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

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Exercises for Chapter 4 241

With deep inelastic scattering, jets in e + e− collisions, the explanation

of the magnetic moments of hadrons, i.e. by studying the electromagnetic in-

teractions of quarks, these acquired a physical reality and their elementarity

was established: they are point-like particles. Then, using weak interactions,

one discovers a fascinating symmetry between quarks and leptons which is

suggestive of a possible underlying uniﬁcation, which would be more gen-

eral and more profound, of the constituents of matter and their fundamental

interactions. It is one of the great hopes of present theoretical physics to con-

struct a complete and uniﬁed theory of all interactions including gravitation.

Superstring theory and the reduction of all corresponding theories to a single

unique “M-Theory ” is a very active ﬁeld of research at present.

4.5 Bibliography

1. F. Halzen and A.D. Martin: Quarks and Leptons (Wiley, New York, 1984)

2. W. Greiner and B. M

ller:Gauge Theory of Weak Interactions (Springer,

Berlin, 2000)

3. U. Mosel: Fields, Symmetries, and Quarks (Springer, Berlin, 1999)

4. Q. Ho-Kim and X.-Y. Pham:Elementary Particles and Their Interactions

(Springer, Berlin, 1998)

5. Introduction to Elementary Particles, D. Griﬃths, (Harper & Row, New

York, 1987).

Exercises

4.1 Discuss the electron energy spectrum from the 2β-decay of

100

Mo shown

in Fig. 4.1. The average total energy taken by the 2 electrons in ∼ 1.2MeV.

How much energy on average is taken by the two neutrinos?

4.2 Charged particles lose energy in when traversing a material by ioniza-

tion. The minimum rate of energy loss (applicable to relativistic particles) is

∼ 2MeVg

−1

. (Multiplying by the density of the medium gives the en-

ergy loss per unit distance in MeV cm

−1

.) Consider the experiment shown in

Fig. 4.1. Estimate the energy loss by the decay electrons before leaving the

foil. Does this energy loss signiﬁcantly aﬀect the energy measurement by the

scintillators?

4.3 Discuss the decay scheme shown in Fig. 4.2. What factors determine the

branching ratios of β-decay to the ground and ﬁrst excited states of

170

Yb?

Estimate the internal conversion fraction for the decay of the 84 keV state by

using (4.61). Is the use of the formula justiﬁed?

242 4. Nuclear decays and fundamental interactions

4.4 Consider the photon spectra shown in Fig. 4.3. From the maximum

Doppler shift of forward going photons, estimate the velocity and energy of

the decaying

Br nuclei. Use the Bethe–Bloch formula (5.32) to estimate

the energy-loss rate of Br ions in nickel and the time for the nuclei to come

to rest. Estimate the lifetime of the decaying state. What are the possible

criticisms of this estimate?

4.5 A commonly used γ-source is

Co which β-decays to an excited state of

Ni as in Fig. 4.32. Use the indicated spins and parities to explain why

does not decay directly to the ground state of

Ni. Explain why two photons

are emitted in the de-excitation of

Ni. Estimate the time delay between the

β-decay and the photon emissions.

1.173 MeV

1.332 MeV

β−

5.271 yr

Fig. 4.32. The decay of

Co.

4.6 What energy photons are emitted after each of the ﬁve β-decays shown

in Fig. 4.12?

4.7 Estimate the energy of the k-capture Auger electrons emitted after the

electron-capture decay of

Ge (Fig. 4.16) by assuming that the inner most

electrons occupy hydrogen-like orbitals.

4.8 Verify the β spectrum in the case of a non-zero neutrino mass (4.101).

4.9 Consider the decay scheme in Fig. 4.19. Explain the large diﬀerence in

1/2

for

152

Eu and the isomer

152m

Eu. Calculate the kinetic energy of the

152

Sm nucleus after the electron-capture decay. What is the energy of the

Exercises for Chapter 4 243

“961 keV” photons that happen to be emitted in the direction opposite to

that of the neutrino? Do they have suﬃcient energy to re-excite

152

Sm nuclei

in the ground state?

4.10 Consider the neutrino event illustrated in Fig. 4.20. The ﬁnal-state

neutron takes only a small amount of the neutrino energy. Estimate the

maximum neutron energy corresponding to positrons emitted opposite to

the direction of the neutrino. What is the cross-section for such neutrons to

elastically scatter on protons (Fig. 3.4). What is the neutron mean-free path

in the scintillator (CH, ρ ∼ 1gcm

−2

)? Using the fact that a neutron loses

on average half of its kinetic energy in a collision with a proton, estimate

the number of collisions necessary for the neutron to be thermalized. After

thermalization, what is the neutron capture cross-section on hydrogen (Fig.

3.4)? What is the mean time be for the neutron is captured?

4.11 In Fig. 4.20, what is the mean-free path of the photons in the scintillator

(CH) assuming that the photons interact mostly via Compton scattering on

electrons, σ ∼ σ

5. Radioactivity and all that

Our galaxy’s stock of heavy elements was mostly produced in supernovae

explosions that eject stable and unstable nuclei into the interstellar medium.

The interstellar clouds containing these elements may later condense to form

stellar systems. By the time this happens, most of the short-lived nuclei would

have decayed, so planets contain, at their birth, only nuclei with lifetimes

greater than, say, 10

yr. On Earth, ∼ 4.5 × 10

yr after its formation, most

of the original unstable nuclei have decayed leaving only those with t

1/2

yr. There is also continuous production of unstable nuclei by cosmic-

rays entering the Earth’s atmosphere. The α and β decays of these nuclei

constitute the natural radioactivity that was discovered by Becquerel at the

end of the nineteenth century.

All living species have evolved in this bath of radiation. What is new on

Earth is the existence of an artiﬁcial radioactivity due to the development

of nuclear technologies, most importantly energy production by nuclear re-

actors. There are now also numerous uses of radioactive nuclei, most notably

in medical treatment and dating.

With these developments, public health concerns have naturally arisen

because of the eﬀect of ionizing radiation on living tissue. Since natural ra-

dioactivity is present everywhere, it should be emphasized that the important

questions concern keeping the ambient radioactivity at a level not much above

its natural level. This obviously requires conﬁnement of the large amounts of

radioactive elements produced in nuclear reactors.

In this chapter, we will ﬁrst discuss the natural sources of radioactivity

and the production of radioactive elements in neutron and charged-particle

beams. We will then discuss how high-energy particles lose energy while

traversing ordinary matter. Particle detectors based on the energy-loss mech-

anisms will be brieﬂy discussed. Finally, we discuss a few applications of

radioactivity.

5.1 Generalities

Most nuclei are inherently unstable, being capable of decaying to other nuclei

by β-decay, α-decay, or by spontaneous ﬁssion. The set of all possible decays

is referred to globally as radioactivity.

246 5. Radioactivity and all that

Let τ be the (mean) lifetime of a nuclear species. The half-life t

1/2

deﬁned by

1/2

≡ τ log 2 ∼ 0.693τ. (5.1)

The name “half-life” refers to the fact that the probability that the nucleus

has decayed after t

1/2

is equal to 1/2. Nuclear physicists traditionally use half-

lives, whereas particle physicists use (mean) lifetimes. The two deﬁnitions

diﬀer by a factor of ln 2 = 0.693. (We hope that the trivial mistakes in

this book that come from forgetting this factor will disappear in the editing

process.)

The activity of a radioactive sample is the number of decays per unit time.

For a source containing N particles the activity is

A = Nλ = N/τ . (5.2)

If one knows A and τ, one can deduce the concentration N of the radioactive

product in the sample.

Activities are measured in Becquerel:

1 Bq = 1 (decay) s

−1

. (5.3)

The previously used unit was the Curie : 1 Curie = 3.7 × 10

Bq. This

corresponds to the activity of 1 g of the radium isotope

226

Ra.

According to the nature of emitted particles, α particles, e

, or photons,

one distinguishes three types of radioactivity: α, β and γ.Theywereprevi-

ously described theoretically in Chap. 4. The types are summarized in Table.

5.1.

5.2 Sources of radioactivity

At the end of the 1930s, one made the distinction between two classes of

radioactivity, natural and artiﬁcial. Natural radioactivity was discovered ac-

cidentally by Henri Becquerel in 1896 who observed that photographic plates

(unexposed to light) are blackened when placed next to uranium sulﬁde crys-

tals. After some time, the primary particles responsible for the blackening

where identiﬁed as

He ions from a chain of decays starting with

238

U →

234

Th α t

1/2

 4.468 × 10

yr . (5.4)

Artiﬁcial radioactivity coming from elements produced in laboratory reac-

tions was discovered by Fr´ed´eric and Ir`ene Joliot-Curie in 1934 who observed

Al → n

P (5.5)

followed by

P →

Si e

1/2

= 150.0 sec . (5.6)

5.2 Sources of radioactivity 247

Table 5.1. Main types of radioactivity. The ﬁrst four, β

, electron-capture, and

2β are A-conserving decays that lead to the one β-stable isobar for the given A.

The next two, α and (spontaneous) ﬁssion change A. All these decays may leave

the daughter nuclei in excited states that decay (generally rapidly) to the ground

states by the last two forms of radioactivity, γ and internal conversion. Two types

of radioactivity, e

−

-capture and internal conversion occur only if the nucleus is

surrounded by atomic electrons.

name reaction energy

spectrum

−

(A, Z) → (A, Z +1)e

−

continuous

(A, Z) → (A, Z − 1)e

continuous

−

-capture e

−

(A, Z) → (A, Z − 1)ν

discrete

+atomic X-rays/Auger e

−

’s

2β (A, Z) → (A, Z ± 2)2e

(2ν

) continuous

α (A, Z) → (A − 4,Z −2)

He discrete

ﬁssion (A, Z) → (A

)(A

) (+neutrons) continuous

γ (A, Z)

∗

→ (A, Z)γ discrete

internal conversion e

−

(A, Z)

∗

→ (A, Z)e

−

discrete

+atomic X-rays/Auger e

−

’s

One can divide natural radioactivity into “fossil” radioactivity due to ele-

ments present at the formation of the Earth, and “cosmogenic” radioactivity

due to elements continually produced in the atmosphere by cosmic-rays.

Techniques for producing artiﬁcial radioactivity can be divided into those

that use neutrons (neutron activation) and those that use charged particle

beams. An example of the latter is reaction (5.5) where the Joliot-Curies used

naturally occurring α radiation (E

∼ 5 MeV) to induce the transmutation

of aluminum to phosphorus. Modern alchemists use artiﬁcially accelerated

ions of arbitrary energies, greatly increasing the possibilities, as discussed in

Sect. 3.1.5.

5.2.1 Fossil radioactivity

Most elements present on Earth condensed from a interstellar cloud about

4.5 × 10

yr ago. This cloud consisted mostly of “primordial”

Hand

that was produced in the ﬁrst minutes after the “Big Bang” (see Chap. 9).

Additionally, the cloud contained heavier elements that had been produced

in earlier generations of stars and dispersed into the interstellar medium by

various processes (see Chap. 8). Most radioactive nuclei were produced in

supernovae that generate a mix of neutron-rich nuclei in a period of time of

248 5. Radioactivity and all that

order a few seconds. When, some millions of years later, the cloud condensed

to form the solar system, a few radioactive nuclear species were still present.

At the present epoch, only nuclei with mean lives greater than, say, 10

yr,

are still present in signiﬁcant numbers.

The nuclei with 10

yr <τ<10

yr are listed in Table 5.2. These long-

lived nuclei involve either highly forbidden β decays (large spin changes) or

are α-decays that happen to have Q-values that place the half-lives in this

range.

Table 5.2. Nuclei with 10

yr <t

1/2

< 10

yr. The “isotopic abundance” is for

the terrestrial mix and the activity for the puriﬁed element corresponds to the

terrestrial isotopic mix. For the activity in the Earth’s crust, the uranium and

thorium activities include the activities of the daughters. Note that three nuclides,

147

Sm and

235

U have lifetimes much less than the age of the Earth (∼ 4.5 ×

yr) and therefore have very small isotopic abundances.

decay half-life isotopic activity activity

(years) abundance (Bq kg

−1

)(Bqkg

−1

)

(percent) (element) (crust)

K →

Ca e

−

89% 1.28 × 10

0.0117 3.0 × 10

6.3 × 10

→

Ar ν

11%

Rb →

Sr e

−

4.75 × 10

27.83 8.8 × 10

8.0 × 10

146

Sm →

142

Nd α 1.03 × 10

< 10

−7

< 1 < 10

−4

147

Sm →

143

Nd α 1.06 × 10

15.1 1.3 × 10

9 × 10

−1

176

Lu →

176

Hf e

−

3.78 × 10

2.61 5.5 × 10

4 × 10

−2

187

Re →

187

Os e

−

4.15 × 10

62.6 1.1 × 10

8 × 10

−4

232

Th →

228

Ra α 1.405 × 10

100 4.05 × 10

3.5 × 10

235

U →

231

Th α 7.038 × 10

0.72 5.7 × 10

1.7 × 10

238

U →

234

Th α 4.468 × 10

99.275 1.2 × 10

4.7 × 10

Most of the nuclei in Table 5.2 are isotopes of rare elements with the ex-

ception of

K. The decay modes of this isotope, shown in Fig. 5.1, yields a β

−

with E

max

=1.31 MeV (BR=89.3%) or a γ with E

=1.46 MeV (BR=10.5%).

Potassium has a chemistry similar to that of sodium and is plentiful in bio-

logical material where it is the dominant source of radioactivity.

5.2 Sources of radioactivity 249

0.00

2.00

EC(0.2%)

β (89.3%)

=1.46MeV

Q=1.31MeV

EC(10.5%)

Qec=0.05MeV

Qec=1.505

−

Fig. 5.1. The β-decay of

K. Being an N-odd,Z-odd nucleus, it has two modes:

electron capture to

Ar and β

−

Ca. The long half life (1.28×10

yr) is explained

by the large angular momentum changes in the decays to the ground states (making

the decays highly forbidden) and the small value of Q

to the 2

state of

Ar.

235

238

232

205

208206

207

209

mostly

1/2

>10

−

Fig. 5.2. The three chains of natural radioactivity:

238

U →

206

Pb;

235

U →

207

Pb;

and

232

Th →

208

Pb. Each chain consists of a series of α− and β

−

-decays. In the

case of the

232

Th chain, a branch occurs at

212

Bi that has a 64% branching ratio

for β

−

-decay and a 36% ratio for α-decay. Similar, though less balanced, branchings

occur throughout the chains and only the primary routes are shown. Also shown is

the recently discovered

209

Bi →

205

Tl chain consisting of a single α decay [52].

250 5. Radioactivity and all that

Table 5.3. Members of the three natural radioactivity chains and their half-lives.

uranium series actinium series thorium series

238

4.468 Gyr

234

24.10 d

234

1.17 m

234

245.5 kyr

235

0.7038 Gyr

230

75.38 kyr

231

25.52 h

232

14.05 Gyr

226

1600 y

231

32760 y

228

5.75 y

222

3.8235 d

227

21.773 y

228

6.15 h

218

3.10 m

227

18.72 d

228

1.9116 y

214

26.8 m

223

11.435 d

224

3.66 d

214

19.9 m

219

3.96 s

220

55.6 s

214

164.3 µs

215

1.781 ms

216

0.145 s

210

22.3 y

211

36.1 m

212

10.64 h

210

5.013 d

211

2.14 m

212

60.55 m

210

138.376 d

207

4.77 m 64%

212

0.299µs

36%

208

3.053 m

206

> 10

207

> 10

208

> 10

The three heavy nuclides,

238

235

Uand

232

Th, are the origins of the

three natural radioactivity chains illustrated in Fig. 5.2 and Table 5.3. The

chains proceed through a series of α and β decays that end at three isotopes

of lead

238

U →

206

Pb (A = 4n + 2) ,

235

U →

207

Pb (A = 4n + 3) ,

232

Th →

208

Pb (A = 4n) ,

The three chains end with an isotope of Z-magic lead. (

208

Pb is doubly

magic, N = 126.) The fourth chain with A =4n + 1 has no long-lived nuclei

heavier than

209

Bi. The chain thus consists of a single decay

209

Bi → α

205

1/2

=1.9 × 10

yr) [52].

As seen in Table 5.3, the lifetimes of the daughter nuclei are all short on

geological timescales so in many applications the cascade can be considered

to pass “instantaneously” to the ﬁnal lead isotope. The fact that

238

235

and

232

Th are long-lived whereas the elements with 209 <A<232 have

much shorter half-lives can be traced to the shell structure of the nucleus

which makes the doubly magic nucleus

208

Pb especially bound. As a result,

the Q-values of α-decay are relatively large for 208 <A<232 (Fig. 2.14)

making the α rates high.

The uranium and thorium chains and

K dominate the radioactivity in-

side the Earth. The activity of the lower atmosphere is mostly due to three

isotopes of the noble gas radon that are daughters of uranium and thorium

5.2 Sources of radioactivity 251

and can diﬀuse out of rocks into the air. It is typically ∼ 20 Bq m

−3

in-

side reasonably ventilated buildings [53]. It should be also noted that the

Earth’s stock of helium originates mostly in the α particles emitted in the

uranium–thorium chains. (Inert primordial helium was not retained during

the formation of the Earth.)

Minerals that contain uranium and thorium also contain their daughters.

Neglecting leakage out of the mineral, all species in the chain

1 → 2 → 3 → 4 ...n

will generate quantities of each nucleus governed by the coupled diﬀerential

equations

= −N

/τ

(5.7)

= N

i−1

/τ

i−1

− N

/τ

i =1. (5.8)

In the case of the three natural chains, the ﬁrst nucleus is by far the longest

lived τ

 τ

i=1,n

. In this case, the solution is

= N

(t = 0)e

−t/τ

(5.9)

= N

i =1,n (5.10)

= N

(t = 0)(1 − e

−t/τ

) . (5.11)

The unstable daughters thus reach an equilibrium abundance proportional

to their half-lives where the activity of each daughter is equal to the activity

of the parent.

It was Marie and Pierre Curie who chemically isolated the daughter nuclei

present in uranium ore. Once uranium ore is reﬁned only the parents

238

and

235

U and the daughter

234

U remain in the chemically pure uranium.

Natural lead ore contains the natural isotopes,

204

Pb,

206

Pb,

207

Pb and

208

Pb, as well as the uranium–thorium daughters due to the decay of any

uranium or thorium present in the lead ore. After reﬁning resulting in the

removal of the uranium–thorium and non-lead daughters, the puriﬁed lead

contains the four natural isotopes as well as the the β-emitting daughters

210

Pb (t

1/2

=22.37 yr),

211

Pb (t

1/2

=36.1 min), and

212

Pb (t

1/2

=10.6hr)

(see Fig. 5.2). Ignoring the two very short-lived daughters, we see that re-

cently mined lead contains a

210

Pb activity that depends on the amount of

238

U present in the ore. It ranges from ∼ 1Bqkg

−1

to 2500 Bq kg

−1

[53] In

ancient lead recovered, for example, from sunken Roman galleys, the

210

has mostly decayed away and activities as small as 0.02 Bq kg

−1

have been

reported [53]. Such ancient lead is used as shielding in measurements where

very low radioactivity is necessary (Exercise 9.7).