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

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282 5. Radioactivity and all that

Exercises

5.1 Assume that a given radioactive nuclide has a decay rate λ

and that it

decays into a daughter nuclide which is itself radioactive with decay rate λ

Assume further that at time t =0thereareN

parent nuclei and N

daughter nuclei. Show that at time t ≥ 0 the number of daughter nuclei is

(t)=N

(1 − e

−(λ

−λ

(λ

− λ

)

. (5.57)

Verify that the limits t → 0, t →∞, λ

 λ

and λ

 λ

are reasonable.

5.2 One gram of cobalt is placed near the core of a nuclear reactor where it is

exposed to a neutron ﬂux of 5 × 10

−2

−1

. The cross-section for thermal

neutron capture on

Co is ∼ 40 b. Calculate the

Co activity after 1 week

of irradiation.

5.3 About 0.3% of the mass of a human is potassium of which 0.012%

K(t

1/2

=1.25 × 10

yr). Use the decay scheme shown in Fig 5.1 to

discuss the annual dose received by a body from the internal

K. Argue that

almost all of the β

−

kinetic energy is absorbed in the body. Use the photon

absorption cross-section in Fig. 5.12 to argue that a signiﬁcant fraction of

the 1.46 MeV γ-rays are also absorbed in the body. Estimate then the annual

dose received from

K decay.

5.4 In 1990, archaeologists found, in Pedra Furada in Brazil, human remains

whose activity in carbon 14 was 19.6 × 10

−4

Bq per gram of carbon. Given

that the lifetime of

Cisτ = 8200 years (half life t

1/2

= 5700 years) and that

the activity of

C in living tissues and in atmospheric CO

is 0.233 Bq (per

gram of C), determine the age of these human remains. Compare the results

with the date when the ﬁrst Neanthropians (i.e. homo sapiens, the species to

which you and all other living human beings on this planet belong) appeared

in Europe, about 35 000 years ago. See Bahn, Paul G.; “50,000-Year-Old

Americans of Pedra Furada,” Nature, 362:114, 1993.

5.5 The half-lives of uranium 234,235 and 238 are respectively 2.5×10

years,

7.110

years and 4.510

years. Their relative natural abundances are re-

spectively 0.0057%, 0.72% and 99.27%. Are these data consistent with the

assumption that these nuclei were formed in equal amounts at the same time?

If not, can one explain the discrepancy knowing that the unstable isotopes

234

Th and

234

Pa exist?

Exercises for Chapter 5 283

5.6 Asimplewayofmakingasourceofneutronsistomixanα-emitter with

beryllium. The α-particles can then produce neutrons via the exothermic

reaction

Be → n

C+5.7MeV . (5.58)

The cross-section for this reactions is ∼ 0.4b for E

∼ 5MeV. For E

4 MeV the cross-section is much smaller because of the Coulomb barrier.

Consider a nucleus emitting 5 MeV α-particles placed in pure beryllium,

(ρ ∼ 1.8g cm

−3

). What is the initial energy-loss rate of the α-particles? How

far does an α-particle travel in the beryllium before losing 1 MeV of energy?

What is the probability that a neutron is created by the (α, n) reaction over

this distance. What α activity is required for 1 Bq of neutron activity?

6. Fission

Fission is the spontaneous or induced breakup of a nucleus. While it is “just

another reaction” from the physics point of view, it has a special role in

public aﬀairs because of its enormous applications in energy production and

weaponry.

In this chapter, we start with a general discussion of nuclear energy pro-

duction. In Sec. 6.2 we catalog the various types of ﬁssion reactions and in

Sec. 6.3 we show how ﬁssion rates can be explained qualitatively by consid-

ering ﬁssion as a barrier penetration process.

The remaining sections consider the basic practical problems in ﬁssion

technology. Section 6.4 lists the nuclei that can be used as nuclear fuel. Section

6.5 presents the general conditions for setting up a ﬁssion chain reaction.

Section 6.6 discusses problems associated with neutrons moderators used in

thermal neutron reactors. Section 6.7 introduces the Boltzmann transport

equation used to calculate the performance of reactors. Finally, in Sect. 6.8

we describe the operation of the major types of ﬁssion reactors.

6.1 Nuclear energy

Nuclear energy is extracted through exothermic nuclear reactions, the two

principal types being ﬁssion and fusion. Of course, we should remember that

radioactivity itself liberates a large amount of energy as ﬁrst recognized by

Pierre Curie and Henri Becquerel. For example, the decay of

238

U →

234

Th α t

1/2

=4.468 × 10

yr , Q

=4.262 MeV ,

generates a power of

P =

238

ln 2

1/2

=8× 10

−9

−1

. (6.1)

This tiny amount of power is increased in nuclear reactors by ten orders of

magnitude through neutron-induced ﬁssion of

235

U. This increases Q from

4.668 MeV to ∼ 200 MeV and decreases the eﬀective lifetime to a few months.

Fission and fusion of nuclei are the main reactions which produce nuclear

energy in large amounts.

286 6. Fission

200100

200

100

Q (MeV)

fission

fusion

Fig. 6.1. The energy release in ﬁssion and self-fusion as predicted by the Bethe–

Weizs¨acker formula (2.13) for β-stable nuclei. Only nuclei with 40 <A<95 are

stable against both ﬁssion and self-fusion. In this ﬁgure, Q

ﬁs

(A, Z) is calculated for

symmetric ﬁssion, A

= A

= A/2 and Z

= Z

= Z/2. Q

fus

(A, Z) is calculated

for the production of a single nucleus of A



=2A and Z



=2Z.

A nucleus

X can breakup, i.e. ﬁssion, in many ways. In the simplest

case, there are two fragments

)

X →

X +

X + Q

ﬁs

(6.2)

The energy release is

ﬁs

= m(A

+ A

+ Z

− [m(A

)+m(A

)]c

= B(A

)+B(A

) − B(A

+ A

+ Z

) . (6.3)

In the rest frame of the initial nucleus, the sum of the kinetic energies of

the ﬁssion fragments is Q

ﬁs

. The diﬀerence in binding energies B(A, Z)of

the initial and ﬁnal nuclei is thus transformed into kinetic energy which is

eventually transfered to the medium, heating it up.

Symmetrically, two nuclei can undergo fusion if, by a nuclear reaction,

they produce a heavier nucleus (plus a light particle y)

→

+ y + Q

fus

. (6.4)

If Q

fus

is positive, the light particle y is necessary to conserve momentum. In

the simplest case, y is a photon so A = A

+ A

, Z = Z

+ Z

and

fus

= B(A

+ A

+ Z

) − B(A

) . (6.5)

If Q

fus

> 0, the reaction can take place with the two initial nuclei at rest,

in which case Q

fus

is the energy of the ﬁnal state photon. As in ﬁssion, the

diﬀerence in binding energies of the initial and ﬁnal nuclei is transformed to

kinetic energy.

6.2 Fission products 287

The curve of binding energies (Fig. 1.2) shows that the binding energy is

maximum for A ∼ 56 implying

• heavy nuclei can ﬁssion ;

• light nuclei can fuse.

Figure 6.1 shows Q for symmetric (A

= A

) fusion and ﬁssion as a func-

tion of A of the initial nucleus. We see that only nuclei with 40 <A<95

are intrinsically stable against both spontaneous ﬁssion and self-fusion. For-

tunately for us, both fusion and ﬁssion are very slow processes under normal

conditions. As we will see in Chap. 7, fusion is inhibited by the Coulomb

barrier between positively charged nuclei. As we will see in the Sect. 6.3,

spontaneous ﬁssion is strongly inhibited by the “ﬁssion barrier” resulting

from the surface term in the Bethe–Weis¨acker formula (2.13).

In practice, we will mostly be concerned with the ﬁssion of heavy nuclei,

A  240. For two ﬁssion fragments with A  120 we can estimate Q

ﬁs

from

Fig. 1.2. For the initial nuclei we have B/A(240)  7.6 MeV, while for the

ﬁssion fragments we have B/A(120) ∼ 8.5 MeV. This gives

ﬁs

∼ 240 × (8.5 − 7.6) ∼ 220 MeV . (6.6)

(This energy is somewhat greater than that shown in Fig. 6.1 because the

ﬁgure shows the Q of the initial ﬁssion while our estimate includes the energy

generated by β-decays of the ﬁssion fragments down to β-stable nuclei.) In

actual ﬁssion processes, there is always a certain number of free neutrons

produced. Let

ν be the average number of such neutrons. Since, by deﬁnition,

they have no binding energy, we have

ﬁs

∼ 220 MeV − ν × 8.5MeV , (6.7)

i.e. for

ν ∼ 2.5, Q

ﬁs

∼ 200 MeV.

6.2 Fission products

A given nucleus can ﬁssion in many ways. For

236

U, one possibility is

236

U →

137

I+

Y + 3n (6.8)

137

I →

137

Xe e

−

1/2

=24.5s

137

Xe →

137

Cs e

−

1/2

=3.818 m

137

Cs →

137

Ba e

−

1/2

=30.07 yr

Y →

Zr e

−

1/2

=5.34 s ,

or, globally

236

U →

137

Ba +

Zr + 3n + 4e

−

. (6.9)

288 6. Fission

The original ﬁssion fragments

137

Iand

Y are transformed through a series

of β-decays to the β-stable

137

Ba and

Zr. Note the long half-life of

137

that makes it eﬀectively stable over a fuel cycle of a nuclear reactor. It is an

example of radioactive waste from a ﬁssion reactor.

Of course the reaction (6.8) is only one of many possible ﬁssion modes. It

is necessary to consider the problem statistically. The observed distribution

of fragments of neutron-induced ﬁssion of

235

U, corresponding roughly to

spontaneous ﬁssion of

236

U, is shown in Fig. 6.2. One observes that ﬁssion is

mainly binary : the probability distribution of ﬁnal products is a two-peak

curve. The ﬁssion fragments have, statistically, diﬀerent masses. This is called

asymmetric ﬁssion. In the case of neutron induced ﬁssion of uranium 235, one

observes:

• A fragment in the “group ” A ∼ 95,Z∼ 36 (Br, Kr, Sr, Zr)

• A fragment in the “group ” A ∼ 140,Z∼ 54 (I, Xe, Ba).

The two groups are near the magic neutron numbers N =50andN =82

Because of the large neutron excess of heavy nuclei, the two ﬁssion frag-

mentsgenerallyarebelowthelineofβ-stability. They therefore β

−

-decay to

the bottom of the stability valley. This process is usually quite rapid, though

in the above example (6.8) the last β-decay of

137

Cs is rather slow. This is

an example of the long-lived radioactivity that is important for the storage

of waste from nuclear reactors.

We see that the ﬁssion process results in the production of a large variety

of particles. They can be classiﬁed as

• Two ﬁssion fragments that are β

−

-unstable.

• Other “prompt” particles, mostly neutrons emitted in the ﬁssion process

and photons emitted by the primary ﬁssion fragments produced in highly

excited states.

• “Delayed” particles mostly e

−

,andγ emittedintheβ

−

-decays of the

primary ﬁssion fragments fragments and their daughters.

Most of the released energy is contained in the initial kinetic energies

of the two ﬁssion fragments. The kinetic energy of each heavy fragment at

the time of fragmentation is of the order of 75 MeV, with initial velocities

of roughly 10

−1

. Given their large masses, their ranges are very small

∼ 10

−6

m. The stopping process transforms the kinetic energy to thermal

energy.

For a given ﬁssile nucleus we call

ν the average number of free neutrons

produced,

µ the number of β-decays, and κ the number of photons. The total

energy balance of a ﬁssion reaction is then

A → B + C +

ν n+µ e

−

+ µ

+ κ γ .

On average, the various components take the following energies

6.2 Fission products 289

A=142

A=95

N=82

0.01/fission

A=140

0.0001/fission

beta stable

0.01/fission

0.0001/fission

beta stable

Z=50

Z=28

N=50

Z=50

N=82

N=50

A=108

235

252

Z=28

Fig. 6.2. The distribution of ﬁssion fragments for neutron induced ﬁssion of

235

and for spontaneous ﬁssion of

252

Cf. The distribution for

235

U is dominated by

asymmetric ﬁssion into a light nucleus (A ∼ 95) and a heavier nucleus (A ∼ 140),

reasonably near the magic neutron numbers N = 50 and N = 82. The distribution

for

252

Cf is broader but still dominated by asymmetric ﬁssion. Because of the large

neutron excess in nuclei with A>230, almost all ﬁssion fragments are below the

line of β-stability and therefore decay by β

−

-emission.

290 6. Fission

MeV

Kinetic energy of fragments 165 ± 5

Energy of prompt photons 7 ± 1

Kinetic energy of neutrons 5 ± 0.5

Energy of β decay electrons 7 ± 1

Energy of β decay antineutrinos 10

Energy of γ decay photons 6 ± 1

Total 200 ± 6

Of the ∼ 200 MeV, only 190 ± 6 MeV are useful, since the neutrinos escape

and do not heat the medium.

The neutrons produced will maintain chain reactions in induced ﬁssion

processes. For

236

U (i.e. for the ﬁssion of

235

U induced by a neutron) the

mean number of produced neutrons is

ν =2.47.

The energy distribution of these neutrons is peaked around E

=0.7

MeV. The mean energy is <E

>∼ 2MeV.

6.3 Fission mechanism, ﬁssion barrier

The complexity of ﬁssion seems out of reach of a detailed theoretical descrip-

tion. However, as early as 1939, the liquid droplet model of Bohr and Wheeler

gave a good qualitative description.

We follow the potential energy of the system as the distance r between

the two fragments A and B varies, as illustrated on Fig. 6.3.

Initially, the two fragments are coalesced in a more or less spherical nu-

cleus. As r increases, the nucleus becomes deformed, its surface area increases

compared to the initial shape. Therefore, this deformation increases the sur-

face tension energy. On the other hand, the increase in the distance A − B

means a decrease of the Coulomb repulsion energy between A and B.There

is a competition between the nuclear forces and the Coulomb repulsion. At

some point, as r varies between r

(initial shape of the nucleus) and inﬁnity

(separated fragments A and B), the potential energy of the system has a

maximum value.

In other words, there is a potential barrier, called the ﬁssion barrier,that

must be crossed in order for the process to occur. One calls activation energy

, the diﬀerence between the maximum of the barrier and the energy of the

initial nucleus in its ground state. For nuclei with A ∼ 240, this energy turns

outtobeoftheorderof6to7MeV.

In order for the original nucleus to decay by spontaneous ﬁssion, the

barrier must be crossed by the quantum tunnel eﬀect. To get an idea of

the factors that determine the diﬃculty of tunneling through the barrier, it

is interesting to compare the surface and Coulomb energies of a spherical

nucleus as predicted by the Bethe–Weizs¨acker semi-empirical mass formula

(2.13)

6.3 Fission mechanism, ﬁssion barrier 291

4πε

Energy

Fig. 6.3. Variation of the energy of a deformed nucleus as a function of the distor-

tion as sketched. For small distortions, the energy increases with increasing distor-

tion because of the increasing surface area. When the two fragments are separated,

the energy falls with increasing separation because of the decreasing Coulomb en-

ergy. An energy barrier E

must be crossed for ﬁssion to occur.

0.7103 MeV Z

−1/3

17.804 MeV A

2/3

25.06

. (6.10)

Nuclei with Z

/A > 25 would be expected to have small barriers because the

Coulomb energy (decreasing function of separation) dominates the surface

energy (increasing function of separation). In fact, by calculating the surface

area and Coulomb energies of a nucleus in the shape of an ellipsoid, it can

be shown that the surface area varies twice as fast as the Coulomb energy as

the nucleus is deformed while keeping the volume constant. This means that

we expect ﬁssion to be instantaneous for

> 2E

⇒

> 50 . (6.11)

Super-heavy nuclei have Z/A ∼ 1/3 implying Z>150 for instantaneous

fusion. This is an absolute upper limit on the size of nuclei.

Figure 6.4 shows the inverse of the rate of spontaneous ﬁssion of selected

nuclei as a function of Z

/A. As expected, the inverse rate decreases rapidly

for increasing Z

/A. For nuclei with Z<92, the lifetime for spontaneous

292 6. Fission

ﬁssion becomes immeasurably large making nuclei with 100 <A<230 eﬀec-

tively stable in spite of the fact that Q

ﬁs

> 0 for these nuclei.

4535

−5

−

(sec)

258

241

255

249

235

250 259

257

239

250

256

258

Fig. 6.4. Spontaneous ﬁssion lifetimes as a function of the ﬁssion parameter Z

for selected nuclei. Circles are for even-Z nuclei, ﬁlled circles for even-even nuclei

and open circles for even-odd nuclei. Squares are for odd-Z nuclei.

The ﬁssion rate can be increased by placing the nucleus in an excited

statesoastoreducetheﬁssionbarrier.Thisismostsimplydonebyphoton

absorption:

X →

∗

→ ﬁssion , (6.12)

where

∗

is any excited state of the nucleus

X. This process is called

photo-ﬁssion.

Figure 6.5 shows the total photo-ﬁssion cross-section on

236

U as a function

of photon energy. The cross-section is negligible for E

< 5 MeV, rapidly

increases in the range 5 MeV <E

< 6 MeV and then slowly increases to

its maximum value of 0.2b at E

∼ 14 MeV. Somewhat arbitrarily we can

deﬁne ∆E

=5.7 MeV as the energy that must be added to the ground state