Basdevant J.-L., Rich J., Spiro M. Fundamentals in Nuclear Physics: From Nuclear Structure to Cosmology

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262 5. Radioactivity and all that

Table 5.4. Some charged-particle detectors and the signal produced for 1 keV of

total energy loss. The largest intrinsic signal is in silicon semiconductor detectors

but that in gas-ionization chambers can be eﬀectively much higher since additional

electrons will be liberated when the primary electrons collide with atoms in the high

electric ﬁelds near collection wires (proportional chambers and Geiger counters).

The signal from scintillators and Cherenkov counters will be reduced because the

photons must be detected, generally by photomultiplier tubes (Figs. 5.14 and 8.14).

The primary advantage of optical detectors is their excellent time resolution, as

good as 0.1ns.

Detector yield

silicon semiconductors ∼ 270e

−

keV

−1

gas-ionization chambers ∼ 50 e

−

keV

−1

organic scintillators < 15 visible photons keV

−1

O Cherenkov counters ∼ 0.1 visible photons keV

−1

(β>1/n n =1.33)

pulse ou

−Q

ion

gas

Fig. 5.8. The principle of operation of a gas-ﬁlled ionization chamber. The charge

liberated by the passage of the particle drifts in the electric ﬁeld toward the elec-

trodes. If the high-voltage supply is removed after charging the electrodes, the

passage of ionizing radiation would simply discharge the chamber. In the early

days of radioactivity, this rate of discharge was a standard monitor of radioactivity

and cosmic radiation. If the high-voltage supply is left in place, after the passage of

the charged particle, the potential diﬀerence between the electrodes ﬁrst drops and

then returns to its nominal value. The resulting pulse signals the passage of a single

particle. This is the principle of proportional counters and Geiger counters.Ifthe

time of passage of the particle is known (e.g. by an external scintillator counter),

the position of the particle can be deduced from the time delay between this time

and the arrival of the ions at the electrodes. This is the principle of the operation

of drift chambers.

5.3 Passage of particles through matter 263

light

photon

dynodes (electron multipliers)

output

scintillator

photo−electron

photocathode (−V)

anode

charged

particle

photomultiplier tube (PMT)

light shield

Fig. 5.9. The principle of a scintillation detector. A charged particle creates a

small amount of visible light while losing energy in the scintillating material. The

photon may be detected by a photomultiplier tube (PMT) where it may liberate

an electron on a photocathode through the photo-electric eﬀect. The photocathode

is held at high (negative) potential so the photoelectron is directed toward a series

of dynodes where each electron creates a certain number of further electrons. The

electrons are ﬁnally collected at the anode.

5.3.2 Particle identiﬁcation

Many nuclear species can be produced in nuclear collisions and we are now in

a position to understand how the identity of individual nuclei can be deter-

mined. A species is characterized by its charge, q, and mass, m, and individual

nuclei are additionally characterized by their velocity, v, or equivalently by

their momentum p or kinetic energy p

/2m. Charged particle detectors give

information on these quantities:

• p/q from the trajectory in a magnetic ﬁeld;

• v from the ﬂight-time t between detector elements separated by a distance

• q

from dE/dx in a thin ionization counter;

• p

/2m from the total energy deposit in a thick ionization counter.

Clearly, a single type of measurement is insuﬃcient to identify the species.

The charge-to-mass ratio q/m can be deduced by combining two measure-

ments, for instance p/q from the magnetic trajectory and v from a time-of-

ﬂight measurement. With perfect precision, this would be suﬃcient to identify

the particle since no two species have exactly the same q/m.

In real experi-

ments, it is generally necessary to have a third type of measurement. Fig. 5.10

shows how dE/dx measurements where combined with magnetic trajectory

and time-of-ﬂight measurements to identify the doubly-magic nuclide

100

Sn.

This is eﬀectively the case in mass-spectrometers (Fig. 1.3) where the magnetic

trajectory is combined with knowledge p

/2m determined by the accelerating

potential. A high precision of ∆(m/q)/(m/q) ∼ 10

−8

generally allows one to

unambiguously identify ionic species.

264 5. Radioactivity and all that

1980

2000

2020

2040 2060

2080

940

960

980

1020

1040

1000

E (arbitrary units)

∆

M/Q x 1000 (u/e)

101

100

102

100

wedge

∆ E

x,y

∆ E

x,y x,y

∆ E

x,y

∆

18 m 36 m

magnets

target

Xe beam

Fig. 5.10. The apparatus used by [56] at the GSI laboratory for the ﬁrst detection

of doubly-magic

100

Sn. A

124

Xe beam of kinetic energy ∼ 1GeV/nucleon is incident

on a thin beryllium target. Particles emerging from the target enter a spectrometer

consisting of four dipole magnets. The magnetic ﬁeld is set so that particles of

A ∼ 100 with kinetic energy ∼ 1GeV/nucleon pass through the spectrometer. As

indicated in the ﬁgure, the spectrometer is equipped with ionization counters that

measure energy loss (∆E), position (x, y), and time of passage (t) for individual

particles. Additionally, a wedge-shaped aluminum block placed at the midpoint of

the spectrometer produced an energy loss that depends on position and therefore

on A. Combining the (x, y) information with the time-of-ﬂight information allows

one to deduce m/q. A scatter plot (bottom panel) of m/q vs. energy loss then

identiﬁes clearly the nuclei. Finally, particles are stopped in a silicon counter where

their decays can be observed.

5.3 Passage of particles through matter 265

5.3.3 Electrons and positrons

100

1000

0.5

1.0

0.05

0.10

0.15

0.20

dE/dx (per radiation length)

−

E (MeV)

( g cm

)

−1

Bremsstrahlung

Ionization

positrons

electrons

Moller

Bhabha

positron

annihilation

Fig. 5.11. Energy loss of electrons and of positrons in lead as a function of the inci-

dent energy [1]. The left scale uses the radiation length of lead (0.56 cm) as the unit

of length. Ionization dominates at low energies while bremsstrahlung dominates

above 10 MeV. Subdominant contributions come from wide-angle elastic scattering

on electrons (Moller for electrons and Bhabba for positrons) and positron annihi-

lation.

Our derivation of the Bethe–Bloch equation (5.30) does not apply to

electrons because we assumed that the ions were suﬃciently massive that

their trajectories are not aﬀected by close encounters with atomic electrons.

In spite of this, for E

< 10 MeV, the stopping power for electrons is to good

approximation given by the Bethe–Bloch equation. Electrons and positrons

at low energy therefore come to rest in a manner similar to that of heavy

ions, though their trajectories are much less straight since they scatter at

large angles from nuclei.

After coming to rest, positrons annihilate with atomic electrons yielding

two photons of energy E

= m

−

→ γγ . (5.35)

266 5. Radioactivity and all that

For E

> 10 MeV, electrons and positrons lose most of their energy by

radiation of photons (bremsstrahlung) that results from their acceleration

in the electron ﬁeld of nuclei. The distance over which an electron loses a

fraction 1/e of its energy through this process is called the radiation length,

. . Because the eﬀect is due to acceleration in the electric ﬁeld of a nucleus,

−1

has a strong dependence on Z, ranging from 61 g cm

−2

in hydrogen

to 6.37 g cm

−2

in lead. For the low-energy electrons and positrons from β-

decay, this eﬀect is generally of secondary importance but rapidly becomes the

dominant process for high-energy electrons and positrons with E>10 MeV.

This can be seen in Fig. 5.11 for lead.

5.3.4 Photons

Photons interact with matter through the following reactions whose cross-

section on carbon and lead are shown in Fig. 5.12:

• Rayleigh-scattering, i.e. elastic scattering from atoms. This process has the

only cross-section that has no energy threshold.

• Compton scattering, i.e. elastic scattering from quasi-free electrons. This

process has a threshold for each shell of bound electrons corresponding to

the ionization energy, I, of the shell. In the range I  E

 m

,the

cross-section is the Thomson cross-section calculated classically in Chap.

=(8π/3)r

I  E

 m

. (5.36)

In this energy range, the diﬀerential cross-section is forward and backward

peaked

dσ

dΩ

cos

2π

, (5.37)

where θ is the scattering angle. For E

>> m

, the cross-section falls like

−2

and is increasingly peaked in the forward direction θ<m

• Photoelectric absorption by atoms

γ atom → e

−

atom

. (5.38)

The threshold for ejection of an electron of a given shell is just the ionization

energy of the shell.

• Pair production , i.e.

γ (A, Z) → e

−

(A, Z) . (5.39)

The threshold is 2m

. This is by far the dominant eﬀect at high energy.

Photons also have a small probability to be absorbed by breaking up

nuclei (photo-nuclear absorption) as described in Sect. 3.1.5.

5.3 Passage of particles through matter 267

−3

Compton

Rayleigh

pair−production

Rayleigh

photo−electric

photon energy (MeV)

−3

photon energy (MeV)

cross−section (barn/atom)

carbon

lead

Compton

Fig. 5.12. Photon cross-sections on carbon and lead [1] as explained in the text.

At low energy, 1 keV <E<100 keV, photo-electric absorption dominates while

electron–positron pair production dominates for E  2m

. Compton scattering

dominates at intermediate energies. Photo-nuclear absorption (Fig. 3.8) is of minor

importance.

268 5. Radioactivity and all that

Photon Detectors. Nuclear physics is mostly concerned with photons in

the range 100 keV − 1 MeV where Compton scattering and photoelectric ab-

sorption are the most important processes. A photon entering a massive de-

tector therefore generates electrons by the sequences shown in Fig. 5.13. In

“photon detectors” it is actually these electrons that are detected. For in-

stance, in germanium diode detectors, a germanium crystal is polarized and

the ionization liberated by the electrons can be sensed as an electric pulse of

amplitude proportional to the deposited electron energy. If the photon is ab-

sorbed, the detector gives a signal proportional to the photon energy. Various

detectors using this principle are listed in Table 5.5.

E(ionization)

full−energy peak

Recoils

Compton

Fig. 5.13. Photons in the MeV range interact with matter by Compton scattering

and photoelectric absorption, as illustrated by photons a, b and c on the left. The

spectrum of ionization energy deposited by the secondary electrons is shown on the

right. Photon a deposits only a fraction of its energy and contributes to the con-

tinuous Compton spectrum. Photons b (photoelectric) and c (Compton followed by

photoelectric) deposit all their energy and lead to the full-energy peak in the energy

spectrum. Photons with E

> 2m

can convert to an electron–positron pair (photon

d). The electron and positron then lose their kinetic energy by ionization and the

positron then annihilates, generally at rest, with another electron. Depending on

whether both, one, or neither of the annihilation photons is photo-absorbed in the

detector, pair production contributes to the full-energy peak or to “single-escape”

or “double escape” peaks 511 keV or 1022 keV below the full-energy peak.

At high energy, E  2m

, the passage in matter of photons and elec-

trons is deeply connected. Photons create electrons and positrons by pair

production, which in turn create photons by bremsstrahlung. The result-

ing cascade of particles is called an “electromagnetic shower” of electrons,

positrons and photons.

5.3 Passage of particles through matter 269

Table 5.5. Some γ- and X-ray detectors and their signal yield. Photons with

E<2m

interact in the detectors mostly by photo-electric absorption and Comp-

ton scattering as shown in Fig. 5.13, and the signal is generated by the secondary

electrons. γ-detectors require the use of high-Z elements to give a high probability

of photoelectric absorption. The eﬀective signal from scintillators is smaller than

that shown below by a factor of order 10 because of limited photon collection eﬃ-

ciency and quantum eﬃciency of photomultipliers. The intrinsic energy resolution

of scintillators is then given by Poisson statistics on the observed number N of

photons, ∆N =

√

N yielding a resolution of no better than 10 keV for a 1 MeV

γ-ray. This resolution is reﬂected in the observed width of the full-energy peak in

Figs. 5.13 and 7.2. The signal from silicon and germanium diodes is so large that

statistical ﬂuctuations are generally small compared to other sources, e.g. ampliﬁer

noise. This results in γ-ray resolutions of typically ∼ 1 keV for germanium diodes,

making them by far the best detectors when high resolution is required. On the

other hand, scintillators are much less expensive and are therefore favored when

large detectors covering a large solid angle are required.

Detector yield

silicon semiconductor diodes (x-rays) 270 e

−

keV

−1

germanium semiconductor diodes (γ-rays) 340 e

−

keV

−1

NaI scintillators (γ-rays) < 40 visible photons keV

−1

BGO scint. ((Bi

)

(GeO

)

(γ-rays) < 6 visible photons keV

−1

5.3.5 Neutrons

Neutrons in the MeV range interact with matter mostly by elastic scattering

on nuclei. This results in a progressive loss of the neutron kinetic energy until

they are thermalized with a mean energy, ∼ kT, given by the temperature

of the medium (Fig. 5.14). The neutron then continues to perform a random

walk with velocity v ∼ 2000 m s

−1

untiltheyareabsorbed,usuallybya(n, γ)

reactions. In a homogeneous medium containing nuclei of mass number A,

the mean time for absorption after thermalization is

τ =

n σv

∼ 6 µs

A gcm

−3

, (5.40)

where n and ρ are the number and mass density of nuclei and where σ is the

thermally averaged cross-section at T = 300K. Note that the absorption time

is substantially sorter than the mean lifetime of a free neutron, ∼ 886.7s.

Neutron detectors. In nuclear physics, neutrons are generally detected by

ﬁrst thermalizing them and then by observing their absorption. Some com-

monly used absorption reactions are listed in Table 5.6. Absorption on

He,

Li and

B yielding charged particles are preferred because the ﬁnal state

particles are easily detected through their ionization. Radiative absorption

157

Gd is sometimes used because of its enormous cross-section for thermal

neutron capture, 2.55 × 10

270 5. Radioactivity and all that

Table 5.6. Some thermal neutron detectors.

Reaction cross-section detector type

He → p

H5.33 × 10

He-ﬁlled proportional chambers

Li →

He 9.42 × 10

Li-doped scintillators

B →

Li 3.8 × 10

bBF

-ﬁlled proportional chambers

157

Gd → γγ

158

Gd 2.55 × 10

b gadolinium-doped scintillators

elastic scatters

3H 4Hen 6Li

photomultiplier

Fig. 5.14. A neutron detector based on the reaction n

Li →

He. The neutron

enters a

Li-doped scintillator. The neutron is thermalized by elastic scatters, after

which it performs a random walk through the material. Eventually, it is absorbed

by a nucleus in the material, in this case by a

Li nucleus. The ﬁnal state particle

H and

He are charged an therefore stop in the medium by depositing their kinetic

energy, mostly by ionization. Scintillation light is detected by the photomultiplier.

If the original neutron is suﬃciently energetic for the ﬁrst few nuclear recoils to

produce a signiﬁcant amount of scintillation light, the event consists of two light

pulses, the ﬁrst for the elastic scatters and the second for the capture.

5.4 Radiation dosimetry

We saw in the previous section that all non-thermalized particles, with the

practical exception of neutrinos, deposit their energy in the medium through

which the propagate. The total energy deposited per unit mass (of the

medium) is called the dose. It has units of Gy (Gray)

1Gy = 1Jkg

−1

=6.25 × 10

MeV kg

−1

. (5.41)

The dose clearly refers to a ﬂux of particles over a speciﬁed period of time or

to a speciﬁed event (e.g. a nuclear accident).

Biological eﬀects depend not only on the total energy deposited but also

on the density of the energy deposit. The equivalent dose is therefore used

to have a better estimate of biological damage caused by the disruption in

cells due to ionization and the resultant breaking of molecular and chemical

bonds. The most important risks involve mutations that can cause cancer.

The unit is the Sievert (Sv)

equivalent dose (Sv) =



× dose (Gy)] , (5.42)

where the factor w

takes into account the long term risks in regular exposi-

tions to weak doses of each type of radiation mentioned in Table 5.7 and the

5.4 Radiation dosimetry 271

sum is over the diﬀerent types of radiation. Heavily ionizing radiation involv-

ing particles with β<1 have risk factors greater than unity. The fact that

are multiples of 5 reﬂects the precision of the biology, not the physics.

Table 5.7. Risk factors in various radiations

Radiation w

X and γ rays, any energy 1

Electrons and muons, any energy 1

Neutrons, E<10 keV 5

Neutrons, 10 <E<100 keV 10

Neutrons, 100 keV < E < 2 MeV 20

Neutrons, 2 <E<20 MeV 10

Neutrons, E>20 MeV 5

Protons, E>20 MeV 5

α particles, ﬁssion fragments, heavy nuclei 20

Table 5.8 shows the typical contribution of various sources of radiation

to the mean equivalent dose we receive annually (in the absence of a local

nuclear event). The primary sources are cosmic rays, 0.26 mSv at sea level,

and natural radioactivity, ∼ 2 mSv. The α− and β−particles from natural

activity are mostly conﬁned to the material containing the radioactivity and

are therefore not dangerous unless ingested. However, γ−rays from β-decays

near the surface of materials radiate us continually. The most energetic of

the photons are the 1460 keV

K and 2620 keV

208

Tl γ-rays. They are often

seen in the background spectra of photon detectors (Fig. 7.2).

More important than the photon emitters is the α-emitting noble gas

Radon produced in the uranium–thorium chains. It diﬀuses out of the ground

and building materials, generating an eﬀective dose of ∼ 1−2 mSv for building

dwellers. This can be an order of magnitude smaller for people living outside

and an order of magnitude higher in poorly ventilated buildings or in granite-

rock areas.

The eﬀective dose derived from ingesting or inhaling radioactive materials

(Table 5.9) is diﬃcult to estimate since it depends critically on where and

for how long the body stores the particular material. The radiotoxicity, R,

associated with a given nuclide is the eﬀective dose (in Sv) resulting from

ingesting or inhaling an activity A (in Bq). It can be written as

R =

Eτ

eﬀ

, (5.43)

where E is the mean energy per decay deposited in the body and w

the associated risk factor. Depending on the nucleus in question, the eﬀective

retention time τ

eﬀ

can be the nuclear mean life for short-lived nuclei, the

biological retention time for elements that are eliminated from the body (e.g.

tritiated water), or the lifetime of the organism itself for long-lived nuclei

that can be permanently attached to the body parts, e.g.

239

Pu in bones.