decay time is proton burn-off due to inelastic pp interaction with $\sigma_{\text{in}} \approx 81$ mbarn, corresponding to pile-up of up to 50 (number of events per individual bunch crossing). In special physics runs with a few bunches and large $\beta^*$, the LHC achieved a head-on beam-beam tune shift of $n_{\text{IP}}|\xi| \approx 0.02$ with $n_{\text{IP}} = 2$, about twice as high as in regular operation.

The Tables also show the LHC luminosity performance in Pb-Pb collisions, which for the ATLAS and CMS experiments well exceeded the design value, while for the ALICE experiment, the luminosity is levelled near the Pb-Pb design value of $10^{27} \text{ cm}^{-2}\text{s}^{-1}$. The LHC can also provide Pb-p collisions as it did in 2013 and 2016, and other ion-ion or ion-proton collisions, at different energies.

In the coming years, an ambitious upgrade program, HL-LHC [30], with the accompanying LHC Injectors Upgrade [31], has as its target an order-of-magnitude increase in integrated luminosity through doubling the proton beam current, the utilization of new larger aperture Nb$_3$Sn superconducting final quadrupoles to allow squeezing the $\beta^*$ to as low as 10 cm, superconducting compact crab cavities and luminosity leveling also for ATLAS and CMS as its key ingredients.

### 31.4.2 Electron-Positron Rings

Asymmetric energies of the two beams allow for the enhancement of B-physics research and for interesting interaction region designs. SuperKEKB operates with 7 GeV electron and 4 GeV positron beams since 2018 and is aiming for luminosities of $8 \times 10^{35} \text{ cm}^{-2}\text{s}^{-1}$ [32]. By summer 2022, a world record luminosity of $4.7 \times 10^{34} \text{ cm}^{-2}\text{s}^{-1}$ has been reached, still at rather low beam currents. Vertical beam-beam tune shifts of $\xi_y \approx 0.05$ for the 4 GeV positron beam, and $\xi_y \approx 0.03$ for the 7 GeV electron beam have been achieved. These values are still about a factor of two lower than at the previous KEKB. Since 2020 SuperKEKB operates with a virtual crab-waist collision scheme, first developed for the FCC-ee design [33]. The original crab-waist scheme, based on additional sextupole magnets, was earlier implemented at DAΦNE [34]. The general crab-waist concept combines a large Piwinski angle $\Phi$, and an extremely low $\beta_y^*$ ($\ll \sigma_z$) with the cancellation of the transverse betatron resonances which occur under conditions of $kQ_x + lQ_y = n$, where $k,l,n$ are integers. The latter is achieved by means of existing or additional electromagnetic sextupoles with special betatron phase advances to the collision point [35]. The crab-waist collision scheme has become a design choice for all proposed future $e^+e^-$ circular colliders.

Beside SuperKEKB and DAΦNE, three other $e^+e^-$ ring colliders currently in operation are VEPP-2000 with $\sqrt{s}$ up to 2.0 GeV, BEPC-II with $\sqrt{s}$ up to 4.6 GeV and VEPP-4M with maximum c.m.e. of 12 GeV [1].

### 31.4.3 RHIC

The Relativistic Heavy Ion Collider [36] employs 3.45 T Nb-Ti superconducting magnets, and collides combinations of fully-stripped ions such as H-H ($p-p$), p-Al, p-Au, d-Au, h-Au, Cu-Cu, Cu-Au, Zr-Zr, Ru-Ru, Au-Au, and U-U over a wide energy range. The high charge per particle (+79 for gold, for instance) makes intra-beam scattering of particles within the bunch a special concern, even for seemingly moderate bunch intensities. In 2012, 3-D stochastic cooling was successfully implemented in RHIC [37] and is now routinely used. With stochastic cooling, steady increases in the bunch intensity, and numerous other upgrades, RHIC now operates at 44 times the Au-Au design average luminosity. Unique among high energy colliders, RHIC heavy ions beams cross the transition energy $\gamma_{\text{rmt}} = 1/\sqrt{\alpha_c}$ during acceleration – see Eqs.(31.11, 31.12) – a point where the derivative with respect to momentum of the revolution period is zero. This period of time is kept as short as allowed by the magnet ramp rate and must be dealt with carefully.

RHIC is also unique in its ability to accelerate and collide polarized proton beams. As proton beam polarization must be maintained from its low-energy source, successful acceleration through
the myriad of depolarizing resonance conditions in high energy circular accelerators has taken years to accomplish [38]. An energy of 255 GeV per proton with 60% final polarization per beam has been realized. As part of a scheme to compensate the head-on beam-beam effect, electron lenses operated routinely during the record high beam-beam parameter polarized proton operation at 100 GeV energy in 2015 [39].

## 31.5 Future High Energy Colliders and Prospects

Modern nuclear physics and high energy particle physics face critical questions which require next-generation high-energy and high-intensity experiments using hadron-hadron, lepton-lepton, and lepton-proton colliding-beam facilities. Understanding the structure of the proton and neutron directly from the dynamics of their quarks and gluons governed by the quantum chromodynamics calls for new ion-ion and electron-ion colliders. Two types of colliders are generally aspired by the HEP community [40, 41]: i) Higgs factories with a c.m.e. of 240–250 GeV in $e^{+}e^{-}$ collisions for precision studies of the Higgs boson ($m_{H}$ = 125 GeV) and exploration of the Higgs sector in greater detail, including measurements of Higgs couplings to fermions and vector bosons, self-coupling, rare decays, mass and width, that can also deliver other electroweak precision physics, e.g. on the Z-pole (91 GeV c.m.e.), at the W-pair threshold (about 160 GeV), and when run as a top quark factory (365–380 GeV); and ii) colliders with c.m.e. levels significantly beyond those of the LHC to explore the energy frontier for potential discoveries through direct searches in $pp, \mu\mu$, and $e^{+}e^{-}$ interactions. In addition, precision physics at future high-luminosity factories operating at the τ-charm energy also provides sensitivity to new physics at multi-TeV energies and beyond. A comprehensive review of the future colliders' projects, ideas, and R&D activities can be found in Ref. [1]; Ref. [42] presents comparative analysis of implementation challenges of numerous future HEP collider proposals. Below we only briefly summarize leading collider proposals for construction over the next several decades which rely mostly on currently available technologies, such as normal-conducting (NC) or SC RF and/or NC or SC magnets, some of them requiring reasonable scope and duration mission-oriented development programs, as well as advanced schemes based on plasma acceleration and other innovative ideas. Tentative parameters of some of the colliders discussed, or mentioned, in this section are summarized in Table 31.1 and Table 31.2.

### 31.5.1 Ion-Ion and Electron-Ion Colliders

NICA (Nucletron-based Ion Collider fAcility) is a new accelerator complex under construction at the Joint Institute for Nuclear Research (JINR, Dubna, Russia) to study properties of hot and dense baryonic matter, strong interactions between quarks and gluons, and spin physics [43]. NICA will provide a variety of beam species, ranging from protons and polarized deuteron to massive gold ions. The 500 m circumference SC magnet based collider is designed for average luminosity in heavy ion and light ion interactions at $\sqrt{s_{NN}}$=4–11 GeV of $1 \times 10^{27} \text{cm}^{-2} \text{s}^{-1}$ for a variety of nuclei up to $^{197}\text{Au}^{79+}$, and for polarized proton and deuteron collisions at $\sqrt{s}$=12–27 GeV with $\mathcal{L}$=(1–10)$\times 10^{31} \text{cm}^{-2} \text{s}^{-1}$. NICA major accelerator challenges include strong intrabeam scattering and space-charge effects which will be mitigated by extensive use of electron and stochastic cooling systems.

The recently announced Electron-Ion Collider (EIC) for nuclear physics research will be built at Brookhaven National Laboratory in the US and arrange collisions between the reconfigured RHIC with a 41–275 GeV proton beam and a 5–18 GeV electron beam stored in a new ring (eRHIC) [44]. The EIC physics requirements [45], include highly polarized ($P_{e,n}$ ~70%) electron and nucleon beams (as the precision of measurements of interest scales as $\mathcal{L}P_{e}^{2}P_{n}^{2}$), a spectrum of ion beams from deuteron to the heaviest nuclei (U or Pb), variable c.m.e. values from $\sqrt{s}$=20 GeV to 140 GeV, high luminosities of up to $10^{33}-10^{34} \text{cm}^{-2} \text{s}^{-1}$, as well as possibilities of having more than one interaction region. Main accelerator design challenges on the path to the required energy, luminosity, and polarization, include the development of SRF crab-cavities and advanced SC magnets for interaction re-

gion focusing, energy-recovery linac (ERL) based electron cooling of hadron beams, essential to attain luminosities two orders of magnitude beyond the predecessor HERA ep collider, and high intensity polarized particle sources, augmented by the development of special magnets and operational techniques to preserve the polarization through the acceleration process to the collisions, including swap-out injection.

### 31.5.2 Higgs/Electroweak Factories

Higgs factory proposals generally aim at improving the precision of coupling measurements of Higgs boson, top quark, W and Z by an order of magnitude or more compared with previous studies. Two proposals for ~100 km circumference circular $e^{+}e^{-}$ colliders have recently gained momentum: the Future Circular Collider (FCC-ee) at CERN [33] and the Circular Electron-Positron Collider (CEPC) in China [46]. Design philosophy of these machines assumes use of the maximum RF power available to compensate $O(100 \text{ MW})$ synchrotron radiation losses $P_{SR} = 2I \cdot \Delta E_{SR}$ and operation at the beam-beam limit $\xi_{y}$ that yields peak luminosity:

$$ \mathcal{L} = \frac{3}{16\pi r_0^2 (m_e c^2)} \frac{P_{SR} \xi_y \rho}{\beta_y^* \gamma^3}, \quad (31.17) $$

that scales approximately as $1/E_{b}^{3.5}$ for practical limits on P, $\xi_{y}$ and $\beta_{y}^{*}$. The short beam lifetime at the high target luminosity, due to radiative Bhabha scattering, requires these machines to be constructed with a full-energy injector ring installed in the same tunnel to top off the electron and positron currents in the collider rings operating at constant energy. Beamstrahlung introduces an additional beam lifetime limitation depending on momentum acceptance (so that achieving sufficient off-momentum dynamic aperture becomes one of the design challenges), as well as some bunch lengthening.

These ambitious, large-scale projects based on well-established technologies are not extendable to TeV or multi-TeV energies, but they offer several important advantages that include the potential for much higher luminosities, and, thus, higher precision, the ability to operate multiple experiments simultaneously, and their ~100 km circular tunnels that could later house O(100 TeV) hadron colliders. The high energy efficiency of circular $e^{+}e^{-}$ colliders is further boosted by advances in RF power sources, by improved SC cavities, and by innovative low-power magnet systems including ones based on high-temperature superconductors (HTS) at moderate magnetic field.

For more than four decades, efforts have been devoted to developing high-gradient RF technology linear $e^{+}e^{-}$ colliders in order to overcome the synchrotron radiation limitations of circular $e^{+}e^{-}$ machines. The International Linear Collider (ILC), with a c.m.e. of 250 GeV in $e^{+}e^{-}$ collisions, has been under consideration for more than two decades and could potentially be upgraded to $\sqrt{s}$=500 GeV and further to 1 TeV. CERN's Compact Linear Collider (CLIC) design, developed since the mid-1980s, also includes possible upgrades, from an initial 380 GeV c.m.e. to ultimately 3 TeV, which would enable searches for new particles of significantly higher masses.

The primary challenge confronting a high energy, high luminosity single pass collider design is the beam power requirement, so that measures must be taken to keep the demand within bounds as illustrated in a transformed Eq. (31.15):

$$ \mathcal{L} = \frac{1}{8\pi\alpha r_0} \frac{P_{\text{wall}}}{\sqrt{s}} \frac{\eta}{\sigma_y^*} N_{\gamma} H_D. \quad (31.18) $$

Here, $P_{\text{wall}}$ is the total wall-plug power of the collider, to be converted into beam power $P_{b}$ = $2f_{0}N E_{b}$ with efficiency $\eta$, $N_{\gamma} \approx 2\alpha r_{0}N/\sigma_{y}^{*}$ is the number of beamstrahlung photons emitted per $e^{\pm}$ ($\alpha \approx 1/137$ denotes the fine-structure constant), and the last factor $H_{D}$, typically between 1 and 2, represents the enhancement of luminosity due to the pinch effect, the additional focusing occurring during the collision of oppositely charged bunches. The management of $P_{\text{wall}}$ leads to an upward push on the bunch population N with an attendant rise in the energy radiated due to the electromagnetic field of one bunch acting on the particles of the other (beamstrahlung). Keeping a significant fraction of the
<table><caption>**Table 31.1:** Tentative parameters of selected future $e^{+}e^{-}$ high-energy colliders. Parameters associated with different beam energy scenarios are comma-separated; H and V indicate horizontal and vertical directions.</caption><thead><tr><th></th><th>FCC-ee</th><th>CEPC</th><th>ILC</th><th>CLIC</th></tr></thead><tbody><tr><td>Species</td><td>e<sup>+</sup>e<sup>-</sup></td><td>e<sup>+</sup>e<sup>-</sup></td><td>e<sup>+</sup>e<sup>-</sup></td><td>e<sup>+</sup>e<sup>-</sup></td></tr><tr><td>Beam energy E<sub>b</sub> (GeV)</td><td>46, 120, 183</td><td>46, 120, 180</td><td>125, 250</td><td>190, 1500</td></tr><tr><td>Circumference or length (km)</td><td>90.66</td><td>100</td><td>20.5, 31</td><td>11, 50</td></tr><tr><td>Interaction regions</td><td>4</td><td>2</td><td>1</td><td>1</td></tr><tr><td>Est. integrated luminosity<br>per experiment (ab<sup>-1</sup>/year)</td><td>17, 0.6, 0.15</td><td>15, 0.65, 0.07</td><td>0.2, 0.3</td><td>0.1, 0.6</td></tr><tr><td>Peak lumi. L/IP (10<sup>34</sup>cm<sup>-2</sup>s<sup>-1</sup>)</td><td>140, 5.0, 1.25</td><td>115, 5.0, 0.5</td><td>1.4, 1.8</td><td>1.5, 6</td></tr><tr><td>Rep/rate (Hz, f<sub>rev</sub> for rings)</td><td>3307</td><td>3000</td><td>5</td><td>50</td></tr><tr><td>Polarization (%)</td><td>&ge;10, 0, 0</td><td>5-10, 0, 0</td><td>80/30 (e<sup>-</sup>/e<sup>+</sup>)</td><td>80/0 (e<sup>-</sup>/e<sup>+</sup>)</td></tr><tr><td>Time between collisions (&micro;s)</td><td>0.025, 0.3, 2.5</td><td>0.025, 0.68, 2.6</td><td>0.55</td><td>0.0005</td></tr><tr><td>Energy spread (rms, 10<sup>-3</sup>)</td><td>1.09, 1.43, 1.92</td><td>1.3, 1.7, 2.0</td><td>e<sup>-</sup>: 1.9, 1.2</td><td>3.5</td></tr><tr><td>Bunch length &sigma;<sub>z</sub> (rms, mm)</td><td>15.5, 4.7, 2.2</td><td>8.7, 3.9, 2.9</td><td>e<sup>+</sup>: 1.5, 0.7</td><td>0.07, 0.044</td></tr><tr><td>IP beam size &sigma;<sup>*</sup> (rms, &micro;m)</td><td>H: 9, 13, 40<br>V: 0.04, 0.04, 0.05</td><td>H: 5.9, 14, 38<br>V: 0.04, 0.04, 0.11</td><td>H: 0.52, 0.47<br>V: 0.008, 0.006</td><td>H: 0.15, 0.04<br>V: 0.003, 0.001</td></tr><tr><td>Emittance, &epsilon;<sub>n</sub> (rms, &micro;m)</td><td>H: 63, 167, 568<br>V: 0.17, 0.3, 0.6</td><td>H: 24, 150, 493<br>V: 0.12, 0.3, 1.7</td><td>H: 5, 10<br>V: 0.035, 0.035</td><td>H: 0.95, 0.66<br>V: 0.03, 0.02</td></tr><tr><td>&beta;<sup>*</sup> at interaction point (cm)</td><td>H: 11, 24, 100<br>V: 0.07, 0.1, 0.16</td><td>H: 13, 33, 104<br>V: 0.09, 0.1, 0.27</td><td>H: 1.3, 1.1<br>V: 0.041, 0.048</td><td>H: 0.8, 0.69<br>V: 0.01, 0.0068</td></tr><tr><td>Full crossing angle &theta;<sub>c</sub> (mrad)</td><td>30</td><td>33</td><td>14</td><td>20</td></tr><tr><td>Crossing scheme</td><td>crab waist</td><td>crab waist</td><td>crab crossing</td><td>crab crossing</td></tr><tr><td>Piinski angle &Phi; = &sigma;<sub>z</sub>&theta;<sub>c</sub>/(2&sigma;<sup>*</sup><sub>x</sub>)</td><td>26, 5.4, 0.8</td><td>24, 4.8, 1.3</td><td>0</td><td>0</td></tr><tr><td>Beam-beam param. &xi;<sub>y</sub> (10<sup>-3</sup>)</td><td>97, 88, 134</td><td>110, 127, 100</td><td>n/a</td><td>n/a</td></tr><tr><td>Disruption parameter D<sub>y</sub></td><td>1.3, 0.9, 2.0</td><td>0.6, 1.3, 0.8</td><td>35, 25</td><td>13, 8</td></tr><tr><td>Average Upsilon &Upsilon; (10<sup>-2</sup>)</td><td>0.01, 0.04, 0.06</td><td>0.02, 0.04, 0.05</td><td>3, 6</td><td>17, 500</td></tr><tr><td>RF frequency f<sub>RF</sub> (MHz)</td><td>400, 400, 800</td><td>650</td><td>1300</td><td>11994</td></tr><tr><td>Particles per bunch N (10<sup>10</sup>)</td><td>21, 12, 16</td><td>14, 13, 20</td><td>2</td><td>0.52, 0.37</td></tr><tr><td>Bunches per beam n<sub>b</sub></td><td>11200, 440, 60</td><td>11951, 249, 35</td><td>1312 (pulse)</td><td>352, 312 (trains at 50 Hz)</td></tr><tr><td>Average beam current I<sub>b</sub> (mA)</td><td>1270, 27, 4.9</td><td>804, 16.7, 3.3</td><td>0.021</td><td>0.014, 0.009</td></tr><tr><td>Injection energy (GeV)</td><td>on E<sub>b</sub> (top off)</td><td>on E<sub>b</sub> (top off)</td><td>5.0 (linac)</td><td>9.0 (linac)</td></tr><tr><td>RF gradient G (MV/m)</td><td>5.7, 10.6, 20.1</td><td>17.4, 24.9, 27.6</td><td>31.5</td><td>72, 100</td></tr><tr><td>Total SR power loss (MW)</td><td>100</td><td>60</td><td>n/a</td><td>n/a</td></tr><tr><td>Total beam power (MW)</td><td>n/a</td><td>n/a</td><td>5.3, 10.5</td><td>5.6, 28</td></tr><tr><td>Key technology</td><td>—</td><td>—</td><td>high grad. SC RF</td><td>two-beam accel.</td></tr></tbody></table>

<table><caption>**Table 31.2:** Tentative parameters of selected future high-energy hadronic and muon colliders. Parameters associated with different particle species for NICA and EIC, and different beam-energy scenarios for a muon collider, are comma-separated. Quantities are, where appropriate, r.m.s.; unless noted otherwise, energies refer to beam energy; H and V indicate horizontal and vertical directions. Parameters of HL-LHC can be found in the High-Energy Collider Parameters review tables.</caption><thead><tr><th></th><th>NICA</th><th>EIC</th><th>FCC-hh</th><th>SPPC</th><th>&mu;&mu; collider</th></tr></thead><tbody><tr><td>Species</td><td>ion-ion, pp</td><td>ep, e-ion</td><td>pp</td><td>pp</td><td>&mu;<sup>+</sup>&mu;<sup>-</sup></td></tr><tr><td>Beam energy E<sub>b</sub> (TeV)</td><td>10<sup>-3</sup>/(4.5/u, 13)</td><td>0.01(e),0.275(p)</td><td>40-58</td><td>62.5</td><td>0.063, 5</td></tr><tr><td>Circumference C (km)</td><td>0.503</td><td>3.834</td><td>90.66</td><td>100</td><td>0.3, 10</td></tr><tr><td>Interaction regions</td><td>2</td><td>1(2)</td><td>4</td><td>2</td><td>1, 2</td></tr><tr><td>Est. integr. luminosity<br>per exp. (ab<sup>-1</sup>/year)</td><td>10<sup>-8</sup>,-3 (ii, pp)</td><td>0.1</td><td>0.2-1.0</td><td>0.6</td><td>0.001, 2.0</td></tr><tr><td>Peak luminosity L (10<sup>34</sup>cm<sup>-2</sup>s<sup>-1</sup>)</td><td>10<sup>-7</sup>,-2 (ii, pp)</td><td>1.05</td><td>5-30</td><td>4.3</td><td>0.008, 20</td></tr><tr><td>Rep/rate (Hz, f<sub>rev</sub> for rings)</td><td>5.9&middot;10<sup>5</sup></td><td>7.8&middot;10<sup>4</sup></td><td>3307</td><td>3000</td><td>15, 5</td></tr><tr><td>Time between collisions (&micro;s)</td><td>0.077</td><td>0.009</td><td>0.025</td><td>0.025</td><td>1, 33</td></tr><tr><td>Energy spread (rms, 10<sup>-3</sup>)</td><td>1.6 (Au)</td><td>0.6 (e), 0.7 (p)</td><td>0.1</td><td>0.1</td><td>0.04, 1</td></tr><tr><td>Bunch length &sigma;<sub>z</sub> (rms, mm)</td><td>600</td><td>7 (e), 60 (p)</td><td>80</td><td>60</td><td>63, 1.5</td></tr><tr><td>IP beam size &sigma;<sup>*</sup> (H/V rms, &micro;m)</td><td>360</td><td>95/8.5</td><td>6.7-3.5 (init.)</td><td>3.0 (init.)</td><td>75, 0.9</td></tr><tr><td>Emittance &epsilon;<sub>n</sub> (H/V rms, mm mrad)</td><td>1.1</td><td>11.3/1.0 (e),<br>9.2/1.6 (p)</td><td>2.2 (init.)</td><td>1.2 (init.)</td><td>200, 25</td></tr><tr><td>Beta function at IP &beta;<sup>*</sup> (H/V cm)</td><td>60</td><td>45/5.6 (e),<br>80/7.2 (p)</td><td>110-30</td><td>50</td><td>1.7, 0.15</td></tr><tr><td>Beam-beam param. &xi; (10<sup>-3</sup> H/V)</td><td>25</td><td>72/100 (e),12 (p)</td><td>5-15</td><td>15</td><td>22, 78</td></tr><tr><td>RF frequency f<sub>RF</sub> (MHz)</td><td>13/39</td><td>591</td><td>400</td><td>400/200</td><td>805/1300</td></tr><tr><td>Particles per bunch N (10<sup>10</sup>)</td><td>0.23</td><td>17.2(e), 6.9(p)</td><td>10</td><td>4</td><td>400, 180</td></tr><tr><td>Bunches per beam n<sub>b</sub></td><td>22</td><td>1160</td><td>9648</td><td>10082</td><td>1</td></tr><tr><td>Average beam current I<sub>b</sub> (mA)</td><td>480</td><td>2500(e),1000 (p)</td><td>500</td><td>190</td><td>640, 9 (peak)</td></tr><tr><td>Injection energy (GeV)</td><td>1-3.8</td><td>on E<sub>b</sub> (e), 25 (p)</td><td>&ge;1000</td><td>3200</td><td>on E<sub>b</sub></td></tr><tr><td>Peak magnetic field B (T)</td><td>1.8</td><td>0.248 (e), 3.80 (p)</td><td>14-20</td><td>20</td><td>10</td></tr><tr><td>Polarization (%)</td><td>0(i), &gt;50(p)</td><td>&gt; 70(e), &gt;70(p)</td><td>0</td><td>0</td><td>0</td></tr><tr><td>SR power loss/beam (MW)</td><td>10<sup>-6</sup></td><td>10(e), &lt; 10<sup>-6</sup>(p)</td><td>1.0-4.25</td><td>2.2</td><td>10<sup>-3</sup>, 0.16</td></tr><tr><td>Key technology</td><td>electron and<br>stoch. cooling</td><td>strong hadron<br>cooling</td><td>Nb<sub>3</sub>Sn/HTS<br>magnets</td><td>HTS<br>magnets</td><td>muon prod.<br>&amp; cooling</td></tr></tbody></table>
luminosity close to the nominal energy represents a design goal, which is met if $N_γ$ does not exceed a value of about 1. A consequence is the use of flat beams, where $N_γ$ is managed by the beam width, and luminosity adjusted by the beam height, thus the explicit appearance of the vertical beam size $σ_y^*$.

The ILC [47, 48] is based on 1.3 GHz standing-wave superconducting accelerating structures operating at 2 K with 31.5 MV/m average gradient, up to 8 nm vertical beam size at the IP, and luminosity comparable to the LHC. Progress toward higher field gradients and $Q$ values continues to be made, with nitrogen-doping techniques being a recent example [49]. CLIC is based on a novel two-beam acceleration scheme [50]. Here, room-temperature NC copper high-gradient 12 GHz accelerating structures are powered by a high-current 1.9 GeV drive beam to efficiently enable accelerating gradients of up to 100 MV/m (though optimal $G=70$ MV/m for the first CLIC stage at $\sqrt{s}=380$ GeV, and for this stage an alternative RF power drive option with 12 GHz klystrons powering is also being considered). To reach their design luminosities, both CLIC and ILC require unprecedented rates of positron production about 40 times the world record set by the SLC positron source, and very tight control of imperfections, such as $O(10~\mu\text{m})$ accuracy of pre-alignment of the main linac and beam delivery system components at the level, suppression of fast vibrations of the quadrupoles due to ground motion to $O(1\text{ nm})$ level at frequencies above 1 Hz, advanced beam-based trajectory tuning, and mitigation of the effect of wakefields [51].

Recent developments of the RF technologies show promise of more compact linear $e^+e^-$ Higgs/Electroweak factories based on either ~70 MV/m travelling-wave 1.3 GHz SRF structures [52] or on 70-120 MV/m high-efficiency, distributed RF coupling, normal-conducting cold copper 5.7 GHz accelerating cavities at liquid nitrogen temperature [53].

There are a number of alternative ideas proposed for studies of the Higgs/Electroweak physics, such as high-energy, high-luminosity $e^+e^-$ collider in a 100 km tunnel using ERLs to accelerate particles to collision energy in 4 to 6 passes and return up to 81% of the energy back into the SRF cavities at deceleration turns, thus, lowering the required facility power several-fold [54]; similar power recovery in one pass can greatly improve efficiency of linear colliders [55-57]; an arrangement of $γγ$ collision through near-IP conversion of high energy electron beams into intense photon beams by backward Compton scattering off a high-power laser [58, 59] or off an FEL photon pulse [60, 61]; $\mu\mu$ Higgs factory with unprecedented 0.004% energy resolution [62], and a high-energy lepton-hadron collider bringing into collision a 60-GeV electron beam from an ERL with the 7 TeV protons circulating in the LHC (LHeC) [63]. At lower energies, Super Tau-Charm Factory proposals aim at the production and precise study of charmonium states and of the tau lepton [64, 65].

### 31.5.3 Energy Frontier Circular Colliders

Several hadron and lepton colliders have been proposed to extend the energy reach beyond the LHC. As noted above, ambitious plans have been proposed to upgrade the FCC and CEPC to hadron colliders – FCC-hh at CERN and Super Proton Proton Collider (SPPC) in China, respectively – by means of next- or next-next generation SC magnets installed in the arc sections of the 100 km rings, so as to enable $\sqrt{s}$ of the order 100 TeV or above [66, 67]. Comparable discovery reach is expected for a circular 10-14 TeV muon collider [68], significantly beyond that of practical $e^+e^-$ linear colliders.

The maximum beam energy Eq.(31.1) is directly proportional to the magnetic field and to the ring circumference, hence, future hadron colliders like FCC-hh and SppC rely on the development of the technology of 14-16 T Nb$_3$Sn dipole magnets [69] or dipole magnets based on high-temperature superconductors (HTS) with a field of up to 18-20 T, including iron-based HTS [70]. Though higher fields are possible with HTS, more cost-effective might be hybrid magnet designs incorporating Nb-Ti, Nb$_3$Sn, and an inner layer of HTS and providing fields of about 20 T. Another important technology is the cryogenic beam vacuum system, which has to cope with unusually high levels of synchrotron radiation (up to 15 MW in total, for FCC-hh) in a cold environment. The beam-screen intercepting the radiation inside the cold bore of the

magnets should operate at or above 50 K — significantly higher temperature than in the LHC.

Design luminosities of these hadron colliders $O(10^{35} \text{ cm}^{-2}\text{s}^{-1})$ will result in a pile-up of events per crossing $O(500)$ (from up to 50 in LHC) and fast intensity drop due to burn-off. Significant radiation damping of beam emittances will naturally level luminosity evolution, though the total beam-beam tune shift $n_{IP}\xi \sim (0.01 - 0.03)$ might need a special control as it will increase during the store [71].

Future hadron colliders are characterized by record high stored beam energy, e.g. 6-9 GJ in FCC-hh, rendering machine protection a paramount concern. A very challenging multi-stage collimation system is needed to avoid local beam loss spikes near cold magnets, which would induce magnet quenches. The primary and secondary collimators of the LHC are based on carbon-carbon composite material. For the future hadron colliders, ever stronger materials are being developed and examined, which also feature higher conductivity and, hence, lower impedance. More advanced options include the use of short bent crystals as primary collimators [72] and the deployment of hollow electron-beam lenses as non-destructible collimators [73].

It is noteworthy that machines like FCC-hh or SPPC can additionally be used for ion-ion/ion-proton collisions; their high-energy proton beams can also be collided with high-intensity $O(60 \text{ GeV})$ electrons from an ERL resulting in c.m.e. of 3.5 TeV.

The lifetime of the muon, $\gamma\tau_0$ where $\tau_0=2.2\mu\text{s}$, is sufficient to allow fast acceleration to high energy before most, or all, of the muons decay, and storage for some 300B turns in a ring with an average bending magnets field of $B$ (in units of Tesla). The muon to electron mass ratio of 210 implies removal of the synchrotron radiation barrier and possibility of a muon collider facility scale to a level compatible with on-site placement at existing accelerator laboratories. High-energy muon colliders, as presently conceived, are predicted to be more compact, more power-efficient and significantly less expensive than the equivalent energy-frontier hadron or $e^+e^-$ machines [74], and a neutrino factory could potentially be realized in the course of their construction [75]. The Higgs production cross-section in the s-channel is enhanced by a factor of $(m_\mu/m_e)^2$ compared to that in $e^+e^-$ collisions.

The average luminosity of a muon collider,

$$ \langle L \rangle = f_0 \gamma^2 \frac{c\tau_0}{2C} \frac{n_b N^2}{4\pi\epsilon_n \beta^*} \mathcal{F} = B P_b \frac{N\tau_0}{4\pi\epsilon_n \beta^*} \left( \frac{c\tau_0 \mathcal{F}}{8\pi e} \right), \quad (31.19) $$

scales with $B$, the total beam power $P_b$, and the beam brightness – the third factor above is nothing but the muon beam-beam tune shift Eq. (31.16). There is an obvious incentive to have all the particles in just one bunch per beam. The beta-function at the two IPs $\beta^* \approx \sigma_z$ scales as $1/\gamma$ within certain range of energies, giving overall scaling $\langle L \rangle \propto \gamma^2$ with other limiting parameters fixed. The main challenges to luminosity achievement with decaying particles are related to production and fast cooling and acceleration of $O(10^{12})$ muons per bunch without emittance degradation. A multi-TeV c.m.e. high luminosity $O(10^{34} \text{ cm}^{-2}\text{s}^{-1})$ muon collider would consist of [76]: (i) a high power proton driver (e.g., 8 GeV 2-4 MW H$^-$ SRF linac), (ii) pre-target accumulation and compressor rings, in which high intensity 1-3 ns long proton bunches are formed, (iii) a liquid-mercury or other high-power target for converting the proton beam into a tertiary muon beam with energy of about 200 MeV, (iv) a multi-stage ionization cooling section that reduces transverse and longitudinal emittances by several orders of magnitude and creates a low emittance beam, similarly to that recently demonstrated [77], (v) a multistage acceleration system, possibly employing either rapid cycling synchrotrons or RLAs to accelerate muons in a modest number of turns up to the final energy using superconducting RF technology, and, finally, (vi) a 3-14 km diameter collider ring, where counter-propagating muon beams are stored and collide over the roughly 3000 turns corresponding to the muon lifetime.

The intense neutrino flux originating from the multi-TeV muon beams decaying in the collider poses another challenge – the need to minimize the environmental impact. The collider complex is usually located underground and when the produced neutrinos emerge at the surface, a small fraction interacts with the rock (and
other material) and produces ionizing radiation dose scaling as $\gamma^3$. The impact of this neutrino-induced radiation can be mitigated, for example, by continually adjusting the orbits of the beams to spread them out on a wider area, by deeper collider tunnels or with a further reduced emittance of the muon beam so that the required luminosity could be obtained using a substantially smaller number of muons.

### 31.5.4 Plasma Acceleration and Other Advanced Concepts

Since about the mid-1950s, it has been understood that collective plasma-based accelerators promise extremely large accelerating gradients, approximately three orders of magnitude greater than ~100 MV/m obtained in conventional breakdown limited RF structures [78]. Ionized plasmas can sustain electron plasma density waves with electric fields in excess of $E_0 = cm_e\omega_p/e$ or

$$E_0 \approx 96 \, [\text{V/m}] \sqrt{n_0[\text{cm}^{-3}]}, \quad (31.20)$$

where $n_0$ denotes the ambient electron number density and $\omega_p = \sqrt{e^2n_0/(m_e\epsilon_0)}$ is the electron plasma frequency [79].

Such gradients can be effectively excited by either powerful external pulses of laser light or by electron bunches if they are shorter than the plasma wavelength $\lambda_p = c/\omega_p \approx 1 \, \text{mm} \times \sqrt{10^{15} \, \text{cm}^{-3}/n_0}$, or by longer beams of protons if their charge density is modulated with the period of $\lambda_p$. In the past decade, we have seen impressive progress in the plasma wakefield acceleration of high-quality beams. Laser-driven electron energy gain of about 8 GeV over 20 cm of plasma with density $3 \times 10^{17} \, \text{cm}^{-3}$ has been demonstrated at the BELLA facility at the Lawrence Berkeley National Laboratory (LBNL) [80]. Short electron bunches were used to boost the energy of externally injected electron bunches by 9 GeV over 1.3 m of ~$10^{17} \, \text{cm}^{-3}$ plasma at the FACET facility in SLAC [81]. The AWAKE experiment at CERN used self-modulating long 450 GeV proton bunches to accelerate electrons to 2 GeV over 10 m of $10^{15} \, \text{cm}^{-3}$ plasma [82].

Whether plasma acceleration will find application in an HEP facility is not yet clear, given the necessity of staging and phase-locking acceleration in multiple plasma chambers [83]. Another critical issue is the power efficiency $\eta$ for a collider based on plasma acceleration, whose luminosity would still be described by Eq. (31.18). Positrons can be accelerated as well but, contrary to electrons, plasma-wakefields will naturally defocus positively charged particles. Such fundamental deficiency can be addressed, e.g., in more complicated plasma acceleration schemes for $e^+$ [84]; or by acceleration of electrons in both linacs with conversion into a $\gamma\gamma$ collider at the IP. In addition, many novel approaches have been proposed to extend the energy or physics reach of future particle colliders, reduce their cost, and improve their luminosity or energy efficiency. Those include: i) dielectric wakefield accelerators in which resonant dielectric accelerating structures are fed by ultra-short RF pulses of wakefields driven by either collinear or preceding high charge electron bunches and withstand 270 MV/m operational accelerating gradients [85]; ii) dielectric laser accelerators – micron-size dielectric accelerating structures driven by a laser and supporting $O(1 \, \text{GV/m})$ accelerating fields [86]; iii) compact linear muon crystal colliders with ultimate energies $O(1-10 \, \text{GeV})$ [87] based on 1–10 TeV/m wakefield acceleration of muons (instead of electrons or hadrons) channeling between the planes in crystals or inside carbon nanotubes (CNT) with charge carrier density ~$10^{20-22} \, \text{cm}^{-3}$ [88]; iv) the Gamma Factory concept [89], where frequent bursts of gamma rays are generated by repeatedly colliding a partially stripped heavy-ion beam circulating in the LHC, or in a future higher-energy hadron storage ring like the FCC-hh, with a conventional laser pulse, profiting from two Lorentz boosts. Active R&D programs are presently underway worldwide to determine the suitability of novel technologies for use in future high-energy colliders.

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## 32. High-Energy Collider Parameters
### High-Energy Collider Parameters: $e^{+}e^{-}$ Colliders (I)

<table><caption>**Table 32.1:** Updated in March 2024 with numbers received from representatives of the colliders (contact E. Pianori, LBNL). The table shows the parameter values achieved. Quantities are, where appropriate, r.m.s.; unless noted otherwise, energies refer to beam energy; H and V indicate horizontal and vertical directions; s.c. stands for superconducting. Parameters for the defunct SPEAR, DORIS, PETRA, PEP, TRISTAN, and VEPP-2M colliders may be found in our 1996 edition (Phys. Rev. D54, 1 July 1996, Part I).</caption><thead><tr><th></th><th>VEPP-2000<br>(Novosibirsk)</th><th>VEPP-4M<br>(Novosibirsk)</th><th>BEPC<br>(China)</th><th>BEPC-II<br>(China)</th><th>DAΦNE<br>(Frascati)</th></tr></thead><tbody><tr><td>Physics start date</td><td>2010</td><td>1994</td><td>1989</td><td>2008</td><td>1999</td></tr><tr><td>Physics end date</td><td>—</td><td>—</td><td>2005</td><td>—</td><td>—</td></tr><tr><td>Maximum beam energy (GeV)</td><td>1.0</td><td>6</td><td>2.5</td><td>1.89 (2.474 max)</td><td>0.510</td></tr><tr><td>Delivered integrated<br>luminosity per exp.<br>(fb<sup>-1</sup>)</td><td>0.25</td><td>0.05</td><td>0.11</td><td>48</td><td>≈ 4.7 in 2001-2007<br>≈ 2.7 w/crab-waist<br>≈ 6.8 2014-2018*<br>≈ 0.4 2021-2022†<br>≈ 1.1 2023-2024‡</td></tr><tr><td>Luminosity (10<sup>30</sup> cm<sup>-2</sup>s<sup>-1</sup>)</td><td>50</td><td>20</td><td>12.6 at 1.843 GeV<br>5 at 1.55 GeV</td><td>1096</td><td>453</td></tr><tr><td>Time between collisions (μs)</td><td>0.04</td><td>0.6</td><td>0.8</td><td>0.006</td><td>0.0027</td></tr><tr><td>Full crossing angle (μ rad)</td><td>0</td><td>0</td><td>0</td><td>2.2 × 10<sup>4</sup></td><td>5 × 10<sup>4</sup></td></tr><tr><td>Energy spread (units 10<sup>-3</sup>)</td><td>0.71</td><td>1</td><td>0.58 at 2.2 GeV</td><td>0.52</td><td>0.40</td></tr><tr><td>Bunch length (cm)</td><td>4</td><td>5</td><td>≈ 5</td><td>1.35</td><td>low current: 1<br>at 15mA: 2</td></tr><tr><td>Beam radius (10<sup>-6</sup> m)</td><td>125 (round)</td><td>H:1000<br>V:30</td><td>H:890<br>V:37</td><td>H:347<br>V:4.5</td><td>H:260<br>V:4.8</td></tr><tr><td>Free space at interaction<br>point (m)</td><td>±0.5</td><td>±2</td><td>±2.15</td><td>±0.63</td><td>±0.295</td></tr><tr><td>Luminosity lifetime (hr)</td><td>continuous</td><td>2</td><td>7-12</td><td>1.5</td><td>0.2</td></tr><tr><td>Turn-around time (min)</td><td>continuous</td><td>18</td><td>32</td><td>4 (topping up)</td><td>2 (topping up)</td></tr><tr><td>Injection energy (GeV)</td><td>0.2-1.0</td><td>1.8</td><td>1.55</td><td>on energy</td><td>on energy</td></tr><tr><td>Transverse emittance<br>(10<sup>-9</sup> m)</td><td>H:150<br>V:150</td><td>H:200<br>V:20</td><td>H:660<br>V:28</td><td>H:121<br>V:1.56</td><td>H:260<br>V:2.6</td></tr><tr><td>β*, amplitude function at<br>interaction point (m)</td><td>H:0.05 - 0.11<br>V:0.05 - 0.11</td><td>H:0.75<br>V:0.05</td><td>H:1.2<br>V:0.05</td><td>H:1.0<br>V:0.0135</td><td>H:0.26<br>V:0.009</td></tr><tr><td>Beam-beam tune shift<br>per crossing (units 10<sup>-4</sup>)</td><td>H:850<br>V:850</td><td>500</td><td>350</td><td>383</td><td>440/894<br>(crab-waist run/test)</td></tr><tr><td>RF frequency (MHz)</td><td>172</td><td>180</td><td>199.53</td><td>499.8</td><td>368.667</td></tr><tr><td>Particles per bunch<br>(units 10<sup>10</sup>)</td><td>8</td><td>15</td><td>20 at 2 GeV<br>11 at 1.55 GeV</td><td>3.8</td><td>e<sup>-</sup>: 3.2<br>e<sup>+</sup>: 2.1</td></tr><tr><td>Bunches per ring<br>per species</td><td>1</td><td>2</td><td>1</td><td>119</td><td>100 to 110<br>(120 buckets)</td></tr><tr><td>Average beam current<br>per species (mA)</td><td>160</td><td>80</td><td>40 at 2 GeV<br>22 at 1.55 GeV</td><td>910</td><td>e<sup>-</sup>: 1250<br>e<sup>+</sup>: 800</td></tr><tr><td>Circumference or length (km)</td><td>0.024</td><td>0.366</td><td>0.2404</td><td>0.23753</td><td>0.098</td></tr><tr><td>Interaction regions</td><td>2</td><td>1</td><td>2</td><td>1</td><td>1</td></tr><tr><td>Magnetic length of dipole (m)</td><td>1.1</td><td>2</td><td>1.6</td><td>outer ring: 1.6<br>inner ring: 1.41</td><td>outer ring: 1.2<br>inner ring: 1</td></tr><tr><td>Length of standard cell (m)</td><td>12</td><td>7.2</td><td>6.6</td><td>outer ring: 6.6<br>inner ring: 6.2</td><td>n/a</td></tr><tr><td>Phase advance per cell (deg)</td><td>H:745<br>V:385</td><td>65</td><td>≈ 60</td><td>60-90<br>non-standard cells</td><td>—</td></tr><tr><td>Dipoles in ring</td><td>8</td><td>78</td><td>40 + 4 weak</td><td>84 + 8 weak</td><td>8</td></tr><tr><td>Quadrupoles in ring</td><td>24 + 4 s.c.</td><td>150</td><td>68</td><td>134+2 s.c.</td><td>48</td></tr><tr><td>Peak magnetic field (T)</td><td>2.4</td><td>0.6</td><td>0.903<br>at 2.8 GeV</td><td>outer ring: 0.677<br>inner ring: 0.766</td><td>1.2</td></tr></tbody></table>

<footnote>*KLOE-2</footnote>

<footnote>†SIDDHARTINO</footnote>

<footnote>‡SIDDHARTA-2</footnote>
High-Energy Collider Parameters: $e^{+}e^{-}$ Colliders (II)

<table><caption>**Table 32.2:** Updated in March 2020 with numbers received from representatives of the colliders (contact E. Pianori, LBNL). The table shows the parameter values achieved. Quantities are, where appropriate, r.m.s.; unless noted otherwise, energies refer to beam energy; *H* and *V* indicate horizontal and vertical directions; s.c. stands for superconducting. ILC and CLIC parameters are documented in the Accelerator physics of colliders review.</caption><thead><tr><th></th><th>CESR<br>(Cornell)</th><th>CESR-C<br>(Cornell)</th><th>LEP<br>(CERN)</th><th>SLC<br>(SLAC)</th></tr></thead><tbody><tr><td>Physics start date</td><td>1979</td><td>2002</td><td>1989</td><td>1989</td></tr><tr><td>Physics end date</td><td>2002</td><td>2008</td><td>2000</td><td>1998</td></tr><tr><td>Maximum beam energy (GeV)</td><td>6</td><td>6</td><td>100 - 104.6</td><td>50</td></tr><tr><td>Delivered integrated luminosity<br>per experiment (fb<sup>-1</sup>)</td><td>41.5</td><td>2.0</td><td>0.221 at Z peak<br>0.501 at 65 - 100 GeV<br>0.275 at &gt;100 GeV</td><td>0.022</td></tr><tr><td>Luminosity (10<sup>30</sup> cm<sup>-2</sup>s<sup>-1</sup>)</td><td>1280 at<br>5.3 GeV</td><td>76 at<br>2.08 GeV</td><td>24 at Z peak<br>100 at &gt; 90 GeV</td><td>2.5</td></tr><tr><td>Time between collisions (μs)</td><td>0.014 to 0.22</td><td>0.014 to 0.22</td><td>22</td><td>8300</td></tr><tr><td>Full crossing angle (μ rad)</td><td>±2000</td><td>±3300</td><td>0</td><td>0</td></tr><tr><td>Energy spread (units 10<sup>-3</sup>)</td><td>0.6 at 5.3 GeV</td><td>0.82 at 2.08 GeV</td><td>0.7→1.5</td><td>1.2</td></tr><tr><td>Bunch length (cm)</td><td>1.8</td><td>1.2</td><td>1.0</td><td>0.1</td></tr><tr><td>Beam radius (μm)</td><td>H:460<br>V:4</td><td>H:340<br>V:6.5</td><td>H:200→300<br>V:2.5→8</td><td>H:1.5<br>V:0.5</td></tr><tr><td>Free space at interaction<br>point (m)</td><td>±2.2 (±0.6<br>to REC quads)</td><td>±2.2 (±0.3<br>to PM quads)</td><td>±3.5</td><td>±2.8</td></tr><tr><td>Luminosity lifetime (hr)</td><td>2-3</td><td>2-3</td><td>20 at Z peak<br>10 at &gt; 90 GeV</td><td>—</td></tr><tr><td>Turn-around time (min)</td><td>5 (topping up)</td><td>1.5 (topping up)</td><td>50</td><td>120 Hz (pulsed)</td></tr><tr><td>Injection energy (GeV)</td><td>1.8-6</td><td>1.5-6</td><td>22</td><td>45.64</td></tr><tr><td>Transverse emittance<br>(10<sup>-9</sup> m)</td><td>210<br>1</td><td>120<br>3.5</td><td>H:20-45<br>V:0.25→1</td><td>H:0.5<br>V:0.05</td></tr><tr><td>β*, amplitude function at<br>interaction point (m)</td><td>1.0<br>0.018</td><td>0.94<br>0.012</td><td>1.5<br>0.05</td><td>0.0025<br>0.0015</td></tr><tr><td>Beam-beam tune shift per<br>crossing (10<sup>-4</sup>) or disruption</td><td>250<br>620</td><td>e<sup>-</sup>: 420 (H), 280 (V)<br>e<sup>+</sup>: 410 (H), 270 (V)</td><td>830</td><td>0.75 (H)<br>2.0 (V)</td></tr><tr><td>RF frequency (MHz)</td><td>500</td><td>500</td><td>352.2</td><td>2856</td></tr><tr><td>Particles per bunch<br>(units 10<sup>10</sup>)</td><td>1.15</td><td>4.7</td><td>45 in collision<br>60 in single beam</td><td>4.0</td></tr><tr><td>Bunches per ring<br>per species</td><td>9 trains<br>of 5 bunches</td><td>8 trains<br>of 3 bunches</td><td>4 trains of 1 or 2</td><td>1</td></tr><tr><td>Average beam current<br>per species (mA)</td><td>340</td><td>72</td><td>4 at Z peak<br>4→6 at &gt; 90 GeV</td><td>0.0008</td></tr><tr><td>Beam polarization (%)</td><td>—</td><td>—</td><td>55 at 45 GeV<br>5 at 61 GeV</td><td>e<sup>-</sup>: 80</td></tr><tr><td>Circumference or length (km)</td><td>0.768</td><td>0.768</td><td>26.66</td><td>1.45 +1.47</td></tr><tr><td>Interaction regions</td><td>1</td><td>1</td><td>4</td><td>1</td></tr><tr><td>Magnetic length of dipole (m)</td><td>1.6-6.6</td><td>1.6-6.6</td><td>11.66/pair</td><td>2.5</td></tr><tr><td>Length of standard cell (m)</td><td>16</td><td>16</td><td>79</td><td>5.2</td></tr><tr><td>Phase advance per cell (deg)</td><td>45-90 (no<br>standard cell)</td><td>45-90 (no<br>standard cell)</td><td>102/90</td><td>108</td></tr><tr><td>Dipoles in ring</td><td>86</td><td>84</td><td>3280 + 24 inj. + 64 weak</td><td>460+440</td></tr><tr><td>Quadrupoles in ring</td><td>101 + 4 s.c.</td><td>101 + 4 s.c.</td><td>520 + 288 + 8 s.c.</td><td>—</td></tr><tr><td>Peak magnetic field (T)</td><td>0.3 / 0.8<br>at 8 GeV</td><td>0.3 / 0.8 at 8 GeV,<br>2.1 wigglers at 1.9 GeV</td><td>0.135</td><td>0.597</td></tr></tbody></table>
High-Energy Collider Parameters: ep, e⁺e⁻ Colliders (III)

<table><caption>**Table 32.3:** Updated in March 2022 with numbers received from representatives of the colliders (contact E. Pianori, LBNL). The table shows the parameter values achieved. Design parameters for SuperKEKB may be found in our 2018 edition (Phys. Rev. D98, 030001 (2018)) Quantities are, where appropriate, r.m.s.; unless noted otherwise, energies refer to beam energy; H and V indicate horizontal and vertical directions; s.c. stands for superconducting.</caption><thead><tr><th></th><th>HERA<br>(DESY)</th><th>KEKB<br>(KEK)</th><th>PEP-II<br>(SLAC)</th><th>SuperKEKB<br>(KEK)</th></tr></thead><tbody><tr><td>Physics start date</td><td>1992</td><td>1999</td><td>1999</td><td>2018</td></tr><tr><td>Physics end date</td><td>2007</td><td>2010</td><td>2008</td><td>—</td></tr><tr><td>Particles collided</td><td>ep</td><td>e<sup>+</sup>e<sup>-</sup></td><td>e<sup>+</sup>e<sup>-</sup></td><td>e<sup>+</sup>e<sup>-</sup></td></tr><tr><td>Maximum beam<br>energy (TeV)</td><td>e: 0.030<br>p: 0.92</td><td>e<sup>-</sup>: 8.33 (8.0 nominal)<br>e<sup>+</sup>: 3.64 (3.5 nominal)</td><td>e<sup>-</sup>: 7-12 (9.0 nominal)<br>e<sup>+</sup>: 2.5-4 (3.1 nominal)</td><td>e<sup>-</sup>: 7<br>e<sup>+</sup>: 4</td></tr><tr><td>Delivered integrated<br>luminosity per exp. (fb<sup>-1</sup>)</td><td>0.8</td><td>1040</td><td>557</td><td>491</td></tr><tr><td>Luminosity (10<sup>30</sup> cm<sup>-2</sup>s<sup>-1</sup>)</td><td>75</td><td>21083</td><td>12069<br>(design: 3000)</td><td>4.71 &times; 10<sup>4</sup></td></tr><tr><td>Time between collisions (ns)</td><td>96</td><td>5.9 or 7.86</td><td>4.2</td><td>4.2</td></tr><tr><td>Full crossing angle (&mu; rad)</td><td>0</td><td>&plusmn;11000*</td><td>0</td><td>&plusmn;41500</td></tr><tr><td>Energy spread (units 10<sup>-3</sup>)</td><td>e: 0.91<br>p: 0.2</td><td>e<sup>-</sup>/e<sup>+</sup>: 0.67/0.73</td><td>e<sup>-</sup>/e<sup>+</sup>: 0.61/0.77</td><td>e<sup>-</sup>/e<sup>+</sup>: 0.64/0.81</td></tr><tr><td>Bunch length (cm)</td><td>e: 0.83<br>p: 8.5</td><td>0.65</td><td>e<sup>-</sup>/e<sup>+</sup>: 1.1/1.0</td><td>e<sup>-</sup>/e<sup>+</sup>: 0.6/0.6</td></tr><tr><td>Beam radius (&mu;m)</td><td>e: 110 (H), 30 (V)<br>p: 111 (H), 30 (V)</td><td>H: 124 (e<sup>-</sup>), 117 (e<sup>+</sup>)<br>V: 1.9</td><td>157<br>4.7</td><td>e<sup>-</sup>: 16.6 (H), 0.22 (V)<br>e<sup>+</sup>: 17.9 (H), 0.22 (V)</td></tr><tr><td>Free space at interaction<br>point (m)</td><td>&plusmn;2</td><td>+0.75/-0.58<br>(+300/-500) mrad cone</td><td>&plusmn;0.2,<br>&plusmn;300 mrad cone</td><td>e<sup>-</sup>: +1.20/-1.28<br>e<sup>+</sup>: +0.78/-0.73<br>(+300/-500) mrad cone</td></tr><tr><td>Initial luminosity decay<br>time, -L/(dL/dt) (hr)</td><td>10</td><td>continuous</td><td>continuous</td><td>continuous</td></tr><tr><td>Turn-around time (min)</td><td>e: 75, p: 135</td><td>continuous</td><td>continuous</td><td>continuous</td></tr><tr><td>Injection energy (GeV)</td><td>e: 12<br>p: 40</td><td>e<sup>-</sup>/e<sup>+</sup>: 8.0/3.5 (nominal)</td><td>e<sup>-</sup>/e<sup>+</sup>: 9.0/3.1 (nominal)</td><td>e<sup>-</sup>/e<sup>+</sup>: 7/4</td></tr><tr><td>Transverse emittance<br>(10<sup>-9</sup> m)</td><td>e: 20 (H), 3.5 (V)<br>p: 5 (H), 5 (V)</td><td>e<sup>-</sup>: 24 (57<sup>†</sup>) (H), 0.61 (V)<br>e<sup>+</sup>: 18 (55<sup>†</sup>) (H), 0.56 (V)</td><td>e<sup>-</sup>: 48 (H), 1.8 (V)<br>e<sup>+</sup>: 24 (H), 1.8 (V)</td><td>e<sup>-</sup>: 4.6 (H), 0.05 (V)<br>e<sup>+</sup>: 4.0 (H), 0.05 (V)</td></tr><tr><td>&beta;<sup>*</sup>, amplitude function at<br>interaction point (m)</td><td>e: 0.6 (H), 0.26 (V)<br>p: 2.45 (H), 0.18 (V)</td><td>e<sup>-</sup>: 1.2 (0.27<sup>†</sup>) (H), 0.0059 (V)<br>e<sup>+</sup>: 1.2 (0.23<sup>†</sup>) (H), 0.0059 (V)</td><td>e<sup>-</sup>: 0.50 (H), 0.012 (V)<br>e<sup>+</sup>: 0.50 (H), 0.012 (V)</td><td>e<sup>-</sup>: 0.060 (H), 1 &times; 10<sup>-3</sup> (V)<br>e<sup>+</sup>: 0.080 (H), 1 &times; 10<sup>-3</sup> (V)</td></tr><tr><td>Beam-beam tune shift<br>per crossing (units 10<sup>-4</sup>)</td><td>e:190 (H), 450 (V)<br>p: 12 (H), 9 (V)</td><td>e<sup>-</sup>: 1020 (H), 900 (V)<br>e<sup>+</sup>: 1270 (H), 1290 (V)</td><td>e<sup>-</sup>: 703 (H), 498 (V)<br>e<sup>+</sup>: 510 (H), 727 (V)</td><td>e<sup>-</sup>: 15 (H), 278 (V)<br>e<sup>+</sup>: 27 (H), 398 (V)</td></tr><tr><td>RF frequency (MHz)</td><td>e:499.7<br>p: 208.2/52.05</td><td>508.887</td><td>476</td><td>508.887</td></tr><tr><td>Particles per bunch<br>(units 10<sup>10</sup>)</td><td>e:3<br>p:7</td><td>e<sup>-</sup>/e<sup>+</sup>: 4.7/6.4</td><td>e<sup>-</sup>/e<sup>+</sup>: 5.2/8.0</td><td>e<sup>-</sup>/e<sup>+</sup>: 3.12/3.81</td></tr><tr><td>Bunches per ring<br>per species</td><td>e: 189<br>p: 180</td><td>1585</td><td>1732</td><td>2249</td></tr><tr><td>Average beam current<br>per species (mA)</td><td>e:40<br>p:90</td><td>e<sup>-</sup>/e<sup>+</sup>: 1188/1637</td><td>e<sup>-</sup>/e<sup>+</sup>: 1960/3026</td><td>e<sup>-</sup>/e<sup>+</sup>: 1118/1393</td></tr><tr><td>Beam polarization (%)</td><td>—</td><td>—</td><td>—</td><td>—</td></tr><tr><td>Circumference or<br>length (km)</td><td>6.336</td><td>3.016</td><td>2.2</td><td>3.016</td></tr><tr><td>Interaction regions</td><td>2 colliding beams<br>1 fixed target (e beam)</td><td>1</td><td>1</td><td>1</td></tr><tr><td>Magnetic length of dipole<br>(m)</td><td>e:9.185;<br>p: 8.82</td><td>e<sup>-</sup>/e<sup>+</sup>: 5.86/0.915</td><td>e<sup>-</sup>/e<sup>+</sup>: 5.4/0.45</td><td>e<sup>-</sup>/e<sup>+</sup>: 5.9/4.0</td></tr><tr><td>Length of standard cell<br>(m)</td><td>e:23.5<br>p:47</td><td>e<sup>-</sup>/e<sup>+</sup>: 75.7/76.1</td><td>15.2</td><td>e<sup>-</sup>/e<sup>+</sup>: 75.7/76.1</td></tr><tr><td>Phase advance per cell<br>(deg)</td><td>e:60<br>p:90</td><td>450</td><td>e<sup>-</sup>/e<sup>+</sup>: 60/90</td><td>450</td></tr><tr><td>Dipoles in ring</td><td>e: 396<br>p: 416</td><td>e<sup>-</sup>/e<sup>+</sup>: 116/112</td><td>e<sup>-</sup>/e<sup>+</sup>: 192/192</td><td>e<sup>-</sup>/e<sup>+</sup>: 116/112</td></tr><tr><td>Quadrupoles in ring</td><td>e:580<br>p:280</td><td>e<sup>-</sup>/e<sup>+</sup>: 452/452</td><td>e<sup>-</sup>/e<sup>+</sup>: 290/326</td><td>e<sup>-</sup>/e<sup>+</sup>: 466/460</td></tr><tr><td>Peak magnetic field (T)</td><td>e: 0.274; p: 5</td><td>e<sup>-</sup>/e<sup>+</sup>: 0.25/0.72</td><td>e<sup>-</sup>/e<sup>+</sup>: 0.18/0.75</td><td>e<sup>-</sup>/e<sup>+</sup>: 0.22/0.19</td></tr></tbody></table>

<footnote>*KEKB was operated with crab crossing from 2007 to 2010.</footnote>

<footnote>†With dynamic beam-beam effect.</footnote>
High-Energy Collider Parameters: $\bar{p}$, pp Colliders

<table><caption>**Table 32.4:** Updated in March 2024 with numbers received from representatives of the colliders (contact E. Pianori, LBNL). The table shows the parameter values achieved. Parameters for the defunct $Sp\bar{p}S$ collider may be found in our 2002 edition (Phys. Rev. D66, 010001 (2002)). Quantities are, where appropriate, r.m.s.; unless noted otherwise, energies refer to beam energy; H and V indicate horizontal and vertical directions; s.c. stands for superconducting.</caption><thead><tr><th></th><th>TEVATRON*</th><th>RHIC<br>Brookhaven</th><th colspan="4">LHC<br>(CERN)</th></tr></thead><tbody><tr><td>Physics start date</td><td>1987</td><td>2001</td><td>2009 (Run 1)</td><td>2015 (Run 2)</td><td>2022 (Run 3)</td><td>2029 (HL-LHC)</td></tr><tr><td>Physics end date</td><td>2011</td><td>—</td><td colspan="4">—</td></tr><tr><td>Particles collided</td><td>p<span style="text-decoration: overline;">p</span></td><td>pp (polarized)</td><td colspan="4">pp</td></tr><tr><td>Maximum beam<br>energy (TeV)</td><td>0.980</td><td>0.255<br>55% polarization</td><td>4.0</td><td>6.5</td><td>6.8</td><td>7.0</td></tr><tr><td>Max. delivered integrated<br>luminosity per exp. (fb<sup>-1</sup>)</td><td>12</td><td>0.38 at 100 GeV<br>1.3 at 250/255 GeV</td><td>23.3 at 4.0 TeV<br>6.1 at 3.5 TeV</td><td>160</td><td>70</td><td>250/y</td></tr><tr><td>Luminosity<br>(10<sup>30</sup> cm<sup>-2</sup>s<sup>-1</sup>)</td><td>431</td><td>245 (pk)<br>160 (avg)</td><td>7.7 &times; 10<sup>3</sup></td><td>2.1 &times; 10<sup>4</sup></td><td>2 &times; 10<sup>4</sup> (leveled)</td><td>5.0 &times; 10<sup>4</sup> (leveled)</td></tr><tr><td>Time between collisions (ns)</td><td>396</td><td>107</td><td>49.90</td><td>24.95</td><td>24.95</td><td>24.95</td></tr><tr><td>Full crossing angle (&mu; rad)</td><td>0</td><td>0</td><td>290</td><td>320 &rarr; 260<sup>&dagger;</sup></td><td>320 &rarr; 270<sup>&dagger;</sup></td><td>500</td></tr><tr><td>Energy spread (units 10<sup>-3</sup>)</td><td>0.14</td><td>0.15</td><td>0.1445</td><td>0.11</td><td>0.11</td><td>0.11</td></tr><tr><td>Bunch length (cm)</td><td>p: 50<br>p&#773;: 45</td><td>60</td><td>9.4</td><td>8</td><td>9.7</td><td>7.6</td></tr><tr><td>Beam radius<br>(10<sup>-6</sup> m)</td><td>p: 28<br>p&#773;: 16</td><td>85</td><td>18.8</td><td>8.5<sup>&ddagger;</sup></td><td>9.3<sup>&ddagger;</sup></td><td>7<sup>&ddagger;</sup></td></tr><tr><td>Free space at interaction<br>point (m)</td><td>&plusmn;6.5</td><td>16</td><td>38</td><td>38</td><td>38</td><td>38</td></tr><tr><td>Initial luminosity decay<br>time, -L/(dL/dt) (hr)</td><td>6 (avg)</td><td>7.5</td><td>&asymp; 6</td><td>8.4</td><td>12 <sup>&sect;</sup></td><td>&asymp; 7.5 (leveled)</td></tr><tr><td>Turn-around time (min)</td><td>90</td><td>25</td><td>180</td><td>150</td><td>180<sup>&para;</sup></td><td>150</td></tr><tr><td>Injection energy (TeV)</td><td>0.15</td><td>0.023</td><td>0.450</td><td>0.450</td><td>0.450</td><td>0.450</td></tr><tr><td>Transverse emittance<br>(10<sup>-9</sup> m)</td><td>p: 3<br>p&#773;: 1</td><td>11</td><td>0.59</td><td>0.29</td><td>0.29</td><td>0.33</td></tr><tr><td>&beta;<sup>*</sup>, ampl. function at<br>interaction point (m)</td><td>0.28</td><td>0.65</td><td>0.6</td><td>0.3 &rarr; 0.25<sup>&parallel;</sup></td><td>1.2 &rarr; 0.3<sup>&parallel;</sup></td><td>1 &rarr; 0.15<sup>&parallel;</sup></td></tr><tr><td>Beam-beam tune shift<br>per crossing (units 10<sup>-4</sup>)</td><td>p: 120<br>p&#773;: 120</td><td>73</td><td>72</td><td>45</td><td>57<sup>&sect;</sup></td><td>104</td></tr><tr><td>RF frequency (MHz)</td><td>53</td><td>accel: 9<br>store: 28</td><td>400.8</td><td>400.8</td><td>400.8</td><td>400.8</td></tr><tr><td>Particles per bunch<br>(units 10<sup>10</sup>)</td><td>p:26<br>p&#773;:9</td><td>18.5</td><td>16</td><td>11</td><td>16</td><td>22</td></tr><tr><td>Bunches per ring<br>per species</td><td>36</td><td>111</td><td>1380</td><td>2556<br>2544 (i.r. 1/5<sup>**</sup>)</td><td>2464<br>2452 (i.r. 1/5<sup>**</sup>)</td><td>2760<br>2748 (i.r. 1/5<sup>**</sup>)</td></tr><tr><td>Average beam current<br>per species (mA)</td><td>p:70<br>p&#773;:24</td><td>257</td><td>400</td><td>510</td><td>710</td><td>1094</td></tr><tr><td>Circumference (km)</td><td>6.28</td><td>3.834</td><td colspan="4">26.659</td></tr><tr><td>Interaction regions</td><td>2 high <span style="text-decoration: overline;">L</span></td><td>6 total, 2 high <span style="text-decoration: overline;">L</span></td><td colspan="4">4 total, 2 high <span style="text-decoration: overline;">L</span></td></tr><tr><td>Magnetic length of dipole (m)</td><td>6.12</td><td>9.45</td><td colspan="4">14.3</td></tr><tr><td>Length of standard cell (m)</td><td>59.5</td><td>29.7</td><td colspan="4">106.90</td></tr><tr><td>Phase advance per cell (deg)</td><td>67.8</td><td>84</td><td colspan="4">90</td></tr><tr><td>Dipoles in ring</td><td>774</td><td>192 per ring<br>+ 12 common</td><td colspan="4">1232<br>main dipoles</td></tr><tr><td>Quadrupoles in ring</td><td>216</td><td>246 per ring</td><td colspan="4">482 2-in-1<br>24 1-in-1</td></tr><tr><td>Magnet types</td><td>s.c., cos&theta;<br>warm iron</td><td>s.c., cos&theta;<br>cold iron</td><td colspan="4">s.c., 2-in-1<br>cold iron</td></tr><tr><td>Peak magnetic field (T)</td><td>4.4</td><td>3.5</td><td colspan="4">8.3<sup>&dagger;&dagger;</sup></td></tr></tbody></table>

<footnote>*Other TEVATRON parameters: $\bar{p}$ source accum. rate: $25 \times 10^{10}$ hr$^{-1}$; max. no. of $\bar{p}$ stored: $3.4 \times 10^{12}$ (Accumulator), $6.1 \times 10^{12}$ (Recycler).</footnote>

<footnote>†Variable crossing angle decreasing during the fill with the reduction in bunch population</footnote>

<footnote>‡Minimum beam radius during levelling</footnote>

<footnote>§Value at start of levelling</footnote>

<footnote>¶Most likely value as observed in 2023</footnote>

<footnote>‖$\beta^*$ levelling</footnote>

<footnote>**Number of bunches colliding at the interaction regions (i.r.) 1 (ATLAS) and 5 (CMS).</footnote>

<footnote>††Value for design beam energy of 7 TeV.</footnote>
High-Energy Collider Parameters: Heavy Ion Colliders

<table><caption>**Table 32.5:** Updated in March 2024 with numbers received from representatives of the collider (contact E. Pianori, LBNL) The table shows the parameter values achieved. For the LHC, only maximum values for the ATLAS and CMS experiments are provided (ALICE and LHCb have different requirements for energy and luminosity). Design values for a high-luminosity upgrade are also given. Quantities are, where appropriate, r.m.s.; unless noted otherwise, energies refer to beam energy; s.c. stands for superconducting. pk and avg denote peak and average values.</caption><thead><tr><th></th><th colspan="3">RHIC<br>(Brookhaven)</th><th colspan="4">LHC<br>(CERN)</th></tr><tr><th></th><th>2000</th><th>2012 / 2018 / 2018 / 2012 / 2004<br>2021 / 2014 / 2002 / 2015 / 2015</th><th>2010</th><th>2012</th><th>2017</th><th>≥ 2029<br>(high lum.)*</th></tr></thead><tbody><tr><td>Physics start date</td><td>2000</td><td>—</td><td>—</td><td>—</td><td>—</td><td>—</td></tr><tr><td>Physics end date</td><td>—</td><td>—</td><td>—</td><td>—</td><td>—</td><td>—</td></tr><tr><td>Particles collided</td><td>Au Au</td><td>U U / Zr / Ru / Cu Au / Cu Cu<br>O O / h Au / d Au / p Au / p Al</td><td>Pb Pb</td><td>p Pb</td><td>Xe Xe</td><td>Pb Pb</td></tr><tr><td>Max. beam energy (TeV/n)</td><td>0.1</td><td>0.1</td><td>2.68</td><td>p:6.5<br>Pb:2.56</td><td>2.72</td><td>2.76</td></tr><tr><td>√s<sub>NN</sub> (TeV)</td><td>0.2</td><td>0.2</td><td>5.36</td><td>8.16</td><td>5.44</td><td>5.5</td></tr><tr><td>Max. delivered int. nucleon-<br>pair lumin. per exp. (pb<sup>-1</sup>)</td><td>2639<br>(at 100 GeV/n)</td><td>21 / 36 / 36.9 / 167 / 60<br>8.2 / 43 / 169 / 124 / 63<br>(all at 100 GeV/n)</td><td>84.8</td><td>194</td><td>0.05</td><td>≈ 121/y</td></tr><tr><td>Luminosity<br>(10<sup>27</sup> cm<sup>-2</sup>s<sup>-1</sup>)</td><td>pk: 15.5<br>avg: 8.7</td><td>pk: 0.4 / 4.8 / 3.8 / 12 / 21<br>450 / 170 / 850 / 880 / 7600<br>avg: 0.6 / 2.2 / 2.1 / 10 / 8<br>230 / 100 / 500 / 450 / 3800</td><td>6.4<sup>†</sup></td><td>900</td><td>0.4</td><td>6.4 (leveled)</td></tr><tr><td>Time between<br>collisions (ns)</td><td>107</td><td>107 / 107 / 107 / 107 / 321<br>107 / 107 / 107 / 107 / 107</td><td>49.9</td><td>99.8 / 149.7</td><td>≈ 5500</td><td>49.9</td></tr><tr><td>Full crossing angle (μ rad)</td><td>0</td><td>0</td><td>200</td><td>280</td><td>300</td><td>340</td></tr><tr><td>Energy spread (units 10<sup>-3</sup>)</td><td>0.75</td><td>0.75</td><td>0.13</td><td>0.11</td><td>0.11</td><td>0.11</td></tr><tr><td>Bunch length (cm)</td><td>30</td><td>30</td><td>9.1</td><td>p / Pb: 9 / 11.5</td><td>11</td><td>7.9</td></tr><tr><td>Beam radius<br>(10<sup>-6</sup> m)</td><td>114<sup>‡</sup></td><td>123<sup>‡</sup> / 87<sup>‡</sup> / 88<sup>‡</sup> / 163<sup>‡</sup> / 145<sup>‡</sup><br>100<sup>‡</sup> / 136<sup>‡</sup> / 124<sup>‡</sup> / 147<sup>‡</sup> / 128<sup>‡</sup></td><td>20</td><td>19</td><td>12</td><td>17</td></tr><tr><td>Free space at inter. point (m)</td><td>16</td><td>16</td><td>38</td><td>38</td><td>38</td><td>38</td></tr><tr><td>Initial luminosity decay<br>time, -L/(dL/dt) (hr)</td><td>1</td><td>-0.35<sup>§</sup> / ∞<sup>¶</sup> / ∞<sup>¶</sup> / ∞<sup>§</sup> / 1.8<br>∞ / 0.6 / ∞<sup>§</sup> / 0.5 / 0.25</td><td>2.8<sup>||</sup></td><td>≈ 2</td><td>≈ 6</td><td>∞</td></tr><tr><td>Turn-around time (min)</td><td>30</td><td>60<sup>**</sup> / 40<sup>**</sup> / 40<sup>**</sup> / 160<sup>**</sup> / 90<sup>**</sup><br>45<sup>**</sup> / 45<sup>**</sup> / 90<sup>**</sup> / 60<sup>**</sup> / 50<sup>**</sup></td><td>144</td><td>150</td><td>180</td><td>≈ 200</td></tr><tr><td>Injection energy (TeV/n)</td><td>0.011</td><td>0.011</td><td>0.177</td><td>p / Pb: 0.45 / 0.177</td><td>0.188</td><td>0.177</td></tr><tr><td>Transverse emittance<br>(10<sup>-9</sup> m)</td><td>19<sup>‡</sup></td><td>22<sup>‡</sup> / 10.7<sup>‡</sup> / 11.2<sup>‡</sup> / 38<sup>‡</sup> / 23<sup>‡</sup><br>12<sup>‡</sup> / 19<sup>‡</sup> / 22<sup>‡</sup> / 26<sup>‡</sup> / 21<sup>‡</sup></td><td>0.80</td><td>0.29</td><td>0.3</td><td>0.5</td></tr><tr><td>β<sup>*</sup>, ampl. function at<br>interaction point (m)</td><td>0.7</td><td>0.7 / 0.7 / 0.7 / 0.7 / 0.9<br>0.7 / 1.0 / 0.7 / 0.8 / 0.8</td><td>0.5</td><td>0.5</td><td>0.4</td><td>0.5</td></tr><tr><td>Beam-beam tune shift<br>per crosssing (units 10<sup>-4</sup>)</td><td>39<sup>‡</sup></td><td>6<sup>‡</sup> / 18<sup>‡</sup> / 21<sup>‡</sup> / 14<sup>‡</sup>, 14<sup>‡</sup> / 30<sup>‡</sup> / 34<sup>‡</sup><br>42<sup>‡</sup>, 22<sup>‡</sup> / 40<sup>‡</sup>, 27<sup>‡</sup> / 53<sup>‡</sup>, 41<sup>‡</sup> / 80<sup>‡</sup>, 59<sup>‡</sup></td><td>3.1</td><td>15</td><td>≈ 10</td><td>4.3</td></tr><tr><td>RF frequency (MHz)</td><td></td><td>accel: 28, store: 197</td><td>400.8</td><td>400.8</td><td>400.8</td><td>400.8</td></tr><tr><td>Particles per bunch<br>(units 10<sup>10</sup>)</td><td>0.20</td><td>0.03 / 0.1 / 0.1 / 0.4, 0.13 / 0.45 / 0.85<br>4.5, 0.13 / 13, 0.20 / 22.5, 0.16 / 24, 1.1</td><td>0.018<br>(r.m.s.)</td><td>p:2.6 Pb:0.022</td><td>0.027</td><td>0.018</td></tr><tr><td>Bunches per ring<br>per species</td><td>111</td><td>111 / 111 / 111 / 111 / 37<br>106 / 111 / 111 / 111 / 111</td><td>960<sup>††</sup></td><td>p:540<br>Pb:684</td><td>16</td><td>1240</td></tr><tr><td>Average beam current<br>per species (mA)</td><td>224</td><td>38 / 56 / 61 / 160,138 / 60 / 95<br>125,143 / 181,213 / 313,176 / 334,199</td><td>25.5</td><td>p:16<br>Pb:15</td><td>0.54</td><td>33</td></tr><tr><td>Circumference (km)</td><td></td><td>3.834</td><td></td><td></td><td>26.659</td><td></td></tr><tr><td>Interaction regions</td><td></td><td>6 total, 2 high Σ</td><td></td><td></td><td>4 total, 3 high Σ</td><td></td></tr><tr><td>Magnetic length of dipole (m)</td><td></td><td>9.45</td><td></td><td></td><td>14.3</td><td></td></tr><tr><td>Length of standard cell (m)</td><td></td><td>29.7</td><td></td><td></td><td>106.90</td><td></td></tr><tr><td>Phase advance per cell (deg)</td><td>93</td><td>84 / 84 / 84 / 84 / 84 / 84<br>93 / 84(d), 93 / 84(p), 93 / 84(p), 93</td><td></td><td></td><td>90</td><td></td></tr><tr><td>Dipoles in ring</td><td></td><td>192 per ring, + 12 common</td><td></td><td></td><td>1232, main dipoles</td><td></td></tr><tr><td>Quadrupoles in ring</td><td></td><td>246 per ring</td><td></td><td></td><td>482 2-in-1, 24 1-in-1</td><td></td></tr><tr><td>Magnet Type</td><td></td><td>s.c. cosθ, cold iron</td><td></td><td></td><td>s.c., 2 in 1, cold iron</td><td></td></tr><tr><td>Peak magnetic field (T)</td><td></td><td>3.5</td><td></td><td></td><td>8.3</td><td></td></tr></tbody></table>

<footnote>*For HL-LHC. Conservative estimates.</footnote>

<footnote>†Highest value achieved in ALICE.</footnote>

<footnote>‡Initial value, increases (descreases) without (with) cooling.</footnote>

<footnote>§Negative or infinite decay time is effect of cooling.</footnote>

<footnote>¶Luminosity leveled to flat after set to target value, with cooling.</footnote>

<footnote>||for a typical good fill without luminosity levelling.</footnote>

<footnote>**measured minimum, not theoretical.</footnote>

<footnote>††1240 bunches per ring were also used, but with lower peak performance due to lower bunch intensity.</footnote>
## 33. Neutrino Beam Lines at High-Energy Proton Synchrotrons

Revised August 2023 with numbers verified by representatives of the synchrotrons (contact C.-J. Lin, LBNL). For existing (future) neutrino beam lines the latest achieved (design) values are given.

The main source of neutrinos at proton synchrotrons is from the decay of pions and kaons produced by protons striking a nuclear target. There are different schemes to focus the secondary particles to enhance neutrino flux and/or tune the neutrino energy profile. In wide-band beams (WBB), the neutrino parent mesons are focused over a wide momentum range to obtain maximum neutrino intensity. In narrow-band beams (NBB), the secondary particles are first momentum-selected to produce a monochromatic parent beam. Another approach to generate a narrow-band neutrino spectrum is to select neutrinos that are emitted off-axis relative to the momentum of the parent mesons. For a comprehensive review of the topic, including other historical neutrino beam lines, see the article by S. E. Kopp, "Accelerator-based neutrino beams," Phys. Rept. 439, 101 (2007).

<table><thead><tr><th></th><th colspan="5">PS<br>(CERN)</th><th colspan="3">SPS<br>(CERN)</th><th>PS<br>(KEK)</th><th colspan="2">Main Ring<br>(JPARC)</th></tr><tr><th></th><th>1963</th><th>1969</th><th>1972</th><th>1983</th><th>1977</th><th>1995</th><th>2006</th><th>1999</th><th>2023</th><th>(2028)</th></tr></thead><tbody><tr><td>Date</td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td></tr><tr><td>Proton Kinetic Energy (GeV)</td><td>20.6</td><td>20.6</td><td>26</td><td>19</td><td>350</td><td>450</td><td>400</td><td>12</td><td>30</td><td>(30)</td></tr><tr><td>Protons per Cycle (10<sup>12</sup>)</td><td>0.7</td><td>0.6</td><td>5</td><td>5</td><td>10</td><td>36</td><td>48</td><td>6</td><td>153</td><td>(330)</td></tr><tr><td>Cycle Time (s)</td><td>3</td><td>2.3</td><td>-</td><td>-</td><td>-</td><td>14.4</td><td>6</td><td>2.2</td><td>1.36</td><td>(1.16)</td></tr><tr><td>Beam Power (kW)</td><td>0.8</td><td>0.9</td><td>-</td><td>-</td><td>-</td><td>180</td><td>510</td><td>5</td><td>540</td><td>(1300)</td></tr><tr><td>Target</td><td>-</td><td>-</td><td>-</td><td>-</td><td>-</td><td>Be</td><td>Graphite</td><td>Al</td><td>Graphite</td><td>(Graphite)</td></tr><tr><td>Target Length (cm)</td><td>-</td><td>-</td><td>-</td><td>-</td><td>-</td><td>290</td><td>130</td><td>66</td><td>91</td><td>(91)</td></tr><tr><td>Secondary Focussing</td><td>1-horn WBB</td><td>3-horn WBB</td><td>2-horn WBB</td><td>bare target</td><td>2-horn WBB</td><td>2-horn WBB</td><td>2-horn WBB</td><td>2-horn WBB</td><td>3-horn off-axis</td><td>(3-horn off-axis)</td></tr><tr><td>Decay Pipe Length (m)</td><td>-</td><td>-</td><td>-</td><td>-</td><td>-</td><td>110</td><td>1090</td><td>200</td><td>96</td><td>(96)</td></tr><tr><td>⟨E<sub>ν</sub>⟩ (GeV)</td><td>1.5</td><td>1.5</td><td>1.5</td><td>1</td><td>20</td><td>24.3</td><td>17</td><td>1.3</td><td>0.6</td><td>(0.6)</td></tr><tr><td>Experiments</td><td>HLBC, Spark Ch.</td><td>HLBC, Spark Ch.</td><td>GGM, Aachen-Padova</td><td>CDHS, CHARM</td><td>GGM,CDHS, CHARM, BEBC</td><td>NOMAD, CHORUS</td><td>OPERA, ICARUS</td><td>K2K</td><td>T2K</td><td>Hyper-K</td></tr></tbody></table>

<table><thead><tr><th></th><th colspan="5">Main Ring<br>(Fermilab)</th><th>Booster<br>(Fermilab)</th><th colspan="4">Main Injector<br>(Fermilab)</th></tr><tr><th></th><th>1974</th><th>1979</th><th>1976</th><th>1991</th><th>1998</th><th>2002, (2022)</th><th>2005</th><th>2017</th><th>2021</th><th>(2031)</th></tr></thead><tbody><tr><td>Date</td><td>1974</td><td>1979</td><td>1976</td><td>1991</td><td>1998</td><td>8</td><td>120</td><td>120</td><td>120</td><td>(60 - 120)</td></tr><tr><td>Proton Kinetic Energy (GeV)</td><td>300</td><td>400</td><td>350</td><td>800</td><td>800</td><td>8</td><td>120</td><td>120</td><td>120</td><td>(60 - 120)</td></tr><tr><td>Protons per Cycle (10<sup>12</sup>)</td><td>10</td><td>10</td><td>13</td><td>10</td><td>12</td><td>4.5</td><td>37</td><td>54</td><td>55</td><td>(75)</td></tr><tr><td>Cycle Time (s)</td><td>-</td><td>-</td><td>-</td><td>60</td><td>60</td><td>0.2</td><td>2</td><td>1.333</td><td>1.2</td><td>(1.2)</td></tr><tr><td>Beam Power (kW)</td><td>-</td><td>-</td><td>-</td><td>20</td><td>25</td><td>29</td><td>350</td><td>720</td><td>840</td><td>(1200)</td></tr><tr><td>Target</td><td>-</td><td>-</td><td>-</td><td>-</td><td>BeO</td><td>Be</td><td>Graphite</td><td>Graphite</td><td>Graphite</td><td>(Graphite)</td></tr><tr><td>Target Length (cm)</td><td>-</td><td>-</td><td>-</td><td>-</td><td>31</td><td>71</td><td>95</td><td>120</td><td>120</td><td>(150-180)</td></tr><tr><td>Secondary Focussing</td><td>dichromatic NBB</td><td>2-horn WBB</td><td>1-horn WBB</td><td>quad trip.</td><td>SSQT WBB</td><td>1-horn WBB</td><td>2-horn WBB</td><td>2-horn off-axis</td><td>2-horn off-axis</td><td>(3-horn WBB)</td></tr><tr><td>Decay Pipe Length (m)</td><td>400</td><td>400</td><td>400</td><td>400</td><td>400</td><td>50</td><td>675</td><td>675</td><td>675</td><td>(220)</td></tr><tr><td>⟨E<sub>ν</sub>⟩ (GeV)</td><td>50,180<sup>†</sup></td><td>25</td><td>100</td><td>90,260</td><td>70,180</td><td>1</td><td>3-20<sup>‡</sup></td><td>2</td><td>2</td><td>(2.5)</td></tr><tr><td>Experiments</td><td>CITF, HPWF, 15' BC</td><td>HPWF, 15' BC</td><td>15' BC, CCFRR</td><td>NuTeV</td><td>MiniBooNE, SciBooNE, MicroBooNE, (SBND,ICARUS)</td><td>MINOS, MINERνA</td><td>NOνA, MINERνA, MINOS+</td><td>NOνA</td><td>DUNE</td><td></td></tr></tbody></table>

<footnote>†Pion and kaon peaks in the momentum-selected channel.</footnote>

<footnote>‡Tunable WBB energy spectrum.</footnote>
# 34. Passage of Particles Through Matter

Revised August 2023 by D.E. Groom (LBNL) and S.R. Klein (NSD LBNL; UC Berkeley).

<table><caption>**Table 34.1:** Summary of variables used in this section. The kinematic variables $\beta$ and $\gamma$ have their usual relativistic meanings.</caption><tbody><tr><td>34.1</td><td>Notation</td><td>575</td></tr><tr><td>34.2</td><td>Electronic energy loss by heavy particles</td><td>575</td></tr><tr><td>34.2.1</td><td>Moments and cross sections</td><td>575</td></tr><tr><td>34.2.2</td><td>Maximum energy transfer to an electron in a single collision</td><td>576</td></tr><tr><td>34.2.3</td><td>Stopping power at intermediate energies</td><td>576</td></tr><tr><td>34.2.4</td><td>Mean excitation energy</td><td>576</td></tr><tr><td>34.2.5</td><td>Density effect</td><td>576</td></tr><tr><td>34.2.6</td><td>Energy loss at low energies</td><td>577</td></tr><tr><td>34.2.7</td><td>Energetic knock-on electrons (δ rays)</td><td>579</td></tr><tr><td>34.2.8</td><td>Restricted energy loss rates for relativistic ionizing particles</td><td>579</td></tr><tr><td>34.2.9</td><td>Fluctuations in energy loss</td><td>579</td></tr><tr><td>34.2.10</td><td>Energy loss in mixtures and compounds</td><td>580</td></tr><tr><td>34.2.11</td><td>Ionization yields</td><td>580</td></tr><tr><td>34.3</td><td>Multiple scattering through small angles</td><td>580</td></tr><tr><td>34.4</td><td>Photon and electron interactions in matter</td><td>581</td></tr><tr><td>34.4.1</td><td>Collision energy losses by $e^{\pm}$</td><td>581</td></tr><tr><td>34.4.2</td><td>Radiation length</td><td>582</td></tr><tr><td>34.4.3</td><td>Bremsstrahlung energy loss by $e^{\pm}$</td><td>582</td></tr><tr><td>34.4.4</td><td>Critical energy</td><td>583</td></tr><tr><td>34.4.5</td><td>Energy loss by photons</td><td>583</td></tr><tr><td>34.4.6</td><td>Bremsstrahlung and pair production at very high energies</td><td>584</td></tr><tr><td>34.4.7</td><td>Photonuclear and electronuclear interactions at still higher energies</td><td>585</td></tr><tr><td>34.5</td><td>Electromagnetic cascades</td><td>585</td></tr><tr><td>34.6</td><td>Muon energy loss at high energy</td><td>586</td></tr><tr><td>34.7</td><td>Cherenkov and transition radiation</td><td>587</td></tr><tr><td>34.7.1</td><td>Optical Cherenkov radiation</td><td>587</td></tr><tr><td>34.7.2</td><td>Coherent radio Cherenkov radiation</td><td>588</td></tr><tr><td>34.7.3</td><td>Transition radiation</td><td>588</td></tr></tbody></table>

This review covers the interactions of photons and electrically charged particles in matter, concentrating on energies of interest for high-energy physics and astrophysics and processes of interest for particle detectors (ionization, Cherenkov radiation, transition radiation). Much of the focus is on particles heavier than electrons ($\pi^{\pm}$, $p$, etc.). Although the charge number $z$ of the projectile is included in the equations, only $z=1$ is discussed in detail. Muon radiative losses are discussed, as are photon/electron interactions at high to ultrahigh energies. Neutrons are not discussed.

## 34.1 Notation

The notation and important numerical values are shown in Table 34.1.

## 34.2 Electronic energy loss by heavy particles

### 34.2.1 Moments and cross sections

The electronic interactions of fast charged particles with speed $v = \beta c$ occur in single collisions with energy losses $W$ [1], leading to ionization, atomic, or collective excitation. Most frequently the energy losses are small (for 90% of all collisions the energy losses are less than 100 eV). In thin absorbers few collisions will take place and the total energy loss will show a large variance [1]; also see Sec. 34.2.9 below. For particles with charge $ze$ more massive than electrons ("heavy" particles), scattering from free electrons is adequately described by the Rutherford differential cross section [2,3].

$$ \frac{d\sigma_R(W; \beta)}{dW} = \frac{2\pi r_e^2 m_e c^2 z^2 (1 - \beta^2 W/W_{\text{max}})}{\beta^2 W^2}, \quad (34.1) $$

where $W_{\text{max}}$, the maximum energy transfer possible in a single collision, is discussed below. It differs from the classical cross section by the factor $(1 - \beta^2 W/W_{\text{max}})$, which arises when the spin of the target electrons is taken into account.

<table><thead><tr><th>Symb.</th><th>Definition</th><th>Value or (usual) units</th></tr></thead><tbody><tr><td><i>m</i><sub>e</sub><i>c</i><sup>2</sup></td><td>electron mass &times; <i>c</i><sup>2</sup></td><td>0.510 998 950 00(15) MeV</td></tr><tr><td><i>r</i><sub>e</sub></td><td>classical electron radius</td><td></td></tr><tr><td></td><td><i>e</i><sup>2</sup>/4&pi;&epsilon;<sub>0</sub><i>m</i><sub>e</sub><i>c</i><sup>2</sup></td><td>2.817 940 3227(19) fm</td></tr><tr><td>&alpha;</td><td>fine structure constant</td><td></td></tr><tr><td></td><td><i>e</i><sup>2</sup>/4&pi;&epsilon;<sub>0</sub>&hbar;<i>c</i></td><td>1/137.035 999 139(31)</td></tr><tr><td><i>N</i><sub>A</sub></td><td>Avogadro's number</td><td>6.022 140 76 &times; 10<sup>23</sup> mol<sup>-1</sup></td></tr><tr><td>&rho;</td><td>density</td><td>g cm<sup>-3</sup></td></tr><tr><td><i>x</i></td><td>mass per unit area</td><td>g cm<sup>-2</sup></td></tr><tr><td><i>M</i></td><td>incident particle mass</td><td>MeV/c<sup>2</sup></td></tr><tr><td><i>E</i></td><td>incident part. energy &gamma;<i>M</i><i>c</i><sup>2</sup></td><td>MeV</td></tr><tr><td><i>T</i></td><td>kinetic energy, (&gamma; &minus; 1)<i>M</i><i>c</i><sup>2</sup></td><td>MeV</td></tr><tr><td><i>W</i></td><td>energy transfer to an electron<br>in a single collision</td><td>MeV</td></tr><tr><td><i>W</i><sub>max</sub></td><td>Maximum possible energy transfer<br>to an electron in a single collision</td><td>MeV</td></tr><tr><td><i>k</i></td><td>bremsstrahlung photon energy</td><td>MeV</td></tr><tr><td><i>z</i></td><td>charge number of incident particle</td><td></td></tr><tr><td><i>Z</i></td><td>atomic number of absorber</td><td></td></tr><tr><td><i>A</i></td><td>atomic mass of absorber</td><td>g mol<sup>-1</sup></td></tr><tr><td><i>K</i></td><td>4&pi;<i>N</i><sub>A</sub><i>r</i><sup>2</sup><i>m</i><sub>e</sub><i>c</i><sup>2</sup><br>(Coefficient for <i>dE</i>/<i>dx</i>)</td><td>0.307 075 MeV mol<sup>-1</sup> cm<sup>2</sup></td></tr><tr><td><i>I</i></td><td>mean excitation energy</td><td>eV (Nota bene!)</td></tr><tr><td>&delta;(&beta;&gamma;)</td><td>density effect correction to ionization energy loss</td><td></td></tr><tr><td>&hbar;&omega;<sub>p</sub></td><td>plasma energy</td><td>&radic;<span style="text-decoration: overline;">&rho;&lang;Z/A&rang; &times; 28.816 eV</span></td></tr><tr><td></td><td>&radic;<span style="text-decoration: overline;">4&pi;N<sub>e</sub>r<sub>e</sub><sup>3</sup><i>m</i><sub>e</sub><i>c</i><sup>2</sup>/&alpha;</span></td><td>&lceil; &rho; in g cm<sup>-3</sup></td></tr><tr><td><i>N</i><sub>e</sub></td><td>electron density</td><td>(units of <i>r</i><sub>e</sub>)<sup>-3</sup></td></tr><tr><td><i>w</i><sub>j</sub></td><td>weight fraction of the <i>j</i>th element in a compound or mixture.</td><td></td></tr><tr><td><i>n</i><sub>j</sub></td><td>&propto; number of <i>j</i>th kind of atoms in a compound or mixture</td><td></td></tr><tr><td><i>X</i><sub>0</sub></td><td>radiation length</td><td>g cm<sup>-2</sup></td></tr><tr><td><i>E</i><sub>c</sub></td><td>critical energy for electrons</td><td>MeV</td></tr><tr><td><i>E</i><sub>&mu;c</sub></td><td>critical energy for muons</td><td>GeV</td></tr><tr><td><i>E</i><sub>s</sub></td><td>scale energy &radic;<span style="text-decoration: overline;">4&pi;/&alpha; <i>m</i><sub>e</sub><i>c</i><sup>2</sup></span></td><td>21.2052 MeV</td></tr><tr><td><i>R</i><sub>M</sub></td><td>Molière radius</td><td>g cm<sup>-2</sup></td></tr></tbody></table>

Bethe's original theory applied only to energies above which atomic effects are not important. The free-electron cross section (Eq. (34.1)) was used to extend the cross section to $W_{\text{max}}$. This free-electron approximation is not valid if $W$ is not large compared to electron binding energies. For this energy regime Bethe [4, 5] used "Born Theorie" to obtain the differential cross section

$$ \frac{d\sigma_B(W; \beta)}{dW} = \frac{d\sigma_R(W, \beta)}{dW} B(W). \quad (34.2) $$

Electronic binding is accounted for by the correction factor $B(W)$. Examples of $B(W)$ and $d\sigma_B/dW$ can be seen in Figs. 5 and 6 of Ref. [1]. For a given material, the correction results in introducing an effective ionization energy $I$ "which is a geometric average of the excitation energies of the medium weighed by the corresponding oscillator strength." [6]. The nontrivial task of finding these values is discussed in Sec. 34.2.4.

At high energies the stopping power is further modified by polarization of the medium, and this "density effect," discussed in Sec. 34.2.5, must also be included.

The mean number of collisions with energy loss between $W$ and $W + dW$ occurring in a distance $\delta x$ is $N_e\delta x$ ($d\sigma/dW)dW$, where $d\sigma(W; \beta)/dW$, where the cross section is the Rutherford formula if free electrons can be assumed and the Bethe form where binding energy is important. It is convenient to define the moments

$$ M_j(\beta) = N_e \delta x \int W^j \frac{d\sigma(W; \beta)}{dW} dW, \quad (34.3) $$

so that $M_0$ is the mean number of collisions in $\delta x$, $M_1$ is the
mean energy loss in $\delta x$, $(M_2 - M_1)^2$ is the variance, etc. The number of collisions is Poisson-distributed with mean $M_0$. $N_e$ is either measured in electrons/g ($N_e = N_A Z/A$) or electrons/cm$^3$ ($N_e = N_A \rho Z/A$). The former is used throughout this chapter, since quantities of interest ($dE/dx$, $X_0$, etc.) vary smoothly with composition when there is no density dependence.

### 34.2.2 Maximum energy transfer to an electron in a single collision

For a point-like particle with mass $M \gg m_e$,

$$W_{\text{max}} = \frac{2m_e c^2 \beta^2 \gamma^2}{1 + 2\gamma m_e / M + (m_e / M)^2} \quad (34.4)$$

In older references [2, 9] the “low-energy” approximation $W_{\text{max}} = 2m_e c^2 \beta^2 \gamma^2$, valid for $2\gamma m_e \ll M$, is often implicit. For a pion in copper, the error thus introduced into $dE/dx$ is greater than 6% at 100 GeV. For $2\gamma m_e \gg M$, $W_{\text{max}} = Mc^2 \beta^2 \gamma$.

At energies of order 100 GeV, the maximum 4-momentum transfer to the electron can exceed 1 GeV/c, where hadronic structure effects modify the cross sections. This problem has been investigated by J.D. Jackson [10], who concluded that for incident hadrons (but not for large nuclei) corrections to $dE/dx$ are negligible below energies where radiative effects dominate. While the cross section for rare hard collisions is modified, the average stopping power, dominated by many softer collisions, is almost unchanged.

### 34.2.3 Stopping power at intermediate energies

The mean rate of energy loss by moderately relativistic charged heavy particles is well described by the “Bethe equation” [2, 4, 5, 9],

$$\left\langle -\frac{dE}{dx} \right\rangle = K z^2 \frac{Z}{A} \frac{1}{\beta^2} \left[ \frac{1}{2} \ln \frac{2m_e c^2 \beta^2 \gamma^2 W_{\text{max}}}{I^2} - \beta^2 - \frac{\delta(\beta\gamma)}{2} \right] \quad (34.5)$$

Eq. (34.5) is valid in the region $0.1 \lesssim \beta\gamma \lesssim 1000$ with an accuracy of a few percent. At $\beta\gamma \sim 0.1$ the projectile speed is comparable to atomic electron “speed,” and at $\beta\gamma \sim 1000$ radiative effects begin to be important (Sec. 34.6). Both limits are $Z$ dependent. A minor dependence on $M$ at high energies is introduced through $W_{\text{max}}$, but for all practical purposes the stopping power in a given material is a function of $\beta$ alone. Small corrections are discussed in Sec. 34.2.6.<fnref n="1" />,<fnref n="2" />

This is the mass stopping power; with the symbol definitions and values given in Table 34.1, the units are MeV g$^{-1}$cm$^2$. As can be seen from Fig. 34.2, $dE/dx$ defined in this way is about the same for most materials, decreasing slowly with $Z$. The linear stopping power, in MeV/cm, is $\rho dE/dx$, where $\rho$ is the density in g/cm$^3$.

The stopping power at first falls as $1/\beta^\alpha$ where $\alpha \approx 1.7-1.5$, decreasing with increasing $Z$, and reaches a broad minimum at $\beta\gamma = 3.8-3.0$ as $Z$ rises from 6 to 82. It then inexorably rises as the argument of the logarithmic term increases. Two independent mechanisms contribute. Two thirds of the rise is produced by the explicit $\beta^2\gamma^2$ dependence through the relativistic flattening and extension of the particle’s electric field. Rather than producing ionization at greater and greater distances, the field polarizes the medium, cancelling the increase in the logarithmic term at high energies. This is taken into account by the density-effect correction $\delta(\beta\gamma)$. The other third is introduced by the $\beta^2\gamma$ dependence of $W_{\text{max}}$, the maximum possible energy transfer to a recoil electron. “Hard collision” events increasingly extend the tail of the energy loss distribution, increasing the mean but with little effect on the position of the maximum, the most probable energy loss.

Few concepts in high-energy physics are as misused as $dE/dx$, since the mean is weighted by rare events with large single-collision energy losses. Even with samples of hundreds of events in a typical detector, the mean energy loss cannot be obtained

dependably. Far better and more easily measured is the most probable energy loss, discussed in Sec. 34.2.9. The most probable energy loss in a typical detector is considerably smaller than the mean given by the Bethe equation. It does not continue to rise with the mean stopping power, but approaches a “Fermi plateau.”

In analysing TPC data (Sec. 35.6.5), the same end is often accomplished by using a restricted energy loss, the mean of 50%–70% of the samples with the smallest signals as the estimator.

Although it must be used with cautions and caveats, $dE/dx$ as described in Eq. (34.5) still forms the basis of much of our understanding of energy loss by charged particles. Extensive tables are available [6, 7] and [pdg.lbl.gov/current/AtomicNuclearProperties/](pdg.lbl.gov/current/AtomicNuclearProperties/).

For heavy projectiles, like ions, additional terms are required to account for higher-order photon coupling to the target, and to account for the finite target radius. These can change $dE/dx$ by a factor of two or more for the heaviest nuclei in certain kinematic regimes [11].

The function as computed for muons on copper is shown as the “Bethe” region of Fig. 34.1. Mean energy loss behavior below this region is discussed in Sec. 34.2.6, and the radiative effects at high energy are discussed in Sec. 34.6. Only in the Bethe region is it a function of $\beta$ alone; the mass dependence is more complicated elsewhere. The stopping power in several other materials is shown in Fig. 34.2. Except in hydrogen, particles with the same speed have similar rates of energy loss in different materials, although there is a slow decrease in the rate of energy loss with increasing $Z$. The qualitative behavior difference at high energies between a gas (He in the figure) and the other materials shown in the figure is due to the density-effect correction, $\delta(\beta\gamma)$, discussed in Sec. 34.2.5. The stopping power functions are characterized by broad minima whose position drops from $\beta\gamma = 3.5$ to 3.0 as $Z$ goes from 7 to 100. The values of minimum ionization as a function of atomic number are shown in Fig. 34.3.

In practical cases, most relativistic particles (e.g., cosmic-ray muons) have mean energy loss rates close to the minimum; they are “minimum-ionizing particles,” or mip’s.

Eq. (34.5) may be integrated to find the total (or partial) “continuous slowing-down approximation” (CSDA) range $R$ for a particle which loses energy only through ionization and atomic excitation. Since $dE/dx$ depends only on $\beta$, $R/M$ is a function of $E/M$ or $pc/M$. In practice, range is a useful concept only for low-energy hadrons ($R \lesssim \lambda_I$, where $\lambda_I$ is the nuclear interaction length), and for muons below a few hundred GeV (above which radiative effects dominate). Fig. 34.4 shows $R/M$ as a function of $\beta\gamma (= p/Mc)$ for a variety of materials.

The mass scaling of $dE/dx$ and range is valid for the electronic losses described by the Bethe equation, but not for radiative losses.

### 34.2.4 Mean excitation energy

“The determination of the mean excitation energy is the principal non-trivial task in the evaluation of the Bethe stopping-power formula” [15]. Recommended values have varied substantially with time. Estimates based on experimental stopping-power measurements for protons, deuterons, and alpha particles and on oscillator-strength distributions and dielectric-response functions were given in ICRU 49 [6]. See also ICRU 37 [12]. These values, shown in Fig. 34.5, have since been widely used. Machine-readable versions can also be found [16].

### 34.2.5 Density effect

As the particle energy increases, its electric field flattens and extends, so that the distant-collision contribution to the logarithmic term in Eq. (34.5) increases as $\beta^2\gamma^2$. However, real media become polarized, limiting the field extension and effectively truncating this part of the logarithmic rise [2, 3, 6, 17, 18]. At very high energies,

$$\delta(\beta\gamma)/2 \rightarrow \ln(\hbar\omega_p/I) + \ln\beta\gamma - 1/2, \quad (34.6)$$

where $\delta(\beta\gamma)/2$ is the density effect correction introduced in Eq. (34.5) and $\hbar\omega_p$ is the plasma energy defined in Table 34.1. A comparison with Eq. (34.5) shows that $dE/dx$ then grows as $\ln T_{\text{max}}$ rather than $\ln \beta^2\gamma^2 T_{\text{max}}$, and that the mean excitation

<footnote n="1">¹For incident spin 1/2 particles, $(W_{\text{max}}/E)^2/4$ is included in the square brackets. Although this correction is within the uncertainties in the total stopping power, its inclusion avoids a systematic bias.</footnote>

<footnote n="2">²In this section, “$dE/dx$” will be understood to mean the mass stopping power “$\langle -dE/dx \rangle$.”</footnote>
<figure><img src="image_171.png" /><figcaption>**Figure 34.1:** Mass stopping power (dE/dx) for positive muons in copper as a function of βγ = p/Mc over nine orders of magnitude in momentum (12 orders of magnitude in kinetic energy). Solid curves indicate the total stopping power. Data below the break at βγ ≈ 0.1 are taken from ICRU 49 [6] assuming only β dependence, and data at higher energies are from [7]. Vertical bands indicate boundaries between different approximations discussed in the text. The short dotted lines labeled “μ⁻” illustrate the “Barkas effect,” the dependence of stopping power on projectile charge at very low energies [8]. dE/dx in the radiative region is not simply a function of β.</figcaption></figure>

energy *I* is replaced by the plasma energy *ħω*<sub>*p*</sub>. An example of the ionization stopping power as calculated with and without the density effect correction is shown in Fig. 34.1. Since the plasma frequency scales as the square root of the electron density, the correction is much larger for a liquid or solid than for a gas, as is illustrated in Fig. 34.2.

The density effect correction is usually computed using Stern-
heimer's parameterization [17]:

$$
\delta(\beta\gamma) = \begin{cases} 2(\ln 10)x - \bar{C} & \text{if } x \ge x_1; \\ 2(\ln 10)x - \bar{C} + a(x_1 - x)^k & \text{if } x_0 \le x < x_1; \\ 0 & \text{if } x < x_0 \text{ (nonconductors)}; \\ \delta_0 10^{2(x-x_0)} & \text{if } x < x_0 \text{ (conductors)} \end{cases} \tag{34.7}
$$

Here $x = \log_{10} \beta\gamma = \log_{10}(p/Mc)$. $\bar{C}$ (the negative of the $C$ used in Ref. [17]) is obtained by equating the high-energy case of Eq. (34.7) with the limit given in Eq. (34.6). The other parameters are adjusted to give a best fit to the results of detailed calculations for momenta below $Mc\exp(x_1)$. For nonconductors the correction is 0 below $\beta\gamma = 10^{x_0}$, corresponding to 100–200 MeV for pions and 1–2 GeV for protons. For conductors it decreases rapidly below this point. Parameters for the elements and nearly 200 compounds and mixtures of interest are published in a variety of places, notably in Ref. [18]. A recipe for finding the coefficients for nontabulated materials is given by Sternheimer and Peierls [19] and is summarized in Ref. [7].

The remaining relativistic rise comes from the β²γ growth of Wmax, which in turn is due to (rare) large energy transfers to a few electrons. When these events are excluded, the energy deposit in an absorbing layer approaches a constant value, the Fermi plateau (see Sec. 34.2.8 below). At even higher energies (e.g., > 332 GeV for muons in iron, and at a considerably higher energy for pro-

tons in iron), radiative effects are more important than ionization
losses. These are especially relevant for high-energy muons, as
discussed in Sec. 34.6.

34.2.6 Energy loss at low energies

The theory of energy loss by ionization and excitation as given by Bethe is based on a first-order Born approximation. It assumes free electrons, and should be valid when the projectile's speed is large compared to that of the atomic electrons. This presents a problem at low energies, where $W_{\text{max}}$ is less than the K shell binding energy. However, Mott showed that the Born approximation can be applied at energies much smaller than atomic binding energies [20]; the incident particle can be treated by classical mechanics since its wavelength is shorter than atomic dimensions. The Born method is actually better justified when its speed is not large compared to the K electron speed [5].

Higher-order corrections must still be made to extend the Bethe
equation (Eq. (34.5)) to low energies. An improved approximation
for the terms in the square brackets of Eq. (34.5) at low energies
is obtained with

$$
L(\beta) = L_a(\beta) - \frac{C(\beta)}{Z} + zL_1(\beta) + z^2 L_2(\beta) . \quad (34.8)
$$

Here $L_a$ is the square-bracketed terms of Eq. (34.5), $C/Z$ is the sum of shell corrections and $zL_1$ and $z^2L_2$ are Barkas and Bloch correction terms [6, 21]. With these corrections, the Bethe treatment is accurate to about 1% down to $\beta \approx 0.05$, or about 1 MeV for protons (0.13 MeV for muons). Values of $L_a$, $C/Z$, $L_1$, and $L_2$ in the range $T = 0.3-30$ MeV for a proton traversing aluminum can be found in Table I of Ref. [21].

Shell correction –C/Z. As the speed of the projectile decreases, the contribution to the stopping power from K shell electrons decreases, and at even lower velocities contributions from L and
<figure><img src="image_172.png" /><figcaption>**Figure 34.2:** Mean energy loss rate in liquid (bubble chamber) hydrogen, gaseous helium, carbon, aluminum, iron, tin, and lead. Radiative effects, relevant for muons and pions, are not included. These become significant for muons in iron for $βγ \gtrsim 1000$, and at lower momenta for muons in higher-Z absorbers. See Fig. 34.23.</figcaption></figure>

<figure><img src="image_173.png" /><figcaption>**Figure 34.3:** Mass stopping power at minimum ionization for the chemical elements. The straight line is fitted for $Z > 6$. A simple functional dependence on $Z$ is not to be expected, since $dE/dx$ also depends on other variables.</figcaption></figure>

higher shells further reduce it. The correction $(C_K+C_L+...)/Z$ is should be included in the square brackets of Eq. (34.5). It is calculated and tabulated (for a few common materials) in a number of places; Refs. [6,12,21] are especially useful. As an example, the shell correction for a 30 MeV proton traversing aluminum is 0.6%, increasing to 9.9% as the proton's energy decreases to 0.3 MeV.

*Barkas correction* $zL_1$. Qualitatively, one might imagine an atom's electron cloud slightly recoiling at the approach of a negative projectile and being attracted toward an approaching positive projectile. Hence the stopping power for negative particles should be slightly smaller than the stopping power for positive particles. In a 1956 paper, *Barkas et al.* noted that negative pions possibly had a longer range than positive pions [8]. The effect has been measured for a number of negative/positive particle pairs, and more recently in detailed studies with antiprotons at the CERN LEAR facility [22]. Since no complete theory exists, an empirical approach is necessary. A 1972 harmonic-oscillator model by Ash-

<figure><img src="image_174.png" /><figcaption>**Figure 34.4:** Range of heavy charged particles in liquid (bubble chamber) hydrogen, helium, carbon, iron, and lead. For example: For a $K^+$ whose momentum is 700 MeV/c, $βγ = 1.42$. For lead we read $R/M \approx 396$, and so the range is $195 \text{ g cm}^{-2}$ (17 cm).</figcaption></figure>

<figure><img src="image_175.png" /><figcaption>**Figure 34.5:** Mean excitation energies (divided by $Z$) as adopted by the ICRU [12]. Those based on experimental measurements are shown by symbols with error flags; the interpolated values are simply joined. The grey point is for liquid $H_2$; the black point at 19.2 eV is for $H_2$ gas. The open circles show more recent determinations by Bichsel [13]. The dash-dotted curve is from the approximate formula of Barkas [14] used in early editions of this Review.</figcaption></figure>

ley *et al.* [23] is often used; it has two parameters determined by experimental data. For protons in aluminum, $L_1/L_a$ is less than 0.1% at 30 MeV, but increases to 17% as $T$ decreases to 0.3 MeV. This correction is indicated in Fig. 34.1.

*Bloch correction* $z^2L_2$. Bloch's extension of Bethe's theory introduced a low-energy correction that takes account of perturbations of the atomic wave functions. The form obtained by Lindhard and Sørensen [11] is used *e.g.* in Refs. [6,21]. For protons in aluminum, $-L_2/L_1$ is less than 0.3% at 3.0 MeV, but rises to 7% when the energy has fallen to 0.3 MeV.
For the interval $0.01 < \beta < 0.05$ there is no satisfactory theory. For protons, one usually relies on the phenomenological fitting formulae developed by Andersen and Ziegler [6,24]. As tabulated in ICRU 49 [6], the nuclear plus electronic proton stopping power in copper is 113 MeV cm² g⁻¹ at $T = 10$ keV ($\beta\gamma = 0.005$), rises to a maximum of 210 MeV cm² g⁻¹ at $T \approx 120$ keV ($\beta\gamma = 0.016$), then falls to 118 MeV cm² g⁻¹ at $T = 1$ MeV ($\beta\gamma = 0.046$). Above 0.5–1.0 MeV the corrected Bethe theory is adequate.

For particles moving more slowly than ≈ 0.01c (more or less the speed of the outer atomic electrons), Lindhard has been quite successful in describing electronic stopping power, which is proportional to $\beta$ [25]. Finally, we note that at even lower energies, e.g., for protons of less than several hundred eV, non-ionizing nuclear recoil energy loss dominates the total energy loss [6, 25, 26].

### 34.2.7 Energetic knock-on electrons (δ rays)

The distribution of secondary electrons with kinetic energies $T \gg I$ is [2]

$$ \frac{d^2 N}{dTdx} = \frac{1}{2} K z^2 \frac{Z}{A} \frac{1}{\beta^2} \frac{F(T)}{T^2} \quad (34.9) $$

for $I \ll T \le W_{max}$, where $W_{max}$ is given by Eq. (34.4). Here $\beta$ is the speed of the primary particle. The factor $F$ is spin-dependent, but is about unity for $T \ll W_{max}$. For spin-0 particles $F(T) = (1 - \beta^2 T/W_{max})$; forms for spins 1/2 and 1 are also given by Rossi [2] (Sec. 2.3, Eqs. 7 and 8). Additional formulae are given in [27]. Equation Eq. (34.9) is inaccurate for $T$ close to $I$ [28].

$\delta$ rays of even modest energy are rare. For a $\beta \approx 1$ particle, for example, on average only one collision with $T_e > 10$ keV will occur along a path length of 90 cm of argon gas [1].

A $\delta$ ray with kinetic energy $T_e$ and corresponding momentum $p_e$ is produced at an angle $\theta$ given by

$$ \cos \theta = (T_e/p_e)(p_{max}/W_{max}), \quad (34.10) $$

where $p_{max}$ is the momentum of an electron with the maximum possible energy transfer $W_{max}$.

### 34.2.8 Restricted energy loss rates for relativistic ionizing particles

Further insight can be obtained by examining the mean energy deposit by an ionizing particle when energy transfers are restricted to $T \le W_{cut} \le W_{max}$. The restricted energy loss rate is

$$ \left. \frac{dE}{dx} \right|_{T < W_{\text{cut}}} = K z^2 \frac{Z}{A} \frac{1}{\beta^2} \left[ \frac{1}{2} \ln \frac{2m_e c^2 \beta^2 \gamma^2 W_{\text{cut}}}{I^2} - \frac{\beta^2}{2} \left( 1 + \frac{W_{\text{cut}}}{W_{\text{max}}} \right) - \frac{\delta}{2} \right] \quad (34.11) $$

This form approaches the normal Bethe function (Eq. (34.5)) as $W_{cut} \to W_{max}$. It can be verified that the difference between Eq. (34.5) and Eq. (34.11) is equal to $\int_{W_{cut}}^{W_{max}} T(d^2N/dTdx)dT$, where $d^2N/dTdx$ is given by Eq. (34.9).

Since $W_{cut}$ replaces $W_{max}$ in the argument of the logarithmic term of Eq. (34.5), the $\beta\gamma$ term producing the relativistic rise in the close-collision part of $dE/dx$ is replaced by a constant, and $|dE/dx|_{T < W_{cut}}$ approaches the constant “Fermi plateau.” (The density effect correction $\delta$ eliminates the explicit $\beta\gamma$ dependence produced by the distant-collision contribution.) This behavior is illustrated in Fig. 34.6, where restricted loss rates for two examples of $W_{cut}$ are shown in comparison with the full Bethe $dE/dx$ and the Landau-Vavilov most probable energy loss (to be discussed in Sec. 34.2.9 below).

“Restricted energy loss” is cut at the total mean energy, not the single-collision energy above $W_{cut}$. It is of limited use. The most probable energy loss, discussed in the next Section, is far more useful in situations where single-particle energy loss is observed.

### 34.2.9 Fluctuations in energy loss

For detectors of moderate thickness $x$ (e.g. scintillators or LAr cells),<fnref n="3" /> the energy loss probability distribution $f(\Delta; \beta\gamma, x)$ is ad-

<figure><img src="image_176.png" /><figcaption>**Figure 34.6:** Bethe $dE/dx$, two examples of restricted energy loss, and the Landau most probable energy per unit thickness in silicon. The change of $\Delta_p/x$ with thickness $x$ illustrates its $a \ln x + b$ dependence. Minimum ionization ($dE/dx|_{min}$) is 1.664 MeV g⁻¹ cm². Radiative losses are excluded. The incident particles are muons.</figcaption></figure>

equately described by the highly-skewed Landau (or Landau-Vavilov) distribution [29, 30].

The most probable energy loss is [31]<fnref n="4" />

$$ \Delta_p = \xi \left[ \ln \frac{2m_e c^2 \beta^2 \gamma^2}{I} + \ln \frac{\xi}{I} + j - \beta^2 - \delta(\beta\gamma) \right], \quad (34.12) $$

where $\xi = (K/2)(Z/A)z^2(x/\beta^2)$ MeV for a detector with a thickness $x$ in g cm⁻², and $j = 0.200$ [31].<fnref n="5" /> While $dE/dx$ is independent of thickness, $\Delta_p/x$ scales as $a \ln x + b$. The density correction $\delta(\beta\gamma)$ was not included in Landau’s or Vavilov’s work, but it was later included by Bichsel [31]. The high-energy behavior of $\delta(\beta\gamma)$ (Eq. (34.6)) is such that

$$ \Delta_p \underset{\beta\gamma \gtrsim 100}{\longrightarrow} \xi \left[ \ln \frac{2m_e c^2 \xi}{(\hbar\omega_p)^2} + j \right] \quad (34.13) $$

Thus the Landau-Vavilov most probable energy loss, like the restricted energy loss, reaches a Fermi plateau. The Bethe $dE/dx$ and Landau-Vavilov-Bichsel $\Delta_p/x$ in silicon are shown as a function of muon energy in Fig. 34.6. The energy deposit in the 1600 µm case is roughly the same as in a 3 mm thick plastic scintillator.

The distribution function for the energy deposit by a 10 GeV muon going through a detector of about this thickness is shown in Fig. 34.7. In this case the most probable energy loss is 62% of the mean $(M_1((\Delta))/M_1(\infty))$. Folding in experimental resolution displaces the peak of the distribution, usually toward a higher value. 90% of the collisions $(M_1((\Delta))/M_1(\infty))$ contribute to energy deposits below the mean. It is the very rare high-energy-transfer collisions, extending to $W_{max}$ at several GeV, that drives the mean into the tail of the distribution. The large weight of these rare events makes the mean of an experimental distribution consisting of a few hundred events subject to large fluctuations and sensitive to cuts. The mean of the energy loss given by the Bethe equation, Eq. (34.5), is thus ill-defined experimentally and is not useful for describing energy loss by single particles.<fnref n="6" /> It rises as $\ln \gamma$ because $W_{max}$ increases as $\gamma$ at high energies. The most probable energy loss should be used.

A practical example: For muons traversing 0.25 inches (0.64 cm) of PVT (polyvinyltolulene) based plastic scintillator,

<footnote>sorber’s thickness, and as such parameterizes the constants describing the Landau distribution. These are fairly insensitive to thickness for $G \lesssim 0.1$, the case for most detectors.</footnote>

<footnote n="4">⁴Practical calculations can be expedited by using the tables of $\delta$ and $\beta$ from the text versions of the muon energy loss tables to be found at pdg.lbl.gov/current/AtomicNuclearProperties.</footnote>

<footnote n="5">⁵Rossi [2], Talman [32], and others give somewhat different values for $j$. The most probable loss is not sensitive to its value.</footnote>

<footnote n="6">⁶It does find application in dosimetry, where only bulk deposit is relevant.</footnote>

<footnote n="3">³“Moderate thickness” means $G \lesssim 0.05–0.1$, where $G$ is given by Rossi Ref. [2], Eq. 2.7(10). It is Vavilov’s $\kappa$ [29]. $G$ is proportional to the ab-</footnote>
<figure><img src="image_177.png" /><figcaption>**Figure 34.7:** Electronic energy deposit distribution for a 10 GeV muon traversing 1.7 mm of silicon, the stopping power equivalent of about 0.3 cm of PVT-based scintillator [1, 13, 33]. The Landau-Vavilov function (dot-dashed) uses a Rutherford cross section without atomic binding corrections but with a kinetic energy transfer limit of $W_{max}$. The solid curve was calculated using Bethe-Fano theory. $M_0(\Delta)$ and $M_1(\Delta)$ are the cumulative 0th moment (mean number of collisions) and 1st moment (mean energy loss) in crossing the silicon. (See Sec. 34.2.1). The fwhm of the Landau-Vavilov function is about $4\xi$ for detectors of moderate thickness. $\Delta_p$ is the most probable energy loss, and $\langle \Delta \rangle$ divided by the thickness is the Bethe $dE/dx$.</figcaption></figure>

<figure><img src="image_178.png" /><figcaption>**Figure 34.8:** Straggling functions in silicon for 500 MeV pions, normalized to unity at the most probable value $\Delta_p/x$. The width $w$ is the full width at half maximum.</figcaption></figure>

the ratio of the most probable E loss rate to the mean loss rate via the Bethe equation is [0.69, 0.57, 0.49, 0.42, 0.38] for $T_\mu$ = [0.01, 0.1, 1, 10, 100] GeV. Radiative losses add less than 0.5% to the total mean energy deposit at 10 GeV, but add 7% at 100 GeV. The most probable E loss rate rises slightly beyond the minimum ionization energy, then is essentially constant.

The Landau distribution fails to describe energy loss in thin absorbers such as gas TPC cells [1] and Si detectors [31], as can be seen e.g. in Fig. 1 of Ref. [1] for an argon-filled TPC cell. Also see Talman [32]. While $\Delta_p/x$ may be calculated adequately with Eq. (34.12), the distributions are significantly wider than the Landau width $w = 4\xi$ Ref. [31], Fig. 15. Examples for 500 MeV pions incident on thin silicon detectors are shown in Fig. 34.8. For very thick absorbers the distribution is less skewed but never approaches a Gaussian.

The most probable energy loss, scaled to the mean loss at minimum ionization, is shown in Fig. 34.9 for several silicon detector thicknesses.

<figure><img src="image_179.png" /><figcaption>**Figure 34.9:** Most probable energy loss in silicon, scaled to the mean loss of a minimum ionizing particle, 388 eV/µm (1.66 MeV g$^{-1}$cm$^2$).</figcaption></figure>

### 34.2.10 Energy loss in mixtures and compounds

A mixture or compound can be thought of as made up of thin layers of pure elements in the right proportion (Bragg additivity). In this case,

$$ \left\langle \frac{dE}{dx} \right\rangle = \sum w_j \left\langle \frac{dE}{dx} \right\rangle_j, \qquad (34.14) $$

where $dE/dx|_j$ is the mean rate of energy loss (in MeV g cm$^{-2}$) in the $j$th element. Eq. (34.5) can be inserted into Eq. (34.14) to find expressions for $\langle Z/A \rangle$, $\langle I \rangle$, and $\langle \delta \rangle$; for example, $\langle Z/A \rangle = \sum w_j Z_j/A_j = \sum n_j Z_j/\sum n_j A_j$. However, $\langle I \rangle$ as defined this way is an underestimate, because in a compound electrons are more tightly bound than in the free elements, and $\langle \delta \rangle$ as calculated this way has little relevance, because it is the electron density that matters. If possible, one uses the tables given in Refs. [18, 34], or the recipes given in [19] (repeated in Ref. [7]), that include effective excitation energies and interpolation coefficients for calculating the density effect correction for the chemical elements and nearly 200 mixtures and compounds. Otherwise, use the recipe for $\delta$ given in Refs. [7, 19], and calculate $\langle I \rangle$ following the discussion in Ref. [15]. (Note the “13%” rule!)

### 34.2.11 Ionization yields

The Bethe equation describes energy loss via excitation and ionization. Many gaseous detectors (proportional counters or TPCs) or liquid ionization detectors count the number of electrons or positive ions from ionization, rather than the ionization energy. As a further complication, the electron liberated in the initial ionization often has enough energy to ionize other atoms or molecules; this process can happen several times. The number of electron-ion pairs per unit length is typically three or more times the original number. Ion or electron counting is a proxy for a direct $dE/dx$ measurement. Calibrations link the number of observed ions to the traversing particle's $dE/dx$.

The details depend on the gases (or liquids) and the particular detector involved. A useful discussion of the physics is provided in Sec.35.6 of this Review.

## 34.3 Multiple scattering through small angles

A charged particle traversing a medium is deflected by many small-angle scatters. Most of this deflection is due to Coulomb scattering from nuclei as described by the Rutherford cross section. (However, for hadronic projectiles, the strong interactions also contribute to multiple scattering.) For many small-angle scatters the net scattering and displacement distributions are Gaussian via the central limit theorem. Less frequent “hard” scatters produce non-Gaussian tails. These Coulomb scattering distributions are well-represented by the theory of Molière [35]. Accessible discussions are given by Rossi [2] and Jackson [3], and exhaustive reviews have been published by Scott [36] and Motz et al. [37]. Experimental measurements have been published by Bichsel [38] (low energy protons) and by Shen et al. [39] (relativistic pions,
kaons, and protons).<fnref n="7" />

If we define

$$ \theta_0 = \theta_{\text{plane}}^{\text{rms}} = \frac{1}{\sqrt{2}} \theta_{\text{space}}^{\text{rms}}, \quad (34.15) $$

then it is sufficient for many applications to use a Gaussian approximation for the central 98% of the projected angular distribution, with an rms width given by Lynch & Dahl [40]:

$$ \begin{aligned} \theta_0 &= \frac{13.6 \text{ MeV}}{\beta cp} z \sqrt{\frac{x}{X_0}} \left[ 1 + 0.088 \log_{10} \left( \frac{x z^2}{X_0 \beta^2} \right) \right] \\ &= \frac{13.6 \text{ MeV}}{\beta cp} z \sqrt{\frac{x}{X_0}} \left[ 1 + 0.038 \ln \left( \frac{x z^2}{X_0 \beta^2} \right) \right] \end{aligned} \quad (34.16) $$

Here p, βc, and z are the momentum, speed, and charge number of the incident particle, and x/X₀ is the thickness of the scattering medium in radiation lengths (defined below). This takes into account the p and z dependence quite well at small Z, but for large Z and small x the β-dependence is not well represented. Further improvements are discussed in Ref. [40].

Eq. (34.16) describes scattering from a single material, while the usual problem involves the multiple scattering of a particle traversing many different layers and mixtures. Since it is from a fit to a Molière distribution, it is incorrect to add the individual θ₀ contributions in quadrature; the result is systematically too small. It is much more accurate to apply Eq. (34.16) once, after finding x and X₀ for the combined scatterer.

<figure><img src="image_180.png" /><figcaption>**Figure 34.10:** Quantities used to describe multiple Coulomb scattering. The particle is incident in the plane of the figure.</figcaption></figure>

The nonprojected (space) and projected (plane) angular distributions are given approximately by [35]

$$ \frac{1}{2\pi \theta_0^2} \exp \left( -\frac{\theta_{\text{space}}^2}{2\theta_0^2} \right) d\Omega, \quad (34.17) $$

$$ \frac{1}{\sqrt{2\pi} \theta_0} \exp \left( -\frac{\theta_{\text{plane}}^2}{2\theta_0^2} \right) d\theta_{\text{plane}}, \quad (34.18) $$

where θ is the deflection angle. In this approximation, θ²ₐₙₐ₉ₐ₅ ≈ (θ²ₚₐₙₐ₅, x + θ²ₚₐₙₐ₅, y), where the x and y axes are orthogonal to the direction of motion, and dΩ ≈ dθₚₐₙₐ₅, x dθₚₐₙₐ₅, y. Deflections into θₚₐₙₐ₅, x and θₚₐₙₐ₅, y are independent and identically distributed. Fig. 34.10 shows these and other quantities sometimes used to describe multiple Coulomb scattering. They are

$$ \psi_{\text{plane}}^{\text{rms}} = \frac{1}{\sqrt{3}} \theta_{\text{plane}}^{\text{rms}} = \frac{1}{\sqrt{3}} \theta_0, \quad (34.19) $$

$$ y_{\text{plane}}^{\text{rms}} = \frac{1}{\sqrt{3}} x \theta_{\text{plane}}^{\text{rms}} = \frac{1}{\sqrt{3}} x \theta_0, \quad (34.20) $$

$$ s_{\text{plane}}^{\text{rms}} = \frac{1}{4\sqrt{3}} x \theta_{\text{plane}}^{\text{rms}} = \frac{1}{4\sqrt{3}} x \theta_0. \quad (34.21) $$

All the quantitative estimates in this section apply only in the limit of small θ rms plane and in the absence of large-angle scatters. The random variables s, ψ, y, and θ in a given plane are correlated.

Obviously, y ≈ xψ. In addition, y and θ have the correlation coefficient ρᵢⱼθ = √3/2 ≈ 0.87. For Monte Carlo generation of a joint (yᵢₙₐ₅, θᵢₙₐ₅) distribution, or for other calculations, it may be most convenient to work with independent Gaussian random variables (z₁, z₂) with mean zero and variance one, and then set

$$ y_{\text{plane}} = z_1 x \theta_0 (1 - \rho_{y\theta}^2)^{1/2} / \sqrt{3} + z_2 \rho_{y\theta} x \theta_0 / \sqrt{3} \quad (34.22a) $$

$$ = z_1 x \theta_0 / \sqrt{12} + z_2 x \theta_0 / 2; \quad (34.22b) $$

$$ \theta_{\text{plane}} = z_2 \theta_0. \quad (34.22c) $$

Note that the second term for yₚₐₙₐ₅ equals xθₚₐₙₐ₅/2 and represents the displacement that would have occurred had the deflection θₚₐₙₐ₅ all occurred at the single point x/2.

For heavy ions the multiple Coulomb scattering has been measured and compared with various theoretical distributions [41].

## 34.4 Photon and electron interactions in matter

At low energies electrons and positrons primarily lose energy by ionization, although other processes (Møller scattering, Bhabha scattering, e⁺ annihilation) contribute, as shown in Fig. 34.11. While ionization loss rates rise logarithmically with energy, bremsstrahlung losses rise nearly linearly (fractional loss is nearly independent of energy), and dominates above the critical energy (Sec. 34.4.4 below), a few tens of MeV in most materials.

### 34.4.1 Collision energy losses by e⁺

Stopping power differs somewhat for electrons and positrons, and both differ from stopping power for heavy particles because of the kinematics, spin, charge, and the identity of the incident electron with the electrons that it ionizes. Complete discussions and tables can be found in Refs. [12, 15, 34].

For electrons, large energy transfers to atomic electrons (taken as free) are described by the Møller cross section. From Eq. (34.4), the maximum energy transfer in a single collision should be the entire kinetic energy, Wₘₐₓ = mₑc²(γ - 1), but because the particles are identical, the maximum is half this, Wₘₐₓ/2. (The results are the same if the transferred energy is ϵ or if the transferred energy is Wₘₐₓ - ϵ. The stopping power is by convention calculated for the faster of the two emerging electrons.) The first moment of the Møller cross section [27] (divided by dx) is the stopping power:

$$ \left\langle -\frac{dE}{dx} \right\rangle = \frac{1}{2} K \frac{Z}{A} \frac{1}{\beta^2} \left[ \ln \frac{m_e c^2 \beta^2 \gamma^2 \{m_e c^2 (\gamma - 1)/2\}}{I^2} + (1 - \beta^2) - \frac{2\gamma - 1}{\gamma^2} \ln 2 + \frac{1}{8} \left(\frac{\gamma - 1}{\gamma}\right)^2 - \delta \right] \quad (34.23) $$

The logarithmic term can be compared with the logarithmic term in the Bethe equation (Eq. (34.2)) by substituting Wₘₐₓ = mₑc²(γ - 1)/2. Electron-positron scattering is described by the fairly complicated Bhabha cross section [27]. There is no identical particle problem, so Wₘₐₓ = mₑc²(γ - 1). The first moment of the Bhabha equation yields

$$ \left\langle -\frac{dE}{dx} \right\rangle = \frac{1}{2} K \frac{Z}{A} \frac{1}{\beta^2} \left[ \ln \frac{m_e c^2 \beta^2 \gamma^2 \{m_e c^2 (\gamma - 1)\}}{2I^2} + 2 \ln 2 - \frac{\beta^2}{12} \left( 23 + \frac{14}{\gamma + 1} + \frac{10}{(\gamma + 1)^2} + \frac{4}{(\gamma + 1)^3} \right) - \delta \right] \quad (34.24) $$

Following ICRU 37 [12], the density effect correction δ has been added to Uehling's equations [27] in both cases.

For heavy particles, shell corrections were developed assuming that the projectile is equivalent to a perturbing potential whose center moves with constant speed. This assumption has no sound theoretical basis for electrons. The authors of ICRU 37 [12] estimated the possible error in omitting it by assuming the correction was twice as great as for a proton of the same speed. At T = 10 keV, the error was estimated to be ≈2% for water, ≈9% for Cu, and ≈21% for Au.

<footnote n="7">⁷Shen et al.'s measurements show that Bethe's simpler methods of including atomic electron effects agrees better with experiment than does Scott's treatment.</footnote>
As shown in Fig. 34.11, stopping powers for $e^-$, $e^+$, and heavy particles are not dramatically different. In silicon, the minimum value for electrons is 1.50 MeV cm$^2$/g (at $\gamma = 3.3$); for positrons, 1.46 MeV cm$^2$/g (at $\gamma = 3.7$), and for muons, 1.66 MeV cm$^2$/g (at $\gamma = 3.58$).

### 34.4.2 Radiation length

High-energy electrons predominantly lose energy in matter by bremsstrahlung, and high-energy photons by $e^+e^-$ pair production. The characteristic amount of matter traversed for these related interactions is called the radiation length $X_0$, usually measured in g cm$^{-2}$. It is the mean distance over which a high-energy electron loses all but $1/e$ of its energy by bremsstrahlung. It is also the appropriate scale length for describing high-energy electromagnetic cascades. $X_0$ has been calculated and tabulated by Y.S. Tsai [42]:

$$ \frac{1}{X_0} = 4\alpha r_e^2 \frac{N_A}{A} \left\{ Z^2 [L_{\text{rad}} - f(Z)] + Z L'_{\text{rad}} \right\}. \quad (34.25) $$

For $A = 1 \text{ g mol}^{-1}$, $4\alpha r_e^2 N_A / A = (716.408 \text{ g cm}^{-2})^{-1}$. $L_{\text{rad}}$ and $L'_{\text{rad}}$ are given in Table 34.2. The function $f(Z)$ is an infinite sum, but for elements up to uranium can be represented to 4-place accuracy by

$$ f(Z) = a^2 \left[ (1 + a^2)^{-1} + 0.20206 - 0.0369 a^2 + 0.0083 a^4 - 0.002 a^6 \right], \quad (34.26) $$

where $a = \alpha Z$ [43].

<table><caption>**Table 34.2:** Tsai's $L_{\text{rad}}$ and $L'_{\text{rad}}$, for use in calculating the radiation length in an element using Eq. (34.25).</caption><thead><tr><th>Element</th><th>Z</th><th>L<sub>rad</sub></th><th>L'<sub>rad</sub></th></tr></thead><tbody><tr><td>H</td><td>1</td><td>5.31</td><td>6.144</td></tr><tr><td>He</td><td>2</td><td>4.79</td><td>5.621</td></tr><tr><td>Li</td><td>3</td><td>4.74</td><td>5.805</td></tr><tr><td>Be</td><td>4</td><td>4.71</td><td>5.924</td></tr><tr><td>Others</td><td>&gt; 4</td><td>ln(184.15 Z<sup>-1/3</sup>)</td><td>ln(1194 Z<sup>-2/3</sup>)</td></tr></tbody></table>

The radiation length in a mixture or compound may be approximated by

$$ 1/X_0 = \sum w_j/X_j, \quad (34.27) $$

where $w_j$ and $X_j$ are the fraction by weight and the radiation length for the jth element.

### 34.4.3 Bremsstrahlung energy loss by $e^\pm$

At very high energies and except at the high-energy tip of the bremsstrahlung spectrum, the cross section can be approximated in the “complete screening case” as [42]

$$ d\sigma/dk = (1/k)4\alpha r_e^2 \left\{ \left(\frac{4}{3} - \frac{4}{3}y + y^2\right)[Z^2(L_{\text{rad}} - f(Z)) + Z L'_{\text{rad}}] + \frac{1}{9}(1-y)(Z^2 + Z) \right\}, \quad (34.28) $$

where $y = k/E$ is the fraction of the electron's energy transferred to the radiated photon. At small $y$ (the “infrared limit”) the term on the second line ranges from 1.7% (low $Z$) to 2.5% (high $Z$) of the total. If it is ignored and the first line simplified with the definition of $X_0$ given in Eq. (34.25), we have

$$ \frac{d\sigma}{dk} = \frac{A}{X_0 N_A k} \left( \frac{4}{3} - \frac{4}{3}y + y^2 \right). \quad (34.29) $$

This cross section (times $k$) is shown by the top curve in Fig. 34.12.

This formula is accurate except near $y = 1$, where screening may become incomplete, and near $y = 0$, where the infrared divergence is removed by the interference of bremsstrahlung amplitudes from

<figure><img src="image_181.png" /><figcaption>**Figure 34.11:** Fractional energy loss per radiation length in lead as a function of electron or positron energy. Electron (positron) scattering is considered as ionization when the energy loss per collision is below 0.255 MeV, and as Møller (Bhabha) scattering when it is above. Adapted from Fig. 3.2 from Messel and Crawford, *Electron-Photon Shower Distribution Function Tables for Lead, Copper, and Air Absorbers*, Pergamon Press, 1970. Messel and Crawford use $X_0(\text{Pb}) = 5.82 \text{ g/cm}^2$, but we have modified the figures to reflect the value given in the Table of Atomic and Nuclear Properties of Materials ($X_0(\text{Pb}) = 6.37 \text{ g/cm}^2$).</figcaption></figure>

<figure><img src="image_182.png" /><figcaption>**Figure 34.12:** The normalized bremsstrahlung cross section $k d\sigma_{LPM}/dk$ in lead versus the fractional photon energy $y = k/E$. The vertical axis has units of photons per radiation length.</figcaption></figure>

nearby scattering centers (the LPM effect) [44, 45] and dielectric suppression [46, 47]. These and other suppression effects in bulk media are discussed in Sec. 34.4.6.

With decreasing energy ($E \lesssim 10 \text{ GeV}$) the high-$y$ cross section drops and the curves become rounded as $y \to 1$. Curves of this familiar shape can be seen in Rossi [2] (Figs. 2.11.2,3); see also the review by Koch & Motz [48].

<figure><img src="image_183.png" /><figcaption>**Figure 34.13:** Two definitions of the critical energy $E_c$.</figcaption></figure>
Except at these extremes, and still in the complete-screening approximation, the number of photons with energies between $k_{min}$ and $k_{max}$ emitted by an electron travelling a distance $d \ll X_0$ is

$$N_{\gamma} = \frac{d}{X_0} \left[ \frac{4}{3} \ln \left( \frac{k_{\max}}{k_{\min}} \right) - \frac{4(k_{\max} - k_{\min})}{3E} + \frac{k_{\max}^2 - k_{\min}^2}{2E^2} \right]. \quad (34.30)$$

<figure><img src="image_184.png" /><figcaption>**Figure 34.14:** Electron critical energy for the chemical elements, using Rossi's definition [2]. The fits shown are for solids and liquids (solid line) and gases (dashed line). The rms deviation is 2.2% for the solids and 4.0% for the gases.</figcaption></figure>

### 34.4.4 Critical energy

An electron loses energy by bremsstrahlung at a rate nearly proportional to its energy, while the ionization loss rate varies only logarithmically with the electron energy. The critical energy $E_c$ is sometimes defined as the energy at which the two loss rates are equal [49]. Among alternate definitions is that of Rossi [2], who defines the critical energy as the energy at which the ionization loss per radiation length is equal to the electron energy. Equivalently, it is the same as the first definition with the approximation $|dE/dx|_{brems} \approx E/X_0$. This form has been found to describe transverse electromagnetic shower development more accurately (see below). These definitions are illustrated in the case of copper in Fig. 34.13.

The accuracy of approximate forms for $E_c$ has been limited by the failure to distinguish between gases and solid or liquids, where there is a substantial difference in ionization at the relevant energy because of the density effect. We distinguish these two cases in Fig. 34.14. Fits were also made with functions of the form $a/(Z+b)^\alpha$, but $\alpha$ was found to be essentially unity. Since $E_c$ also depends on $A$, $I$, and other factors, such forms are at best approximate.

Values of $E_c$ for both electrons and positrons in more than 300 materials at pdg.lbl.gov/current/AtomicNuclearProperties.

### 34.4.5 Energy loss by photons

Contributions to the photon cross section in a light element (carbon) and a heavy element (lead) are shown in Fig. 34.15. At low energies it is seen that the photoelectric effect dominates, although Compton scattering, Rayleigh scattering, and photonuclear absorption also contribute. The photoelectric cross section is characterized by discontinuities (absorption edges) as thresholds for photoionization of various atomic levels are reached. Photon attenuation lengths for a variety of elements are shown in Fig. 34.16, and data for 30 eV < $k$ < 100 GeV for all elements are available from the web pages given in the caption. Here $k$ is the photon energy.

The increasing domination of pair production as the energy increases is shown in Fig. 34.17. Using approximations similar to those used to obtain Eq. (34.29), Tsai's formula for the differential

<figure><img src="image_185.png" /><figcaption>**Figure 34.15:** Photon total cross sections as a function of energy in carbon and lead, showing the contributions of different processes [50]:</figcaption></figure>

$\sigma_{p.e.}$ = Atomic photoelectric effect (electron ejection, photon absorption)

$\sigma_{Rayleigh}$ = Rayleigh (coherent) scattering-atom neither ionized nor excited

$\sigma_{Compton}$ = Incoherent scattering (Compton scattering off an electron)

$\kappa_{nuc}$ = Pair production, nuclear field

$\kappa_e$ = Pair production, electron field

$\sigma_{g.d.r.}$ = Photonuclear interactions, most notably the Gi-ant Dipole Resonance [51]. In these interactions, the target nucleus is usually broken up.

Original figures through the courtesy of John H. Hubbell (NIST).

cross section [42] reduces to

$$\frac{d\sigma}{dx} = \frac{A}{X_0 N_A} \left[ 1 - \frac{4}{3}x(1-x) \right] \quad (34.31)$$

in the complete-screening limit valid at high energies. Here $x = E/k$ is the fractional energy transfer to the pair-produced electron (or positron), and $k$ is the incident photon energy. The cross
<figure><img src="image_186.png" /><figcaption>**Figure 34.16:** The photon mass attenuation length (or mean free path) $\lambda = 1/(\mu/\rho)$ for various elemental absorbers as a function of photon energy. The mass attenuation coefficient is $\mu/\rho$, where $\rho$ is the density. The intensity $I$ remaining after traversal of thickness $t$ (in mass/unit area) is given by $I = I_0 \exp(-t/\lambda)$. The accuracy is a few percent. For a chemical compound or mixture, $1/\lambda_{\text{eff}} \approx \sum_{\text{elements}} w_Z/\lambda_Z$, where $w_Z$ is the proportion by weight of the element with atomic number $Z$. The processes responsible for attenuation are given in Fig. 34.11. Since coherent processes are included, not all these processes result in energy deposition. The data for $30 \text{ eV} < E < 1 \text{ keV}$ are from Ref. [52], those for $1 \text{ keV} < E < 100 \text{ GeV}$ from Ref. [53].</figcaption></figure>

<figure><img src="image_187.png" /><figcaption>**Figure 34.17:** Probability $P$ that a photon interaction will result in conversion to an $e^+e^-$ pair. Except for a few-percent contribution from photonuclear absorption around 10 or 20 MeV, essentially all other interactions in this energy range result in Compton scattering off an atomic electron. For a photon attenuation length $\lambda$ (Fig. 34.16), the probability that a given photon will produce an electron pair (without first Compton scattering) in thickness $t$ of absorber is $P[1 - \exp(-t/\lambda)]$.</figcaption></figure>

section is very closely related to that for bremsstrahlung, since the Feynman diagrams are variants of one another. The cross section is of necessity symmetric between $x$ and $1-x$, as can be seen by the solid curve in Fig. 34.18. See the review by Motz, Olsen, & Koch for a more detailed treatment [54]. Eq. (34.31) may be integrated to find the high-energy limit for the total $e^+e^-$ pair-production cross section:

$$ \sigma = \frac{7}{9} (A/X_0 N_A). \qquad (34.32) $$

Equation Eq. (34.32) is accurate to within a few percent down to energies as low as 1 GeV, particularly for high-Z materials.

### 34.4.6 Bremsstrahlung and pair production at very high energies

At ultrahigh energies, Eqns. 34.28–34.32 will fail because of quantum mechanical interference between amplitudes from different scattering centers. Since the longitudinal momentum transfer to a given center is small ($\propto k/E(E-k)$, in the case of bremsstrahlung), the interaction is spread over a comparatively long distance called the formation length ($\propto E(E-k)/k$) via the uncertainty principle. In alternate language, the formation length is the distance over which the highly relativistic electron and the photon “split apart.” The interference is usually destructive. Calculations of the “Landau-Pomeranchuk-Migdal” (LPM) effect may be made semi-classically based on the average multiple scattering, or more rigorously using a quantum transport approach [44, 45].

<figure><img src="image_188.png" /><figcaption>**Figure 34.18:** The normalized pair production cross section $d\sigma_{LPM}/dx$, versus fractional electron energy $x = E/k$.</figcaption></figure>

In amorphous media, bremsstrahlung is suppressed if the pho-
ton energy k is less than $E^2/(E + E_{LPM})$ [45], where<fnref n="8" />

$$E_{LPM} = (m_e c^2)^2 \alpha \frac{X_0}{4\pi\hbar c\rho} = (7.7 \text{ TeV/cm}) \times \frac{X_0}{\rho} . \quad (34.33)$$

Since physical distances are involved, $X_0/\rho$, in cm, appears. The energy-weighted bremsstrahlung spectrum for lead, $k d\sigma_{LPM}/dk$, is shown in Fig. 34.12. With appropriate scaling by $X_0/\rho$, other materials behave similarly.

For photons, pair production is reduced for $E(k - E) > k E_{LPM}$. The pair-production cross sections for different photon energies are shown in Fig. 34.18.

If $k \ll E$, several additional mechanisms can also produce suppression. When the formation length is long, even weak factors can perturb the interaction. For example, the emitted photon can coherently forward scatter off of the electrons in the media. Because of this, for $k < \omega_p E/m_e \sim 10^{-4}$, bremsstrahlung is suppressed by a factor $(km_e/\omega_p E)^2$ [47]. Magnetic fields can also suppress bremsstrahlung.

In crystalline media, the situation is more complicated, with coherent enhancement or suppression possible. The cross section depends on the electron and photon energies and the angles between the particle direction and the crystalline axes [56].

### 34.4.7 Photonuclear and electronuclear interactions at still higher energies

At still higher photon and electron energies, where the bremsstrahlung and pair production cross-sections are heavily suppressed by the LPM effect, photonuclear and electronuclear interactions predominate over electromagnetic interactions.

At photon energies above about $10^{20}$ eV, for example, photons usually interact hadronically. The exact cross-over energy depends on the model used for the photonuclear interactions. These processes are illustrated in Fig. 34.19. At still higher energies ($\gtrsim 10^{23}$ eV), photonuclear interactions can become coherent, with the photon interaction spread over multiple nuclei. Essentially, the photon coherently converts to a $\rho^0$, in a process that is somewhat similar to kaon regeneration [57].

<figure><img src="image_189.png" /><figcaption>**Figure 34.19:** Interaction length for a photon in ice as a function of photon energy for the Bethe-Heitler (BH), LPM (Mig) and photonuclear ($\gamma$A) cross sections [57]. The Bethe-Heitler interaction length is $9X_0/7$, and $X_0$ is 0.393 m in ice.</figcaption></figure>

Similar processes occur for electrons. As electron energies increase and the LPM effect suppresses bremsstrahlung, electronuclear interactions become more important. At energies above $10^{21}$ eV, these electronnuclear interactions dominate electron energy loss [57].

### 34.5 Electromagnetic cascades

When a high-energy electron or photon is incident on a thick absorber, it initiates an electromagnetic cascade as pair production and bremsstrahlung generate more electrons and photons with

lower energy. The longitudinal development is governed by the high-energy part of the cascade, and therefore scales as the radiation length in the material. Electron energies eventually fall below the critical energy, and then dissipate their energy by ionization and excitation rather than by the generation of more shower particles. In describing shower behavior, it is therefore convenient to introduce the scale variables

$$t = x/X_0, \quad y = E/E_c, \quad (34.34)$$

so that distance is measured in units of radiation length and energy in units of critical energy.

<figure><img src="image_190.png" /><figcaption>**Figure 34.20:** An EGS4 simulation of a 30 GeV electron-induced cascade in iron. The histogram shows fractional energy deposition per radiation length, and the curve is a gamma-function fit to the distribution. Circles indicate the number of electrons with total energy greater than 1.5 MeV crossing planes at $X_0/2$ intervals (scale on right) and the squares the number of photons with $E \ge 1.5$ MeV crossing the planes (scaled down to have same area as the electron distribution).</figcaption></figure>

Longitudinal profiles from an EGS4 [58] simulation of a 30 GeV electron-induced cascade in iron are shown in Fig. 34.20. The number of particles crossing a plane (very close to Rossi's $\Pi$ function [2]) is sensitive to the cutoff energy, here chosen as a total energy of 1.5 MeV for both electrons and photons. The electron number falls off more quickly than energy deposition. This is because, with increasing depth, a larger fraction of the cascade energy is carried by photons. Exactly what a calorimeter measures depends on the device, but it is not likely to be exactly any of the profiles shown. In gas counters it may be very close to the electron number, but in glass Cherenkov detectors and other devices with “thick” sensitive regions it is closer to the energy deposition (total track length). In such detectors the signal is proportional to the “detectable” track length $T_d$, which is in general less than the total track length $T$. Practical devices are sensitive to electrons with energy above some detection threshold $E_d$, and $T_d = T F(E_d/E_c)$. An analytic form for $F(E_d/E_c)$ obtained by Rossi [2] is given by Fabjan in Ref. [59]; see also Amaldi [60].

The mean longitudinal profile of the energy deposition in an electromagnetic cascade is reasonably well described by a gamma distribution [61]:

$$\frac{dE}{dt} = E_0 b \frac{(bt)^{a-1} e^{-bt}}{\Gamma(a)} \quad (34.35)$$

The maximum $t_{max}$ occurs at $(a-1)/b$. We have made fits to shower profiles in elements ranging from carbon to uranium, at energies from 1 GeV to 100 GeV. The energy deposition profiles are well described by Eq. (34.35) with

$$t_{max} = (a - 1)/b = 1.0 \times (\ln y + C_j), \quad j = e, \gamma, \quad (34.36)$$

where $C_e = -0.5$ for electron-induced cascades and $C_\gamma = +0.5$ for photon-induced cascades. To use Eq. (34.35), one finds $(a-1)/b$ from Eq. (34.36) and Eq. (34.34), then finds $a$ either by assuming $b \approx 0.5$ or by finding a more accurate value from Fig. 34.21. The results are very similar for the electron number profiles, but there

<footnote n="8">⁸This definition differs from that of Ref. [55] by a factor of two. $E_{LPM}$ scales as the 4th power of the mass of the incident particle, so that $E_{LPM} = (1.4 \times 10^{10} \text{ TeV/cm}) \times X_0/\rho$ for a muon.</footnote>
is some dependence on the atomic number of the medium. A similar form for the electron number maximum was obtained by Rossi in the context of his “Approximation B,” [2] (see Fabjan’s review in Ref. [59]), but with $C_e = -1.0$ and $C_γ = -0.5$; we regard this as superseded by the EGS4 result.

<figure><img src="image_191.png" /><figcaption>**Figure 34.21:** Fitted values of the scale factor *b* for energy deposition profiles obtained with EGS4 for a variety of elements for incident electrons with $1 \le E_0 \le 100$ GeV. Values obtained for incident photons are essentially the same.</figcaption></figure>

The “shower length” $X_s = X_0/b$ is less conveniently parameterized, since *b* depends upon both *Z* and incident energy, as shown in Fig. 34.21. As a corollary of this *Z* dependence, the number of electrons crossing a plane near shower maximum is underestimated using Rossi’s approximation for carbon and seriously overestimated for uranium. Essentially the same *b* values are obtained for incident electrons and photons. For many purposes it is sufficient to take $b \approx 0.5$.

The length of showers initiated by ultra-high energy photons and electrons is somewhat greater than at lower energies since the first or first few interaction lengths are increased via the mechanisms discussed above.

The gamma function distribution is very flat near the origin, while the EGS4 cascade (or a real cascade) increases more rapidly. As a result Eq. (34.35) fails badly for about the first two radiation lengths; it was necessary to exclude this region in making fits.

Because fluctuations are important, Eq. (34.35) should be used only in applications where average behavior is adequate. Grindhammer *et al.* have developed fast simulation algorithms in which the variance and correlation of *a* and *b* are obtained by fitting Eq. (34.35) to individually simulated cascades, then generating profiles for cascades using *a* and *b* chosen from the correlated distributions [62].

The transverse development of electromagnetic showers in different materials scales fairly accurately with the *Molière radius* $R_M$, given by [63, 64]

$$R_M = X_0 E_s / E_c, \qquad (34.37)$$

where $E_s \approx 21$ MeV (Table 34.1), and the Rossi definition of $E_c$ is used.

In a material containing a weight fraction $w_j$ of the element with critical energy $E_{cj}$ and radiation length $X_j$, the *Molière radius* is given by

$$\frac{1}{R_M} = \frac{1}{E_s} \sum \frac{w_j E_{cj}}{X_j}. \qquad (34.38)$$

Measurements of the lateral distribution in electromagnetic cascades are shown in Refs. [63, 64]. On the average, only 10% of the energy lies outside the cylinder with radius $R_M$. About 99% is contained inside of $3.5R_M$, but at this radius and beyond composition effects become important and the scaling with $R_M$ fails. The distributions are characterized by a narrow core, and broaden

as the shower develops. They are often represented as the sum of two Gaussians.

At high enough energies, the LPM effect (Sec. 34.4.6) reduces the cross sections for bremsstrahlung and pair production, and hence can cause significant elongation of electromagnetic cascades [45].

## 34.6 Muon energy loss at high energy

At sufficiently high energies, radiative processes become more important than ionization for all charged particles. For muons and pions in materials such as iron, this “critical energy” occurs at several hundred GeV. (There is no simple scaling with particle mass, but for protons the “critical energy” is much, much higher.) Radiative effects dominate the energy loss of energetic muons found in cosmic rays or produced at the newest accelerators. These processes are characterized by small cross sections, hard spectra, large energy fluctuations, and the associated generation of electromagnetic and (in the case of photonuclear interactions) hadronic showers [65–73] As a consequence, at these energies the treatment of energy loss as a uniform and continuous process is for many purposes inadequate.

It is convenient to write the average rate of muon energy loss as [74]

$$\langle -dE/dx \rangle = a(E) + b(E) E. \qquad (34.39)$$

Here $a(E)$ is the ionization energy loss given by Eq. (34.5), and $b(E)$ is the sum of $e^+e^-$ pair production, bremsstrahlung, and photonuclear contributions. To the approximation that these slowly-varying functions are constant, the mean range $x_0$ of a muon with initial energy $E_0$ is given by

$$x_0 \approx (1/b) \ln(1 + E_0 / E_{\mu c}), \qquad (34.40)$$

where $E_{\mu c} = a/b$.

Fig. 34.22 shows contributions to $b(E)$ for iron. Since $a(E) \approx 0.002$ GeV g$^{-1}$ cm$^2$, $b(E)E$ dominates the energy loss above several hundred GeV, where $b(E)$ is nearly constant. The rates of energy loss for muons in hydrogen, uranium, and iron are shown in Fig. 34.23 [7].

<figure><img src="image_192.png" /><figcaption>**Figure 34.22:** Contributions to the fractional energy loss by muons in iron due to $e^+e^-$ pair production, bremsstrahlung, and photonuclear interactions, as obtained from Groom *et al.* [7] except for post-Born corrections to the cross section for direct pair production from atomic electrons.</figcaption></figure>

The “muon critical energy” $E_{\mu c}$ can be defined more exactly as the energy at which radiative and ionization losses are equal, and can be found by solving $E_{\mu c} = a(E_{\mu c})/b(E_{\mu c})$. This definition corresponds to the solid-line intersection in Fig. 34.13, and is different from the Rossi definition we used for electrons. It serves the same function: below $E_{\mu c}$ ionization losses dominate, and above $E_{\mu c}$ radiative effects dominate. The dependence of $E_{\mu c}$ on atomic number *Z* is shown in Fig. 34.24.
<figure><img src="image_193.png" /><figcaption>**Figure 34.23:** The average energy loss of a muon in hydrogen, iron, and uranium as a function of muon energy. Contributions to $dE/dx$ in iron from ionization and pair production, bremsstrahlung and photonuclear interactions are also shown.</figcaption></figure>

<figure><img src="image_194.png" /><figcaption>**Figure 34.24:** Muon critical energy for the chemical elements, defined as the energy at which radiative and ionization energy loss rates are equal [7]. The equality comes at a higher energy for gases than for solids or liquids with the same atomic number because of a smaller density effect reduction of the ionization losses. The fits shown in the figure exclude hydrogen. Alkali metals fall 3-4% above the fitted function, while most other solids are within 2% of the function. Among the gases the worst fit is for radon (2.7% high).</figcaption></figure>

<figure><img src="image_195.png" /><figcaption>**Figure 34.25:** The momentum distribution of 1 TeV/c muons after traversing 3 m of iron as calculated by S.I. Striganov [7].</figcaption></figure>

The radiative cross sections are expressed as functions of the fractional energy loss $\nu$. The bremsstrahlung cross section goes

roughly as $1/\nu$ over most of the range, while for the pair production case the distribution goes as $\nu^{-3}$ to $\nu^{-2}$ [75]. “Hard” losses are therefore more probable in bremsstrahlung, and in fact energy losses due to pair production may very nearly be treated as continuous. The simulated momentum distribution of an incident 1 TeV/c muon beam after it crosses 3 m of iron is shown in Fig. 34.25 [7]. The most probable loss is 8 GeV, or 3.4 MeV $g^{-1}cm^2$. The full width at half maximum is 9 GeV/c, or 0.9%. The radiative tail is almost entirely due to bremsstrahlung, although most of the events in which more than 10% of the incident energy lost experienced relatively hard photonuclear interactions. The latter can exceed detector resolution [76], necessitating the reconstruction of lost energy. Tables in Ref. [7] list the stopping power as 9.82 MeV $g^{-1}cm^2$ for a 1 TeV muon, so that the mean loss should be 23 GeV (≈ 23 GeV/c), for a final momentum of 977 GeV/c, far below the peak. This agrees with the indicated mean calculated from the simulation. Electromagnetic and hadronic cascades in detector materials can obscure muon tracks in detector planes and reduce tracking efficiency [77].

## 34.7 Cherenkov and transition radiation [3, 78, 79]

A charged particle radiates if its speed is greater than the local phase speed of light (Cherenkov radiation) or if it crosses suddenly from one medium to another with different optical properties (transition radiation). Neither process is important for energy loss, but both are used in high-energy and cosmic-ray physics detectors.

### 34.7.1 Optical Cherenkov radiation

The angle $\theta_c$ of Cherenkov radiation, relative to the particle's direction, for a particle with speed $\beta c$ in a medium with index of refraction $n$ is

$$ \cos \theta_c = (1/n\beta) \\ \text{or} \\ \tan \theta_c = \sqrt{\beta^2 n^2 - 1} \\ \approx \sqrt{2(1 - 1/n\beta)} \quad \text{for small } \theta_c, \text{ e.g. in gases.} \quad (34.41) $$

The threshold speed $\beta_t$ is $1/n$, and $\gamma_t = 1/(1 - \beta_t^2)^{1/2}$. Therefore, $\beta_t\gamma_t = 1/(2\delta + \delta^2)^{1/2}$, where $\delta = n - 1$. Values of $\delta$ for various commonly used gases are given as a function of pressure and wavelength in Ref. [80]. See its Table 6.1 for values at atmospheric pressure. Data for other commonly used materials are given in Ref. [81].

<figure><img src="image_196.png" /><figcaption>**Figure 34.26:** Cherenkov light emission and wavefront angles. In a dispersive medium, $\theta_c + \eta \neq 90^\circ$.</figcaption></figure>

Practical Cherenkov radiator materials are dispersive. Let $\omega$ be the photon's frequency, and let $k = 2\pi/\lambda$ be its wavenumber. The photons propagate at the group speed $v_g = d\omega/dk = c/[n(\omega) + \omega(dn/d\omega)]$. In a non-dispersive medium, this simplifies to $v_g = c/n$.

In his classical paper, Tamm [82] showed that for dispersive media the radiation is concentrated in a thin conical shell whose
vertex is at the moving charge, and whose opening half-angle $\eta$ is

$$ 
\begin{aligned}
\cot \eta &= \left[ \frac{d}{d\omega} (\omega \tan \theta_c) \right]_{\omega_0} \\
&= \left[ \tan \theta_c + \beta^2 \omega n(\omega) \frac{dn}{d\omega} \cot \theta_c \right]_{\omega_0},
\end{aligned}
\quad (34.42) 
$$

where $\omega_0$ is the central value of the small frequency range under consideration. (See Fig. 34.26.) This cone has a opening half-angle $\eta$, and, unless the medium is non-dispersive ($dn/d\omega = 0$), $\theta_c + \eta \neq 90^\circ$. The Cherenkov wavefront ‘sideslips’ along with the particle [83]. This effect has timing implications for ring imaging Cherenkov counters [84], but it is probably unimportant for most applications.

The number of photons produced per unit path length of a particle with charge ze and per unit energy interval of the photons is

$$ 
\begin{aligned}
\frac{d^2 N}{dE dx} &= \frac{\alpha z^2}{\hbar c} \sin^2 \theta_c = \frac{\alpha^2 z^2}{r_e m_e c^2} \left(1 - \frac{1}{\beta^2 n^2(E)}\right) \\
&\approx 370 \sin^2 \theta_c(E) \text{eV}^{-1} \text{cm}^{-1} \quad (z = 1),
\end{aligned}
\quad (34.43) 
$$

or, equivalently,

$$ \frac{d^2 N}{dx d\lambda} = \frac{2\pi\alpha z^2}{\lambda^2} \left(1 - \frac{1}{\beta^2 n^2(\lambda)}\right). \quad (34.44) $$

The index of refraction $n$ is a function of photon energy $E = \hbar\omega$, as is the sensitivity of the transducer used to detect the light. For practical use, Eq. (34.43) must be multiplied by the transducer response function and integrated over the region for which $\beta n(\omega) > 1$. Further details are given in the discussion of Cherenkov detectors in the Particle Detectors section (Sec. 35.5 of this Review).

When two particles are close together (lateral separation $\lesssim 1$ wavelength), the electromagnetic fields from the particles may add coherently, affecting the Cherenkov radiation. Because of their opposite charges, the radiation from an $e^+e^-$ pair at close separation is suppressed compared to two independent leptons [85].

### 34.7.2 Coherent radio Cherenkov radiation

Coherent Cherenkov radiation is produced by many charged particles with a non-zero net charge moving through matter on an approximately common “wavefront”—for example, the electrons and positrons in a high-energy electromagnetic cascade. The signals can be visible for energies above $10^{16}$ eV; see Sec. 36.3.3 for more details. The phenomenon is called the Askaryan effect [86]. Near the end of a shower, when typical particle energies are below $E_c$ (but still relativistic), a charge imbalance develops. Photons can Compton-scatter atomic electrons, and positrons can annihilate with atomic electrons to contribute even more photons which can in turn Compton scatter. These processes result in a roughly 20% excess of electrons over positrons in a shower. The net negative charge leads to coherent radio Cherenkov emission. The radiation includes a component from the decelerating charges (as in bremsstrahlung). Because the emission is coherent, the electric field strength is proportional to the shower energy, and the signal power increases as its square. The electric field strength also increases linearly with frequency, up to a maximum frequency determined by the lateral spread of the shower. This cutoff occurs at about 1 GHz in ice, and scales inversely with the Moliere radius. At low frequencies, the radiation is roughly isotropic, but, as the frequency rises toward the cutoff frequency, the radiation becomes increasingly peaked around the Cherenkov angle. The radiation is linearly polarized in the plane containing the shower axis and the photon direction. A measurement of the signal polarization can be used to help determine the shower direction. The characteristics of this radiation have been nicely demonstrated in a series of experiments at SLAC [87]. A detailed discussion of the radiation can be found in Ref. [88].

### 34.7.3 Transition radiation

The energy radiated when a particle with charge ze crosses the boundary between vacuum and a medium with plasma frequency $\omega_p$ is

$$ I = \alpha z^2 \gamma \hbar \omega_p / 3, \quad (34.45) $$

where

$$ \hbar\omega_p = \sqrt{4\pi N_e r_e^3} m_e c^2 / \alpha = \sqrt{\rho (\text{in g/cm}^3) \langle Z/A \rangle} \times 28.81 \text{ eV}. \quad (34.46) $$

For styrene and similar materials, $\hbar\omega_p \approx 20$ eV; for air it is 0.7 eV.

<figure><img src="image_197.png" /><figcaption>**Figure 34.27:** X-ray photon energy spectra for a radiator consisting of 200 25 µm thick foils of Mylar with 1.5 mm spacing in air (solid lines) and for a single surface (dashed line). Curves are shown with and without absorption. Adapted from Ref. [89].</figcaption></figure>

The number spectrum $dN_\gamma/d(\hbar\omega)$ diverges logarithmically at low energies and decreases rapidly for $\hbar\omega/\gamma\hbar\omega_p > 1$. About half the energy is emitted in the range $0.1 \le \hbar\omega/\gamma\hbar\omega_p \le 1$. Inevitable absorption in a practical detector removes the divergence. For a particle with $\gamma = 10^3$, the radiated photons are in the soft x-ray range 2 to 40 keV. The $\gamma$ dependence of the emitted energy thus comes from the hardening of the spectrum rather than from an increased quantum yield.

The number of photons with energy $\hbar\omega > \hbar\omega_0$ is given by the answer to problem 13.15 in Ref. [3],

$$ N_{\gamma}(\hbar\omega > \hbar\omega_0) = \frac{\alpha z^2}{\pi} \left[ \left( \ln \frac{\gamma \hbar \omega_p}{\hbar \omega_0} - 1 \right)^2 + \frac{\pi^2}{12} \right], \quad (34.47) $$

within corrections of order $(\hbar\omega_0/\gamma\hbar\omega_p)^2$. The number of photons above a fixed energy $\hbar\omega_0 \ll \gamma\hbar\omega_p$ thus grows as $(\ln \gamma)^2$, but the number above a fixed fraction of $\gamma\hbar\omega_p$ (as in the example above) is constant. For example, for $\hbar\omega > \gamma\hbar\omega_p/10$, $N_\gamma = 2.519 \alpha z^2 / \pi = 0.59\% \times z^2$.

The particle stays “in phase” with the x ray over a distance called the formation length, $d(\omega) = (2c/\omega)(1/\gamma^2 + \theta^2 + \omega_p^2/\omega^2)^{-1}$. Most of the radiation is produced in this distance. Here $\theta$ is the x-ray emission angle, characteristically $1/\gamma$. For $\theta = 1/\gamma$ the formation length has a maximum at $d(\gamma\omega_p/\sqrt{2}) = \gamma c/\sqrt{2}\omega_p$. In practical situations it is tens of µm.

Since the useful x-ray yield from a single interface is low, in practical detectors it is enhanced by using a stack of $N$ foil radiators—foils $L$ thick, where $L$ is typically several formation lengths—separated by gas-filled gaps. The amplitudes at successive interfaces interfere to cause oscillations about the single-interface spectrum. At increasing frequencies above the position of the last interference maximum ($L/d(\omega) = \pi/2$), the formation zones, which have opposite phase, overlap more and more and the
spectrum saturates, $dI/d\omega$ approaching zero as $L/d(\omega) \to 0$. This is illustrated in Fig. 34.27 for a realistic detector configuration.

For regular spacing of the layers fairly complicated analytic solutions for the intensity have been obtained [89,90]. Although one might expect the intensity of coherent radiation from the stack of foils to be proportional to $N^2$, the angular dependence of the formation length conspires to make the intensity $\propto N$.

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[nist.gov/pml/xcom-photon-cross-sections-database](http://nist.gov/pml/xcom-photon-cross-sections-database).
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