34. Passage of Particles Through Matter

1

34. Passage of Particles Through Matter
Revised August 2023 by D.E. Groom (LBNL) and S.R. Klein (NSD LBNL; UC Berkeley).

34.1
34.2

Notation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Electronic energy loss by heavy particles . . . . . . . . . . . . . . . . . . . . . . .
34.2.1 Moments and cross sections . . . . . . . . . . . . . . . . . . . . . . . . . . . .
34.2.2 Maximum energy transfer to an electron in a single collision . . . . . . . . . .
34.2.3 Stopping power at intermediate energies . . . . . . . . . . . . . . . . . . . . .
34.2.4 Mean excitation energy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
34.2.5 Density effect . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
34.2.6 Energy loss at low energies . . . . . . . . . . . . . . . . . . . . . . . . . . . .
34.2.7 Energetic knock-on electrons (δ rays) . . . . . . . . . . . . . . . . . . . . . . .
34.2.8 Restricted energy loss rates for relativistic ionizing particles . . . . . . . . . .
34.2.9 Fluctuations in energy loss . . . . . . . . . . . . . . . . . . . . . . . . . . . .
34.2.10 Energy loss in mixtures and compounds . . . . . . . . . . . . . . . . . . . . .
34.2.11 Ionization yields . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
34.3 Multiple scattering through small angles . . . . . . . . . . . . . . . . . . . . . . . .
34.4 Photon and electron interactions in matter . . . . . . . . . . . . . . . . . . . . . .
34.4.1 Collision energy losses by e± . . . . . . . . . . . . . . . . . . . . . . . . . . .
34.4.2 Radiation length . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
34.4.3 Bremsstrahlung energy loss by e± . . . . . . . . . . . . . . . . . . . . . . . .
34.4.4 Critical energy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
34.4.5 Energy loss by photons . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
34.4.6 Bremsstrahlung and pair production at very high energies . . . . . . . . . . .
34.4.7 Photonuclear and electronuclear interactions at still higher energies . . . . . .
34.5 Electromagnetic cascades . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
34.6 Muon energy loss at high energy . . . . . . . . . . . . . . . . . . . . . . . . . . . .
34.7 Cherenkov and transition radiation . . . . . . . . . . . . . . . . . . . . . . . . . . .
34.7.1 Optical Cherenkov radiation . . . . . . . . . . . . . . . . . . . . . . . . . . .
34.7.2 Coherent radio Cherenkov radiation . . . . . . . . . . . . . . . . . . . . . . .
34.7.3 Transition radiation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

1
2
2
3
4
6
7
10
11
11
11
14
15
15
17
17
18
19
20
21
22
25
27
29
33
33
34
35

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 ofinterest for particle detectors (ionization, Cherenkov radiation, transition radiation). Much of the
focus is on particles heavier than electrons (π ± , 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 notdiscussed.

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

S. Navas et al. (Particle Data Group), Phys. Rev. D 110, 030001 (2024)
31st May, 2024 10:18am

34. Passage of Particles Through Matter2

Table 34.1: Summary of variables used in this section. The kinematic
variables β and γ have their usual relativistic meanings.
Symb.

Definition

Value or (usual) units

me c2 electron mass × c2
reclassical electron radius
e2 /4π0 me c2αfine structure constante2 /4π0 ~c
NA
Avogadro’s number

0.510 998 950 00(15) MeV

ρ
x
M
E
T
W

g cm−3g cm−2
MeV/c2MeV
MeV
MeV

densitymass per unit area
incident particle mass
incident part. energy γM c2
kinetic energy, (γ − 1)M c2
energy transfer to an electron

2.817 940 3227(19) fm
1/137.035 999 139(31)6.022 140 76 × 1023 mol−1

in a single collision
Wmax Maximum possible energy transfer MeV
to an electron in a single collisionkbremsstrahlung photon energy
MeVzcharge number of incident particle
Z
atomic number of absorberA
atomic mass of absorberg mol−1K
4πNA re2 me c2
0.307 075 MeV mol−1 cm2(Coefficient for dE/dx)
I
mean excitation energyeV (Nota bene!)
δ(βγ) density effect correction to ionization energy lossp
~ωp plasma energyρ hZ/Ai × 28.816 eVp|−→ ρ in g cm−323
4πN r m c /α
e e

Ne
wj
nj
X0
Ec
Eµc
Es
RM

e

electron density(units of re )−3weight fraction of the jth element in a compound or mixt.
∝ number of jth kind of atoms in a compound or mixtureradiation lengthg cm−2
critical energy for electrons
MeV
critical energy for muonsGeVpscale energy 4π/α me c2
21.2052 MeV
Molière radiusg cm−234.2 Electronic energy loss by heavy particles
34.2.1 Moments and cross sections
The electronic interactions of fast charged particles with speed v = βc 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”
31st May, 2024

3

34. Passage of Particles Through Matterparticles), scattering from free electrons is adequately described by the Rutherford differential
cross section [2, 3].
dσR (W ; β)2πre2 me c2 z 2 (1 − β 2 W/Wmax )=,(34.1)dW
β2
W2
where Wmax , the maximum energy transfer possible in a single collision, is discussed below. It
differs from the classical cross section by the factor (1 − β 2 W/W max), which arises when the spinof the target electrons is taken into account.
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 Wmax . 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 sectiondσB (W ; β)dσR (W, β)=
B(W ) .
dW
dW

(34.2)

Electronic binding is accounted for by the correction factor B(W ). Examples of B(W ) and dσB /dW
can be seen in Figs. 5 and 6 of Ref. [1]. For a given material, the correction results in introducing aneffective 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 distanceδx is Ne δx (dσ/dW )dW , where dσ(W ; β)/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 isconvenient to define the momentsZ

Mj (β) = Ne δx

Wjdσ(W ; β)dW ,dW(34.3)

so that M0 is the mean number of collisions in δx, M1 is the mean energy loss in δx, (M2 −M1 )2 is thevariance, etc. The number of collisions is Poisson-distributed with mean M0 . Ne is either measured
in electrons/g (Ne = NA Z/A) or electrons/cm3 (Ne = NA ρZ/A). The former is used throughout
this chapter, since quantities of interest (dE/dx, X0 , 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 >> me ,Wmax =

2me c2 β 2 γ 2.
1 + 2γme /M + (me /M )2(34.4)

In older references [2,9] the “low-energy” approximation Wmax = 2me c2 β 2 γ 2 , valid for 2γme  M ,is often implicit. For a pion in copper, the error thus introduced into dE/dx is greater than 6% at
100 GeV. For 2γme  M , Wmax = M c2 β 2 γ.
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 thecross section for rare hard collisions is modified, the average stopping power, dominated by many
softer collisions, is almost unchanged.
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34. Passage of Particles Through Matter

Mass stopping power [MeV cm2/g]

4

μ+ on Cu

100

μ−

Bethe

10

LindhardScharff

AndersenZiegler

1

Radiative

Minimum
ionization

NuclearlossesRadiative
effects
reach 1%

Eμc

Radiative
lossesWithout δ

0.001

0.01

0.1

1

10

0.1

1

10

100

1

[MeV/c]

βγ

100

1000

10

4

10

10

100

1

10

[GeV/c]Muon momentum

5[TeV/c]100

Figure 34.1: Mass stopping power (dE/dx) for positive muons in copper as a function of βγ =p/M c 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 fromICRU 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.
34.2.3 Stopping power at intermediate energies
The mean rate of energy loss by moderately relativistic charged heavy particles is well describedby the “Bethe equation” [2, 4, 5, 9],


dE
−
dx



"

#

1 1 2me c2 β 2 γ 2 Wmax
δ(βγ)
= Kz
ln
− β2 −
.
2
2
Aβ 2
I
2
2Z

(34.5)

Eq. (34.5) is valid in the region 0.1 . βγ . 1000 with an accuracy of a few percent. At βγ ∼ 0.1
the projectile speed is comparable to atomic electron “speed,” and at βγ ∼ 1000 radiative effects
begin to be important (Sec. 34.6). Both limits are Z dependent. A minor dependence on M athigh energies is introduced through Wmax , but for all practical purposes the stopping power in agiven material is a function of β alone. Small corrections are discussed in Sec. 34.2.6.1,2
This is the mass stopping power; with the symbol definitions and values given in Table 34.1,
the units are MeV g−1 cm2 . 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, isρ dE/dx, where ρ is the density in g/cm3 .
1
For incident spin 1/2 particles, (Wmax /E)2 /4 is included in the square brackets. Although this correction iswithin the uncertainties in the total stopping power, its inclusion avoids a systematic bias.
2
In this section, “dE/dx” will be understood to mean the mass stopping power “h−dE/dxi.”

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534. Passage of Particles Through Matter

The stopping power at first falls as 1/β α where α ≈ 1.7–1.5, decreasing with increasing Z, andreaches a broad minimum at βγ = 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. Twothirds of the rise is produced by the explicit β 2 γ 2 dependence through the relativistic flattening
and extension of the particle’s electric field. Rather than producing ionization at greater andgreater 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 δ(βγ). The otherthird is introduced by the β 2 γ dependence of Wmax , the maximum possible energy transfer to arecoil 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 byrare 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 restrictedenergy 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/.
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 afactor 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 β alone; themass 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 ofenergy loss in different materials, although there is a slow decrease in the rate of energy loss withincreasing 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, δ(βγ), discussed
in Sec. 34.2.5. The stopping power functions are characterized by broad minima whose position
drops from βγ = 3.5 to 3.0 as Z goes from 7 to 100. The values of minimum ionization as a functionof 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 β, R/M is a function of E/M or pc/M . In practice,range is a useful concept only for low-energy hadrons (R . λI , where λI 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 βγ (= p/M c) 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.
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34. Passage of Particles Through Matter

610〈– dE/dx〉 (MeV g —1cm2)8
65

H2 liquid4
He gas

3
2

1
0.1

0.1

Sn
Pb

1.0

C

1000

10 000

0.1

1.0
10
100
Muon momentum (GeV/c)

1000

0.1

1.0
10
100
Pion momentum (GeV/c)

1000

10
100
1000
Proton momentum (GeV/c)

10 0001.010100βγ = p/McFe

Al

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 & 1000, and at lower momenta for muons inhigher-Z absorbers. See Fig. 34.23.
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 substantiallywith 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
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34. Passage of Particles Through Matter

7

2.5– dE/dx min (MeV g —1cm 2 )H2 gas: 4.10
H2 liquid: 3.97

2.0

2.35 — 0.28 ln(Z)1.5Solids
Gases

1.0

0.5

H

He

1

2

Li Be B C NO Ne
5

10

Z

Fe
20

Sn

U

50

100

Figure 34.3: Mass stopping power at minimum ionization for the chemical elements. The straightline is fitted for Z > 6. A simple functional dependence on Z is not to be expected, since dE/dx
also depends on other variables.

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 distantcollision contribution to the logarithmic term in Eq. (34.5) increases as β 2 γ 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,

δ(βγ)/2 → ln(~ωp /I) + ln βγ − 1/2 ,(34.6)

where δ(βγ)/2 is the density effect correction introduced in Eq. (34.5) and ~ωp is the plasma energy
defined in Table 34.1. A comparison with Eq. (34.5) shows that dE/dx then grows as ln Tmax rather
than ln β 2 γ 2 Tmax , and that the mean excitation energy I is replaced by the plasma energy ~ωp .
An example of the ionization stopping power as calculated with and without the density effectcorrection is shown in Fig. 34.1. Since the plasma frequency scales as the square root of theelectron density, the correction is much larger for a liquid or solid than for a gas, as is illustrated
in Fig. 34.2.
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34. Passage of Particles Through Matter

8

50000
20000
10000

Pb

R /M (g cm −2 GeV −1 )

5000

Fe

C

2000
1000

He gas500

H 2 liquid

200
100
50
20
10
5
2
1

0.1

2

5

1.0

0.02

0.05

0.1

0 .0 2

0.1

0.05

0.2

2

5

10.0

2

5

100.0

0.2

0.5

1.0

2.0

5.0

10.0

βγ = p/Mc

Muon momentum (GeV/c)
0.1

0.2

0 .5

1.0

Pion momentum (GeV/c)

0.5

1.0

2.0

5.0

2.0

10.0

Proton momentum (GeV/c)

5.0

20.0

10.0
50.0

Figure 34.4: Range of heavy charged particles in liquid (bubble chamber) hydrogen, helium gas,carbon, iron, and lead. For example: For a K + whose momentum is 700 MeV/c, βγ = 1.42. For
lead we read R/M ≈ 396, and so the range is 195 g cm−2 (17 cm).

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934. Passage of Particles Through Matter

Figure 34.5: Mean excitation energies (divided by Z) as adopted by the ICRU [12]. Those basedon experimental measurements are shown by symbols with error flags; the interpolated values are
simply joined. The grey point is for liquid H2 ; the black point at 19.2 eV is for H2 gas. The
open circles show more recent determinations by Bichsel [13]. The dash-dotted curve is from theapproximate formula of Barkas [14] used in early editions of this Review.
The density effect correction is usually computed using Sternheimer’s parameterization [17]:2(ln 10)x − C2(ln 10)x − C + a(x − x)k1
δ(βγ) =
0
2(x−x0 )δ0 10

if x ≥ x1 ;if x0 ≤ x < x1 ;if x < x0 (nonconductors);
if x < x0 (conductors)(34.7)

Here x = log10 βγ = log10 (p/M c). 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 areadjusted to give a best fit to the results of detailed calculations for momenta below M c exp(x1 ).
For nonconductors the correction is 0 below βγ = 10x0 , corresponding to 100–200 MeV for pions
and 1–2 GeV for protons. For conductors it decreases rapidly below this point. Parameters for theelements 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 β 2 γ 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
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1034. Passage of Particles Through Mattereven higher energies (e.g., > 332 GeV for muons in iron, and at a considerably higher energy for
protons in iron), radiative effects are more important than ionization losses. These are especiallyrelevant 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 islarge compared to that of the atomic electrons. This presents a problem at low energies, where
Wmax 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(β) = La (β) −C(β)+ zL1 (β) + z 2 L2 (β) .
Z(34.8)

Here La is the square-bracketed terms of Eq. (34.5), C/Z is the sum of shell corrections and zL1 andz 2 L2 are Barkas and Bloch correction terms [6, 21]. With these corrections, the Bethe treatment isaccurate to about 1% down to β ≈ 0.05, or about 1 MeV for protons (0.13 MeV for muons). Valuesof La , C/Z, L1 , and L2 in the range T = 0.3–30 MeV for a proton traversing aluminum can befound 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 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 ofplaces; 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 zL1 . 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 numberof 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 Ashley et al. [23] is often used; it has two parameters determinedby experimental data. For protons in aluminum, L1 /La 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 2 L2 . Bloch’s extension of Bethe’s theory introduced a low-energy correctionthat 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,−L2 /L| 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 < β < 0.05 there is no satisfactory theory. For protons, one usually relieson 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 (βγ = 0.005), rises to a maximum of 210 MeV cm2 g−1 at T ≈ 120 keV (βγ = 0.016),
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34. Passage of Particles Through Matter11

then falls to 118 MeV cm2 g−1 at T = 1 MeV (βγ = 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 β [25]. Finally, we note that at even lower energies, e.g., for protons of less thanseveral 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  I is [2]d2 N
1Z 1 F (T )= Kz 2(34.9)dT dx
2
A β2 T 2
for I  T ≤ Wmax , where Wmax is given by Eq. (34.4). Here β is the speed of the primaryparticle. The factor F is spin-dependent, but is about unity for T  Wmax . For spin-0 particles
F (T ) = (1 − β 2 T /Wmax ); 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].
δ rays of even modest energy are rare. For a β ≈ 1 particle, for example, on average only one
collision with Te > 10 keV will occur along a path length of 90 cm of argon gas [1].
A δ ray with kinetic energy Te and corresponding momentum pe is produced at an angle θ givenbycos θ = (Te /pe )(pmax /Wmax ) ,(34.10)where pmax is the momentum of an electron with the maximum possible energy transfer Wmax .
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 ≤ Wcut ≤ Wmax . The restricted energy loss rate isdE
Z 1 1 2me c2 β 2 γ 2 Wcut β 2
Wcut−= Kz 2ln−
1+
2
2
dx T <Wcut
Aβ 2
I
2
Wmax






δ
−
.
2


(34.11)This form approaches the normal Bethe function (Eq. (34.5)) as WRcut → Wmax . It can be verified
Wmax
T (d2 N/dT dx)dT , wherethat the difference between Eq. (34.5) and Eq. (34.11) is equal to W
cut
d2 N/dT dx is given by Eq. (34.9).
Since Wcut replaces Wmax in the argument of the logarithmic term of Eq. (34.5), the βγ term
producing the relativistic rise in the close-collision part of dE/dx is replaced by a constant, and|dE/dx|T <Wcut approaches the constant “Fermi plateau.” (The density effect correction δ eliminates
the explicit βγ dependence produced by the distant-collision contribution.) This behavior is illustrated in Fig. 34.6, where restricted loss rates for two examples of Wcut are shown in comparison
with the full Bethe dE/dx and the Landau-Vavilov most probable energy loss (to be discussed inSec. 34.2.9 below).
“Restricted energy loss” is cut at the total mean energy, not the single-collision energy above
Wcut 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),3 the energy loss probability
distribution f (∆; βγ, x) is adequately described by the highly-skewed Landau (or Landau-Vavilov)
distribution [29, 30].
3“Moderate thickness” means G . 0.05–0.1, where G is given by Rossi Ref. [2], Eq. 2.7(10). It is Vavilov’sκ [29]. G is proportional to the absorber’s thickness, and as such parameterizes the constants describing the Landau
distribution. These are fairly insensitive to thickness for G . 0.1, the case for most detectors.
31st May, 2024

34. Passage of Particles Through Matter

12

MeV g−1 cm2 (Electonic loses only)

3.0
2.5

Silicon
Bethe

2.0

Restricted energy loss for :Tcut = 10 dE/dx|min
Tcut = 2 dE/dx|min

1.5

Landau/Vavilov/Bichsel Δp /x for:x/ρ = 1600 μm
320 μm
80 μm

1.0
0.5
0.1

1.0

10.0

100.0

Muon kinetic energy (GeV)

1000.0Figure 34.6: Bethe dE/dx, two examples of restricted energy loss, and the Landau most probableenergy per unit thickness in silicon. The change of ∆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.
The most probable energy loss is [31]4"

2me c2 β 2 γ 2ξ∆p = ξ ln+ ln + j − β 2 − δ(βγ)
I
I

#

,

(34.12)where ξ = (K/2) hZ/Ai z 2 (x/β 2 ) MeV for a detector with a thickness x in g cm−2 , and j = 0.200[31].5 While dE/dx is independent of thickness, ∆p /x scales as a ln x + b. The density correction
δ(βγ) was not included in Landau’s or Vavilov’s work, but it was later included by Bichsel [31].
The high-energy behavior of δ(βγ) (Eq. (34.6)) is such that"

2me c2 ξ∆p −→ ξ ln+jβγ&100(~ωp )2#.

(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 ∆p /x in silicon are shown as a functionof 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 the4

Practical calculations can be expedited by using the tables of δ and β from the text versions of the muon energy
loss tables to be found at pdg.lbl.gov/curremt/AtomicNuclearProperties.
5
Rossi [2], Talman [32], and others give somewhat different values for j. The most probable loss is not sensitive
to its value.

31st May, 2024

34. Passage of Particles Through Matter

13

Energy loss [MeV cm2/g]
1.4

1.8

10 GeV muon
1.7 mm Si

150

f(Δ) [MeV−1]

1.6

2.0

2.2

2.4

1.0

M 0(Δ)/M 0(∞)
0.8

Μ 1(Δ)/Μ 1(∞)

0.6

Landau-Vavilov
Bichsel (Bethe-Fano theory)

0.4

100

fwhm
50

0
0.4

0.5

0.2

<Δ >

Δp
0.6

0.7

0.8

Mj (Δ)/Mj (∞)

1.2

0.9

0.0
1.0

Electronic energy loss Δ [MeV]
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 LandauVavilov function (dot-dashed) uses a Rutherford cross section without atomic binding corrections
but with a kinetic energy transfer limit of Wmax . The solid curve was calculated using Bethe-Fanotheory. M0 (∆) and M1 (∆) 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 LandauVavilov function is about 4ξ for detectors of moderate thickness. ∆p is the most probable energy
loss, and h∆i divided by the thickness is the Bethe dE/dx.
mean (M1 (h∆i)/M1 (∞)). Folding in experimental resolution displaces the peak of the distribution,
usually toward a higher value. 90% of the collisions (M1 (h∆i)/M1 (∞)) contribute to energy deposits
below the mean. It is the very rare high-energy-transfer collisions, extending to Wmax 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 tolarge 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.6 It rises as ln γ because Wmax increases as γ 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, 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 Tµ = [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 essentiallyconstant.
The Landau distribution fails to describe energy loss in thin absorbers such as gas TPC cells [1]6

It does find application in dosimetry, where only bulk deposit is relevant.

31st May, 2024

34. Passage of Particles Through Matter

14

0.50

(MeV g −1 cm 2 )

∆ /x

1.00

1.50

1.0

2.50

500 MeV pion in silicon

640 μ m (149 mg/cm2 )320 μ m (74.7 mg/cm 2 )160 μ m (37.4 mg/cm 2 )80 μ m (18.7 mg/cm 2 )f (∆ /x )

0.8
0.6

w

0.4

Mean energy
loss rate∆ p /x

0.2
0.0
100

2.00

200

300

∆ /x

400

(eV/μm)500

600

Figure 34.8: Straggling functions in silicon for 500 MeV pions, normalized to unity at the mostprobable value ∆p /x. The width w is the full width at half maximum.
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 ∆p /x may be calculated adequately with Eq. (34.12), the distributions are
significantly wider than the Landau width w = 4ξ 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 distributionis 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.
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 theright proportion (Bragg additivity). In this case,


dE
dx



=

X



wj

dE
,
dx j


(34.14)

where dE/dx|j is the mean rate of energy loss (in MeV g cm−2 ) in the jth element. Eq. (34.5)
can be inserted into Eq. (34.14) to find expressions for hZ/Ai, hI i, and hδi; for example, hZ/Ai =PPP
wj Zj /Aj = nj Zj / nj Aj . However, hI i as defined this way is an underestimate, because in
a compound electrons are more tightly bound than in the free elements, and hδi as calculated thisway has little relevance, because it is the electron density that matters. If possible, one uses thetables given in Refs. [18,34], or the recipes given in [19] (repeated in Ref. [7]), that include effective
31st May, 2024

34. Passage of Particles Through Matter

15

1.00
0.95
0.90

(∆ p /x) / dE/dx min

0.85
0.80

2)
x = 640 µ m (149 mg/cm 2)
320 µ m (74.7 mg/cm
2)160 µ m (37.4 mg/cm
2)80 µ m (18.7 mg/cm

0.75
0.70
0.65
0.60
0.55
0.50
0 .3

1

3

10

30

βγ (= p/m )

100

300

1000

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 cm2 ).
excitation energies and interpolation coefficients for calculating the density effect correction for thechemical elements and nearly 200 mixtures and compounds. Otherwise, use the recipe for δ given
in Refs. [7, 19], and calculate hIi 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 typicallythree 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 thisdeflection 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 thecentral 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
31st May, 2024

34. Passage of Particles Through Matter

16

protons) and by Shen et al. [39] (relativistic pions, kaons, and protons).7
If we define
1 rms
θ0 = θ rms
plane = √ θspace ,2(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]:"13.6 MeV
θ0 =
z
βcp

r

x
x z2
1 + 0.088 log10 ()
X0
X0 β 2

13.6 MeV
=
z
βcp

r

x
x z2
1 + 0.038 ln()
X0
X0 β 2"##(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 accountthe p and z dependence quite well at small Z, but for large Z and small x the β-dependence is notwell represented. Further improvements are discussed in Ref. [40].
Eq. (34.16) describes scattering from a single material, while the usual problem involves themultiple 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; theresult is systematically too small. It is much more accurate to apply Eq. (34.16) once, after finding
x and X0 for the combined scatterer.

xx/2

ΨplanesplaneyplaneθplaneFigure 34.10: Quantities used to describe multiple Coulomb scattering. The particle is incident
in the plane of the figure.
The nonprojected (space) and projected (plane) angular distributions are given approximatelyby [35]
2
θspace
1



 dΩ,


exp −(34.17)
2π θ022θ022

1
 θplane  dθplane ,
√
exp 
−

 2θ 2 

2π θ0
0



7



(34.18)Shen et al.’s measurements show that Bethe’s simpler methods of including atomic electron effects agrees betterwith experiment than does Scott’s treatment.

31st May, 2024

17

34. Passage of Particles Through Matter222
where θ is the deflection angle. In this approximation, θspace≈ (θplane,x+ θplane,y), where the x
and y axes are orthogonal to the direction of motion, and dΩ ≈ dθplane,x dθplane,y . Deflections into
θplane,x and θplane,y are independent and identically distributed. Fig. 34.10 shows these and otherquantities sometimes used to describe multiple Coulomb scattering. They are

1 rms
ψ rms
plane = √ θ plane31= √ θ0 ,3(34.19)

1
rms
y rms
plane = √ x θ plane31= √ x θ0 ,3

(34.20)

1
1
rms
s rms
plane = √ x θ plane = √ x θ0 .
4 3
4 3

(34.21)

All the quantitative estimates in this section apply only in the limit of small θ rms
plane and in theabsence 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 ρyθ = 3/2 ≈ 0.87. For
Monte Carlo generation of a joint (y plane , θplane ) distribution, or for other calculations, it may bemost convenient to work with independent Gaussian random variables (z1 , z2 ) with mean zero andvariance one, and then set√√
yplane =z1 x θ0 (1 − ρ2yθ )1/2 / 3 + z2 ρyθ x θ0 / 3√=z1 x θ0 / 12 + z2 x θ0 /2;(34.22b)

θplane =z2 θ0 .

(34.22c)(34.22a)Note that the second term for y plane equals x θplane /2 and represents the displacement that would
have occurred had the deflection θplane 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 otherprocesses (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 shouldbe the entire kinetic energy, Wmax = me c2 (γ − 1), but because the particles are identical, themaximum is half this, Wmax /2. (The results are the same if the transferred energy is  or if thetransferred energy is Wmax − . 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
31st May, 2024

34. Passage of Particles Through Matter18
the stopping power:


dE
−
dx



"

1 Z 1
me c2 β 2 γ 2 {me c2 (γ − 1)/2}= K
ln+ (1 − β 2 )2 A β2I2
1
2γ − 1ln 2 +
−
2
γ
8



γ−1
γ

#

2

−δ

(34.23)The logarithmic term can be compared with the logarithmic term in the Bethe equation(Eq. (34.2)) by substituting Wmax = me c2 (γ − 1)/2. Electron-positron scattering is describedby the fairly complicated Bhabha cross section [27]. There is no identical particle problem, so
Wmax = me c2 (γ − 1). The first moment of the Bhabha equation yields


dE
−
dx



"

1 Z 1
me c2 β 2 γ 2 {me c2 (γ − 1)}= Kln+ 2 ln 2
2 A β2
2I 2
β2
14
10
4
−
23 +
+
+
2
12
γ + 1 (γ + 1)
(γ + 1)3


#



−δ

.

(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 soundtheoretical 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.
As shown in Fig. 34.11, stopping powers for e− , e+ , and heavy particles are not dramaticallydifferent. In silicon, the minimum value for electrons is 1.50 MeV cm2 /g (at γ = 3.3); for positrons,
1.46 MeV cm2 /g (at γ = 3.7), and for muons, 1.66 MeV cm2 /g (at γ = 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 X0 , usually measured in g cm−2 . It is the mean distanceover 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. X0 has been
calculated and tabulated by Y.S. Tsai [42]:o1NA n 2 = 4αre2
Z Lrad − f (Z) + Z L0rad .
X0
A(34.25)For A = 1 g mol−1 , 4αre2 NA /A = (716.408 g cm−2 )−1 . Lrad and L0rad 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-placeaccuracy byf (Z) =a2 (1 + a2 )−1 + 0.20206 − 0.0369 a2 + 0.0083 a4 − 0.002 a6 ,(34.26)

where a = αZ [43].
The radiation length in a mixture or compound may be approximated by
1/X0 =

X

wj /Xj ,where wj and Xj are the fraction by weight and the radiation length for the jth element.
31st May, 2024

(34.27)

34. Passage of Particles Through Matter19

Table 34.2: Tsai’s Lrad and L0rad , for use in calculating the radiation
length in an element using Eq. (34.25).
Element Z

Lrad

L0rad

H
1
He
2
Li
3
Be
4
Others > 4

5.31
4.79
4.74
4.71
ln(184.15 Z −1/3 )

6.144
5.621
5.805
5.924
ln(1194 Z −2/3 )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. Adaptedfrom 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 X0 (Pb) =5.82 g/cm2 , but we have modified the figures to reflect the value given in the Table of Atomic andNuclear Properties of Materials (X0 (Pb) = 6.37 g/cm2 ).
34.4.3 Bremsstrahlung energy loss by e±
At very high energies and except at the high-energy tip of the bremsstrahlung spectrum, thecross section can be approximated in the “complete screening case” as [42]dσ/dk = (1/k)4αre2 ( 43 − 34 y + y 2 )[Z 2 (Lrad − f (Z)) + Z L0rad ] + 19 (1 − y)(Z 2 + Z) ,

(34.28)where y = k/E is the fraction of the electron’s energy transferred to the radiated photon. At smally (the “infrared limit”) the term on the second line ranges from 1.7% (low Z) to 2.5% (high Z) of
31st May, 2024

34. Passage of Particles Through Matter20

the total. If it is ignored and the first line simplified with the definition of X0 given in Eq. (34.25),
we have


dσ
A
4
4
2 .
−
y
+
y
=
(34.29)dk
X0 NA k 3 3This cross section (times k) is shown by the top curve in Fig. 34.12.

( X0 NA/A) ydσLPM/dy

10 GeV

Bremsstrahlung

1.2
100 GeV
1 TeV
0.8

10 TeV

100 TeV

0.4

1 PeV
10 PeV
0
0

0.25

0.5

0.75

1

y = k/E
Figure 34.12: The normalized bremsstrahlung cross section k dσLP M /dk in lead versus the fractional photon energy y = k/E. The vertical axis has units of photons per radiation length.
This formula is accurate except near y = 1, where screening may become incomplete, and neary = 0, where the infrared divergence is removed by the interference of bremsstrahlung amplitudes
from nearby scattering centers (the LPM effect) [44, 45] and dielectric suppression [46, 47]. Theseand other suppression effects in bulk media are discussed in Sec. 34.4.6.
With decreasing energy (E . 10 GeV) the high-y cross section drops and the curves becomerounded as y → 1. Curves of this familar shape can be seen in Rossi [2] (Figs. 2.11.2,3); see also
the review by Koch & Motz [48].
Except at these extremes, and still in the complete-screening approximation, the number ofphotons with energies between kmin and kmax emitted by an electron travelling a distance d  X0
is
"
#


2
2
d 4
kmax
4(kmax − kmin ) kmax
− kmin
Nγ =
ln
−
+
.
(34.30)
X0 3
kmin
3E
2E 2
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 Ec
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 ≈ E/X0 . This form has been found to describe
31st May, 2024

34. Passage of Particles Through Matter

21

200

20

rah
lun

rem
sst

ct
b

Ex
a

em

30

E

Rossi:
Ionization per X 0= electron energy

50
40

al

t
To
s≈

70

Br

d E /dx × X 0 (MeV)

100g

Copper
X 0 = 12.86 g cm−2
E c = 19.63 MeV

Ionization

Brems = ionization
10

2

5

10
20
50
Electron energy (MeV)

100

200

Figure 34.13: Two definitions of the critical energy Ec .

transverse electromagnetic shower development more accurately (see below). These definitions areillustrated in the case of copper in Fig. 34.13.
The accuracy of approximate forms for Ec 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 withfunctions of the form a/(Z + b)α , but α was found to be essentially unity. Since Ec also depends
on A, I, and other factors, such forms are at best approximate.
Values of Ec 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 varietyof 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
31st May, 2024

34. Passage of Particles Through Matter

22

400
200

Ec (MeV)100
710 MeV________
Z + 0.92

610 MeV________
Z + 1.24

50

Solids
Gases

20
10
5

H

He

Li Be B C N O Ne

1

2

5

10
Z

Fe

20

Sn

50

100

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.
cross section [42] reduces to
i
A h
dσ
=
1 − 34 x(1 − x)
dx
X0 NA

(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 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 thesolid 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 crosssection:σ = 97 (A/X0 NA ).
(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 toa given center is small (∝ k/E(E − k), in the case of bremsstrahlung), the interaction is spread overa comparatively long distance called the formation length (∝ E(E − k)/k) via the uncertainty principle. In alternate language, the formation length is the distance over which the highly relativistic
31st May, 2024

34. Passage of Particles Through Matter

23(a) Carbon ( Z = 6)
- experimental σtot

Cross section (barns/atom)

1 Mb

σp.e.
1 kb

σRayleigh

1b

κ nuc
σCompton

κe

10 mb

(b) Lead ( Z = 82)

Cross section (barns/atom)

1 Mb- experimental σtot

σp.e.

σRayleigh
1 kb

κ nuc
σg.d.r.
1bκeσCompton
10 mb
10 eV

1 keV

1 MeV

1 GeV

100 GeV

Photon Energy

Figure 34.15: Photon total cross sections as a function of energy in carbon and lead, showing thecontributions of different processes [50]:σp.e. = Atomic photoelectric effect (electron ejection, photon absorption)
σRayleigh = Rayleigh (coherent) scattering–atom neither ionized nor excitedσCompton = Incoherent scattering (Compton scattering off an electron)
κnuc = Pair production, nuclear fieldκe = Pair production, electron field
σg.d.r. = Photonuclear interactions, most notably the Giant Dipole Resonance [51]. In theseinteractions, the target nucleus is usually broken up.
Original figures through the courtesy of John H. Hubbell (NIST).

electron and the photon “split apart.” The interference is usually destructive. Calculations of the
31st May, 2024

34. Passage of Particles Through Matter

24
100

Absorption length

λ (g/ cm 2 )

10
1

Si

0.1
0.01

H

C

1 keV

10 keV

Fe

Sn

Pb

0.001
10

–4

10

–5

10

–6

10 eV

100 eV

100 keV

1 MeV
10 MeV
Photon energy

100 MeV

1 GeV

10 GeV

100 GeVFigure 34.16: The photon mass attenuation length (or mean free path) λ = 1/(µ/ρ) for various
elemental absorbers as a function of photon energy. The mass attenuation coefficient is µ/ρ, whereρ is the density. The intensity I remaining after traversal of thickness t (in mass/unit area) is
given by I = I0 exp(−t/λ). The accuracy is a few percent. For a chemical compound or mixture,P
1/λeff ≈ elements wZ /λZ , where wZ 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 areincluded, not all these processes result in energy deposition. The data for 30 eV < E < 1 keV are
from Ref. [52], those for 1 keV < E < 100 GeV from Ref. [53].
“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].
In amorphous media, bremsstrahlung is suppressed if the photon energy k is less than E 2 /(E +ELP M ) [45], where8
X0
X0
ELP M = (me c2 )2 α= (7.7 TeV/cm) ×.
(34.33)4π~cρρSince physical distances are involved, X0 /ρ, in cm, appears. The energy-weighted bremsstrahlung
spectrum for lead, k dσLP M /dk, is shown in Fig. 34.12. With appropriate scaling by X0 /ρ, othermaterials behave similarly.
For photons, pair production is reduced for E(k − E) > k ELP M . The pair-production crosssections for different photon energies are shown in Fig. 34.18.
If k  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 cancoherently forward scatter off of the electrons in the media. Because of this, for k < ωp E/me ∼10−4 , bremsstrahlung is suppressed by a factor (kme /ωp 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].
8This definition differs from that of Ref. [55] by a factor of two. ELP M scales as the 4th power of the mass of theincident particle, so that ELP M = (1.4 × 1010 TeV/cm) × X0 /ρ for a muon.

31st May, 2024

34. Passage of Particles Through Matter

Probability

25

Photon energy (MeV)
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 λ (Fig. 34.16), the probability that a given photon willproduce an electron pair (without first Compton scattering) in thickness t of absorber is P [1 −exp(−t/λ)].
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 1020 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 (& 1023 eV), photonuclear interactions
can become coherent, with the photon interaction spread over multiple nuclei. Essentially, thephoton coherently converts to a ρ0 , in a process that is somewhat similar to kaon regeneration [57].
Similar processes occur for electrons. As electron energies increase and the LPM effect suppresses bremsstrahlung, electronuclear interactions become more important. At energies above
1021 eV, these electronuclear interactions dominate electron energy loss [57].

31st May, 2024

34. Passage of Particles Through Matter26( X0 NA/A) dσLPM/dx

1.00

Pair production

0.75

1 TeV
10 TeV0.50
100 TeV

1 EeV

0.25
1 PeV
10 PeV

100 PeV
0

0

0.25

0.5

0.75

1

x = E/kFigure 34.18: The normalized pair production cross section dσLP M /dx, versus fractional electronenergy x = E/k.

log 10 (Interaction Length) [m]

5
4
3

σBH
σMig
σγA
σMig + σγA

2
1
0

−1

10

12

14

16

18

20

22

24

26

log 10 k [eV]
Figure 34.19: Interaction length for a photon in ice as a function of photon energy for the BetheHeitler (BH), LPM (Mig) and photonuclear (γA) cross sections [57]. The Bethe-Heitler interaction
length is 9X0 /7, and X0 is 0.393 m in ice.
31st May, 2024

34. Passage of Particles Through Matter

27

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 withlower 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 thegeneration of more shower particles. In describing shower behavior, it is therefore convenient tointroduce the scale variablest = x/X0 ,y = E/Ec ,(34.34)so that distance is measured in units of radiation length and energy in units of critical energy.

0.125
30 GeV electron
incident on iron

(1/E 0 ) dE/dt

0.100

80

0.075
Energy0.050

40Photons
× 1/ 6.8

0.025
0.000

60

20

Electrons
0

5
10
15
t = depth in radiation lengths

20

Number crossing plane

1000Figure 34.20: An EGS4 simulation of a 30 GeV electron-induced cascade in iron. The histogramshows 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 X0 /2 intervals (scale on right) and the squares the number of photons withE ≥ 1.5 MeV crossing the planes (scaled down to have same area as the electron distribution).
Longitudinal profiles from an EGS4 [58] simulation of a 30 GeV electron-induced cascade iniron are shown in Fig. 34.20. The number of particles crossing a plane (very close to Rossi’s Πfunction [2]) is sensitive to the cutoff energy, here chosen as a total energy of 1.5 MeV for bothelectrons and photons. The electron number falls off more quickly than energy deposition. Thisis 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 inglass 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 Td , which is in general less than the total track length T . Practical devices are sensitive to
31st May, 2024

28

34. Passage of Particles Through Matterelectrons with energy above some detection threshold Ed , and Td = T F (Ed /Ec ). An analytic form
for F (Ed /Ec ) 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]:dE(bt)a−1 e−bt
= E0 bdt
Γ (a)(34.35)The maximum tmax 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 arewell described by Eq. (34.35) withtmax = (a − 1)/b = 1.0 × (ln y + Cj ),j = e, γ,(34.36)

where Ce = −0.5 for electron-induced cascades and Cγ = +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 assumingb ≈ 0.5 or by finding a more accurate value from Fig. 34.21. The results are very similar forthe electron number profiles, but there 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 Ce = −1.0 and Cγ = −0.5; weregard this as superseded by the EGS4 result.
The “shower length” Xs = X0 /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 numberof 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 forincident electrons and photons. For many purposes it is sufficient to take b ≈ 0.5.
The length of showers initiated by ultra-high energy photons and electrons is somewhat greaterthan 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 realcascade) 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 inwhich the variance and correlation of a and b are obtained by fitting Eq. (34.35) to individuallysimulated 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 RM , given by [63, 64]
RM = X0 Es /Ec ,(34.37)

where Es ≈ 21 MeV (Table 34.1), and the Rossi definition of Ec is used.
In a material containing a weight fraction wj of the element with critical energy Ecj and radiation
length Xj , the Molière radius is given by
1
1 X wj Ecj
=
.
RM
Es
Xj

(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 RM . About 99% is
31st May, 2024

34. Passage of Particles Through Matter

29

0.8

0.7

Carbon0.6

Aluminum

b

Iron
0.5
Uranium
0.4

0.3
10

100

1000

10 000y = E/Ec
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 ≤ E0 ≤ 100 GeV. Values obtained for incident
photons are essentially the same.

contained inside of 3.5RM , but at this radius and beyond composition effects become importantand the scaling with RM 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 atseveral hundred GeV. (There is no simple scaling with particle mass, but for protons the “criticalenergy” 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 generationof electromagnetic and (in the case of photonuclear interactions) hadronic showers [65–73] As aconsequence, at these energies the treatment of energy loss as a uniform and continuous process isfor many purposes inadequate.
31st May, 2024

34. Passage of Particles Through Matter30

It is convenient to write the average rate of muon energy loss as [74]h−dE/dxi = a(E) + b(E) E.

(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 theseslowly-varying functions are constant, the mean range x0 of a muon with initial energy E0 is givenbyx0 ≈ (1/b) ln(1 + E0 /Eµc ),(34.40)where Eµc = a/b.
Fig. 34.22 shows contributions to b(E) for iron. Since a(E) ≈ 0.002 GeV g−1 cm2 , b(E)E
dominates the energy loss above several hundred GeV, where b(E) is nearly constant. The rates ofenergy loss for muons in hydrogen, uranium, and iron are shown in Fig. 34.23 [7].

9

10 6 b(E ) (g −1 cm 2 )

8
7

Iron

b total6
5
4b pair3b bremsstrahlung

2
1
0
1

b nuclear

10

10 3
10 2
Muon energy (GeV)

10 4

10 5

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.
The “muon critical energy” Eµc can be defined more exactly as the energy at which radiativeand ionization losses are equal, and can be found by solving Eµc = a(Eµc )/b(Eµc ). This definition
corresponds to the solid-line intersection in Fig. 34.13, and is different from the Rossi definition we
31st May, 2024

31

34. Passage of Particles Through Matter

H2 (gas) total

Figure 34.23: The average energy loss of a muon in hydrogen, iron, and uranium as a function ofmuon energy. Contributions to dE/dx in iron from ionization and pair production, bremsstrahlung
and photonuclear interactions are also shown.
used for electrons. It serves the same function: below Eµc ionization losses dominate, and aboveEµc radiative effects dominate. The dependence of Eµc on atomic number Z is shown in Fig. 34.24.
The radiative cross sections are expressed as functions of the fractional energy loss ν. The
bremsstrahlung cross section goes roughly as 1/ν over most of the range, while for the pair production case the distribution goes as ν −3 to ν −2 [75]. “Hard” losses are therefore more probablein bremsstrahlung, and in fact energy losses due to pair production may very nearly be treatedas continuous. The simulated momentum distribution of an incident 1 TeV/c muon beam after itcrosses 3 m of iron is shown in Fig. 34.25 [7]. The most probable loss is 8 GeV, or 3.4 MeV g−1 cm2 .
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 cm2 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 meancalculated from the simulation. Electromagnetic and hadronic cascades in detector materials canobscure muon tracks in detector planes and reduce tracking efficiency [77].

31st May, 2024

34. Passage of Particles Through Matter

32
4000

7980 GeV(Z + 2.03) 0.879

2000

E µc (GeV)

1000

5700 GeV(Z + 1.47) 0.838

700
400

Gases
Solids

200
100

H

He

Li Be B C N O Ne

1

2

5

Fe

10

Sn

20

Z

50

100

Figure 34.24: Muon critical energy for the chemical elements, defined as the energy at whichradiative and ionization energy loss rates are equal [7]. The equality comes at a higher energy forgases than for solids or liquids with the same atomic number because of a smaller density effectreduction 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 thegases the worst fit is for radon (2.7% high).
0.10

MediandN/dp [1/(GeV/c)]

0.08

1 TeV muons
on 3 m Fe

987 GeV/c

0.06

Mean
977 GeV/c

0.04

FWHM
9 GeV/c
0.02

0.00
950

960

970

980

990

1000

Final momentum p [GeV/c]
Figure 34.25: The momentum distribution of 1 TeV/c muons after traversing 3 m of iron as
calculated by S.I. Striganov [7].

31st May, 2024

33

34. Passage of Particles Through Matter

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 θc of Cherenkov radiation, relative to the particle’s direction, for a particle with speed
βc in a medium with index of refraction n iscos θc = (1/nβ)ortan θc =≈qq

β 2 n2 − 1

2(1 − 1/nβ)

for small θc , e.g. in gases.

(34.41)

The threshold speed βt is 1/n, and γt = 1/(1 − βt2 )1/2 . Therefore, βt γt = 1/(2δ + δ 2 )1/2 , whereδ = n − 1. Values of δ for various commonly used gases are given as a function of pressure andwavelength in Ref. [80]. See its Table 6.1 for values at atmospheric pressure. Data for othercommonly used materials are given in Ref. [81].

γc
er

en

v=

vg

Ch

θc

ko

η

v

wa

ve

fro

nt

Particle velocity v = βcFigure 34.26: Cherenkov light emission and wavefront angles. In a dispersive medium, θc + η 6=
900 .
Practical Cherenkov radiator materials are dispersive. Let ω be the photon’s frequency, and let
k = 2π/λ be its wavenumber. The photons propage at the group speed vg = dω/dk = c/[n(ω) +ω(dn/dω)]. In a non-dispersive medium, this simplies to vg = c/n.
In his classical paper, Tamm [82] showed that for dispersive media the radiation is concentrated
31st May, 2024

3434. Passage of Particles Through Matterin a thin conical shell whose vertex is at the moving charge, and whose opening half-angle η isdcot η =
(ω tan θc )
dω
ω0






= tan θc + β 2 ω n(ω)

dncot θc,dωω0

(34.42)where ω0 is the central value of the small frequency range under consideration. (See Fig. 34.26.)
This cone has a opening half-angle η, and, unless the medium is non-dispersive (dn/dω = 0),
θc + η 6= 900 . 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 formost 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
d2 N
αz 2
α2 z 21=
sin2 θc =1− 2 2dEdx~cre me c2
β n (E)≈ 370 sin2 θc (E) eV−1 cm−1
(z = 1) ,(34.43)or, equivalently,
d2 N
2παz 2
=
dxdλ
λ2



1
1− 2 2β n (λ).

(34.44)The index of refraction n is a function of photon energy E = ~ω, as is the sensitivity of thetransducer used to detect the light. For practical use, Eq. (34.43) must be multiplied by the thetransducer response function and integrated over the region for which β n(ω) > 1. Further details
are given in the discussion of Cherenkov detectors in the Particle Detectors section (Sec. 35.5 ofthis Review).
When two particles are close together (lateral separation . 1 wavelength), the electromagnetic
fields from the particles may add coherently, affecting the Cherenkov radiation. Because of theiropposite 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 andpositrons in a high-energy electromagnetic cascade. The signals can be visible for energies above
1016 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 Ec (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 fromthe decelerating charges (as in bremsstrahlung). Because the emission is coherent, the electricfield 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, andscales 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 around31st May, 2024

34. Passage of Particles Through Matter35

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 seriesof 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 anda medium with plasma frequency ωp is
I = αz 2 γ~ωp /3,
where
~ωp =

q

4πNe re3 me c2 /α =q(34.45)ρ (in g/cm3 ) hZ/Ai × 28.81 eV.

(34.46)For styrene and similar materials, ~ωp ≈ 20 eV; for air it is 0.7 eV.

Without absorptiondS/d( ω), differential yield per interface (keV/keV)

10−2

25 μm Mylar/1.5 mm air
γ = 2 ×104

Single interface
200 foils

10−3With absorption

10−4

10−5

1

10
100
x-ray energy ω (keV)

1000

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 areshown with and without absorption. Adapted from Ref. [89].
The number spectrum dNγ /d(~ω diverges logarithmically at low energies and decreases rapidly
for ~ω/γ~ωp > 1. About half the energy is emitted in the range 0.1 ≤ ~ω/γ~ωp ≤ 1. Inevitable
31st May, 2024

36

34. Passage of Particles Through Matter

absorption in a practical detector removes the divergence. For a particle with γ = 103 , the radiatedphotons are in the soft x-ray range 2 to 40 keV. The γ dependence of the emitted energy thus comesfrom the hardening of the spectrum rather than from an increased quantum yield.
The number of photons with energy ~ω > ~ω0 is given by the answer to problem 13.15 inRef. [3],
"
#
2
αz 2
π2
γ~ωp
Nγ (~ω > ~ω0 ) =
−1 +
ln
,
(34.47)π~ω0
12within corrections of order (~ω0 /γ~ωp )2 . The number of photons above a fixed energy ~ω0  γ~ωpthus grows as (ln γ)2 , but the number above a fixed fraction of γ~ωp (as in the example above) isconstant. For example, for ~ω > γ~ωp /10, Nγ = 2.519 αz 2 /π = 0.59% × z 2 .
The particle stays “in phase” with the x ray over a distance called the formation length, d(ω) =(2c/ω)(1/γ 2 + θ2 + ωp2 /ω 2 )−1 . Most of the radiation is produced in this distance. Here θ is the
x-ray emission
angle,√√ characteristically 1/γ. For θ = 1/γ the formation length has a maximum atd(γωp / 2) = γc/ 2 ωp . 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(w) = π/2), the formation zones, which have opposite phase, overlap more
and more and the spectrum saturates, dI/dω approaching zero as L/d(ω) → 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 beenobtained [89, 90]. Although one might expect the intensity of coherent radiation from the stack offoils to be proportional to N 2 , the angular dependence of the formation length conspires to makethe intensity ∝ N .
References[1] H. Bichsel, Nucl. Instrum. Meth. A562, 154 (2006).
[2] B. Rossi, High Energy Particles, Prentice-Hall, Inc., Englewood Cliffs, NJ, 1952.
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