Braibant S., Giacomelli G., Spurio M. Particles and Fundamental Interactions: An Introduction to Particle Physics

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14.8 ˇ Decay 443

ΔM (MeV / c

)

ΔM (MeV / c

)

Mo Tc Ru Rh Pd Ag Cd In Sn Mo Tc Ru Rh Pd Ag Cd In Sn

A = 101

A=106

–

Fig. 14.11 ˇ decay mass parabola (a) for a nucleus with odd-A (A D 101)and(b) for a nucleus

with even-A (A D 106). The ordinate scale shows the relative mass difference between states,

corresponding to the energy available for the .e



; 

/ or .e

;

/ pairs in the ˇ decays denoted by

arrows. The stable states are

101

Ru, with BE =A D 8:601 MeV/nucleon and

106

Pd, with BE=A D

8:579 MeV/nucleon [14w2]

coupling constant) that

106

Cd can be considered as stable. The odd-odd nuclei with

A>14are never stable, as they always have a near even-even nucleus more strongly

bound. Only the light nuclei

H ,

Li,

N arestable(seeTable14.1), as the

increase of the energy asymmetry term would exceed the decrease of the energy

term due to the conﬁguration.

14.8.1 Elementary Theory of Nuclear ˇ-Decay

The transition probability of the neutron ˇ decay (which corresponds to the

elementary process (14.35)) has already been calculated in Sect. 8.3; the same

formalism with a few appropriate changes can be applied to (14.36). The transition

probability is given by

.p/ D

2

„

jMj

.Q  T

.Z; E/dp (14.38)

(note the dependence on [Energy

]). The square matrix element jMj

is dimen-

sionless; it depends on the wave functions of neutrons and protons in nuclei. The

Pauli exclusion principle, the isospin multiplicity of the new nuclear state and the

multiplicity of nuclear spin states must be taken into account in the calculation of

jMj. Compared to the expression for the neutron ˇ decay (Sect. 8.3), a corrective

function F

.Z; E/ depending on the atomic number and electron energy was

introduced in (14.38). It takes into account the effect of electron interaction with

444 14 Fundamental Aspects of Nucleon Interactions

the Coulomb ﬁeld of the nucleus. The effect is different for the ˇ



and ˇ

decays,

for which the potential is attractive in the former and repulsive in the latter. The term

.Z; E/, which relates to the density of ﬁnal states, was ﬁrst calculated by Fermi.

The lifetime  for the ˇ decay can be derived by integrating (14.38)



2

„

jMj

.Z; E/: (14.39)

The dimensional dependence on [Energy

] has been absorbed in .m

.Z; E/ is a (dimensionless) function which depends on F

.Z; E/, on the nu-

clear electric charge and on the integration upper limit for the electron momentum.

Although it can be calculated on the basis of nuclear models, when the decay energy

is E

max

 m

, the electron has a large momentum and F

.Z; E/ ' 1.Inthis

case, p

max

c ' E

max

, and the approximation f

.Z; E/ D .E

max

=30 (as

for the neutron) can be used. If E

max

 mc

, the nuclear lifetime depends on the

ﬁfth power of the maximum energy available in the ﬁnal state, in agreement with

the Sargent rule:



max

60

„

jMj

: (14.40)

This strong dependence on E

max

explains why the ˇ decay lifetime ranges from

fractions of seconds to billions of years.

14.9 Nuclear Reactions and Nuclear Fission

In a nuclear reaction, two particles or two nuclei change their state as a result of

their interaction

a C b ! c C d C Q: (14.41)

Q indicates the mass difference between the initial and ﬁnal states, Q D .m

 m

. Reactions with Q>0are called exothermic:massisconverted

into kinetic energy of the ﬁnal state. Reactions with Q<0are endothermic:

kinetic energy is converted into mass. Due to the short range of the nuclear

interaction and if the initial particles have electric charges, energy must be provided

to overcome the Coulomb repulsion. In nuclear reactions, in addition to energy

and momentum, the angular momentum, the electric charge, the baryonic number,

the strong isospin, the charge conjugation and the parity are conserved. The ﬁrst

nuclear reaction causing a substance transmutation was observed by Rutherford in

1919 using ˛ particles (emitted in polonium decays) with a kinetic energy sufﬁcient

to compensate the negative value of Q and the Coulomb repulsion in the reaction

˛ C

N !

O C pI Q D1:19 MeV.

14.9 Nuclear Reactions and Nuclear Fission 445

An important experimental breakthrough came in 1932 when the neutron was

discovered by Chadwick in the reaction ˛ C

Be !

C C nI Q DC5:71 MeV.

This discovery opened new possibilities for investigation of the nuclear structure

and of the nuclear interactions. Indeed, neutrons are not affected by the Coulomb

repulsion and nuclear reactions can occur with very little kinetic energy. In addition

to nuclear interaction, there are reactions caused by weak or electromagnetic

interactions which play a key role in the mechanism of nucleosynthesis and energy

production in stars.

14.9.1 Nuclear Fission

The neutron discovery was followed by an intense activity to produce nuclear

reactions initiated by neutrons. E. Fermi (Nobel laureate in 1938 for this work)

studied neutron capture reactions to produce heavy nuclei and their ˇ decays. In

1938, O. Hahn and F. Strassmann observed that elements with atomic number

approximately equal to half that of uranium are produced in collisions of neutrons

with uranium nuclei, e.g.,

n C

U !

Ba C

Kr: (14.42)

In 1939, L. Meitner and O. Frisch hypothesized that the production of elements

with intermediate atomic number was due to the heavy nuclear ﬁssion induced by

neutron capture.

Spontaneous ﬁssion. For elements in the periodic table, the spontaneous ﬁssion

(i.e., not induced by external factors)

N !

Aa

Zz

X C

Y C Q (14.43)

is preventedby the nuclear attractive potential. This can be deduced from the nuclear

drop model which predicts that only nuclei with Z

=A > 47 may be subject to

spontaneous ﬁssion.

Fission induced by neutrons. The situation can change drastically if a heavy

nucleus captures a neutron. In this case, the nucleus can split into two. The decrease

in the nuclear binding energy (14.15), due to variations of the negative terms, allow

solutions for which the nuclear ﬁssion from

N to two nuclei

Y (with

C A

D A, Z

C Z

D Z) is energetically favored.

Bohr and Wheeler determined some properties of spontaneous ﬁssion and n

induced ﬁssion using the nuclear drop model:

• For nuclei with Z

=A > 47, i.e., Z  130–140, the energy difference after

the ﬁssion is positive; the nuclei are unstable and undergo spontaneous ﬁssion

(indeed, they do not exist in nature).

446 14 Fundamental Aspects of Nucleon Interactions

• For nuclei near the instability threshold, i.e., with A ' 100, the energy change

due to neutron capture is not sufﬁcient to cause ﬁssion.

• For heavier stable nuclei, A  240, the energy variation induced by neutron

capture is sufﬁcient to cause ﬁssion.

• The activation energy EB Q needed to ignite the ﬁssion can be calculated; it is

different for odd-A and even-A nuclei.

Uranium ﬁssion. Natural uranium is composed of two isotopes:

238

U and

235

U ,

with relative abundances of 99.28% and 0.72%. The ﬁssion of

235

U is initiated by

capture of thermal

neutrons with a cross-section: 

235

' 580 b. The ﬁssion of

238

U is initiated by neutrons with kinetic energy T

>1:8MeV with a much smaller

cross-section: 

238

' 0:5 b. The typical energy released in a nuclear ﬁssion is Q D

210 MeV, yielding an efﬁciency Q=M

D 210 MeV=220 GeV ' 10

3

Nuclei with A

 90, A

 145 are produced in the ﬁnal state, for example,

one has

n C

235

U !

Br C

148

La C n

n C

235

U !

Rb C

141

CsC n C n:

In addition, an average number

n  2:5 of immediate neutrons with kinetic

energy T

' 2 MeV are also produced. The term immediate neutrons indicates that

ﬁssion occurs with very short reaction times (10

16

–10

14

s) and that the neutron

emission is accompanied by the emission of photons with energy E



' 8 MeV.

The remaining energy is in the form of kinetic energy of the two emitted nuclei.

Usually, these nuclei have an excess of neutrons and reach the stability valley in

the A  Z plane through ˇ emission. The energy released in the ˇ decays is on

average E

 20 MeV, from which approximately 12 MeV are carried away by

the antineutrinos. Therefore, the ﬁssion reaction is a source of neutrons, photons,

electrons and antineutrinos.

One may wonder why the

235

U is more efﬁcient than other isotopes in the ﬁssion

reaction induced by neutron capture. In the case of

235

U , the intermediate state

236

is formed; the mass difference is

M

235

D M



235



C m

 M



236



D 6:5 MeV:

The activation energy of the

235

U ﬁssion is 6.2 MeV; therefore, the neutrons

do not need to have kinetic energy and the

235

U ﬁssion is obtained with thermal

neutrons. In the case of the

238

U isotope, the intermediate state

239

U is formed; the

mass difference is

M

238

D M



238



C m

 M



239



D 4:8 MeV:

Thermal neutrons are obtained slowing down neutrons; after a number of elastic collisions with

light nuclei in a neutron moderator at a given temperature, neutrons arrive at about this energy

level, provided that they are not absorbed.

14.9 Nuclear Reactions and Nuclear Fission 447

The activation energy for the

238

U ﬁssion is 6.6 MeV; hence, to enable the

ﬁssion of this isotope, neutrons with kinetic energy T

>1:8MeV are required.

In the case of uranium, the term a

in (14.15) plays a decisive role. In the neutron

capture reaction, the even-even nuclei, for example,

236

U and

238

U , become even-

odd nuclei. An additional energy equal to a

1=2

D 12:6 MeV=

236 ' 0:8 MeV

should be provided. Instead, the nucleus

235

U becomes an even-even nucleus

after neutron capture, but the same amount of energy ('0.8 MeV) now becomes

available. Note that the difference between the binding energies in the two cases

reﬂects the difference between M

235

 M

238

14.9.2 Fission Nuclear Reactors

A ﬁssion reaction typically produces an average number of neutrons >1; these neu-

trons may subsequently induce new ﬁssion reactions. Conditions for self-powered

chain reactions can be created to produce energy by ﬁssion. The ﬁrst nuclear pile

was built by Fermi and collaborators in 1942. A nuclear reactor based on controlled

ﬁssion reactions are used to produce: (1) energy; (2) neutron sources for research

and medical purposes; (3) radio-isotopes or other ﬁssile substances, for example,

239

Pu or

233

U . Uncontrolled chain reactions are the basis for nuclear bombs.

A typical nuclear reactor for energy production is based on chain reactions

in uranium. By

235

U ﬁssion reaction, each produces an average of ' 200 MeV

and 2.5 neutrons. The cross-section of neutron capture is small and only thermal

neutrons are efﬁciently captured by

235

U to produce successive ﬁssion reactions.

For this reason, it is necessary to moderate neutrons, forcing them to lose energy

in successive collisions with light nuclei. From the construction point of view, the

core of a uranium nuclear reactor consists of a moderator material and of enriched

uranium, i.e., a kind of uranium in which the percent composition of

235

U is

increased typically at 3% level (in nature, this isotope has an abundance of 0.7%).

To moderate neutrons, materials containing low Z nuclei are used, e.g., H

O (the so-called heavy water)orC . Carbon is not very efﬁcient, but can be

distributed effectively in the fuel. The Chernobyl energy power plant in the former

USSR used carbon as a moderator. The water (normal or heavy) is also an efﬁcient

heat exchanger to cool the reactor. The hydrogen nuclei in the molecule of water

are also very efﬁcient to moderate neutrons; there is a disadvantage due to the high

cross-section of the competitive reaction n Cp !

H C, that reduces the neutron

budget. The deuterium nucleus present in heavy water (instead of hydrogen) has a

much smaller cross-section for the process n C

H !

H C , but it produces

radioactive tritium that must be ﬁltered out from the cooling system.

With a proper combination of fuel and moderator, the situation in which there

is on average one thermal neutron per ﬁssion is reached. This is the so-called

critical reactor situation. Preventing that this factor exceeds unity and that the

reactor reaches the super-critical situation with the risk of failures, materials with

448 14 Fundamental Aspects of Nucleon Interactions

high capture cross-section for thermal neutrons must be inserted. The most suitable

material is cadmium.

One gram of

235

U produces, in a critical reactor,

1g.

235

U/D

6  10

235

 200 MeV ' 0:8  10

J; (14.44)

approximately three times the energy produced in the combustion of a ton of coal.

The energy associated with nuclear processes is in fact 6 orders of magnitude

higher than the energy associated with chemical (D electromagnetic) reactions that

govern combustion processes.

14.10 Nuclear Fusion in Astrophysical Environments

In a fusion reaction, two light nuclei combine to form a nucleus with larger atomic

weight. The binding energy of nuclei BE (14.15) as a function of the mass number

A shows that the fusion reaction,

X C

Aa

Zz

Y !

W C Q; (14.45)

produces free energy Q>0if d.BE/=dA >0, corresponding to nuclei with

A<60(i.e., nuclei lighter than iron). Due to the short range of nuclear interaction,

the fusion reaction can only occur if the nuclei have sufﬁcient initial kinetic energy

to compensate the Coulomb repulsion in order to bring the two nuclei in contact.

For example, the reaction

C C

C !

Mg C 13:9 MeV

is exothermic, but requires that the two carbon nuclei initially have an energy

that overcomes the Coulomb repulsion. The activation energy E

corresponds to

(A D 12; Z D 6; R

D 1:2 fm)



1=3

D 9 MeV: (14.46)

Spending 9.0, 13.9MeV are produced. The efﬁciency (Q=2M

'610

4

in this

example) is inversely proportional to the mass of the merging nuclei. Energy

released in fusion reactions transforms in kinetic energy of the nuclei; if there

is a force ﬁeld capable of conﬁning the nuclei, the temperature increases un-

til equilibrium is reached. In the equilibrium phase, the fusion reactions are

self-sustained and produce energy. Stable conﬁnement situations are present in the

cores of stars, and (hopefully) in future nuclear fusion reactors.

14.10 Nuclear Fusion in Astrophysical Environments 449

For light nuclei with Z ' A=2, the required temperature to compensate the

Coulomb repulsion (14.46)is

 kT D

5=3

' 0:14  A

5=3

MeV T ' 1:6 10

 A

5=3

K: (14.47)

However, the energy of the nuclei in the Maxwell velocity distribution tail and the

tunneling probability make nuclear fusion possible even at lower temperatures.

14.10.1 Fusion in Stars

The Sun (like all main sequence stars) produces energy by fusion (see Fig. 12.11a):

four protons form a helium nucleus releasing about 26 MeV with a very high

efﬁciency, 26 MeV=.2m

C 2m

/ D 0:007. The temperature inside the Sun is

T ' 1:5  10

K, corresponding to a proton kinetic energy of E

' 1:3 keV, much

lower than the Coulomb repulsion energy ' 0:8 MeV. The force ﬁeld that holds the

nuclei in the star core is due to the gravitational pressure of matter present in the

outer layers. A star is a system in equilibrium between pressure due to gravity and

pressure due to radiation produced by fusion reactions in the core.

The material formed in the early phase of the universe, at the time of the

primordial nucleosynthesis (Sect. 13.6), was in mass by 3/4 of protons, 1/4 helium

nuclei and traces of heavier Li; Be; B elements. Gravitational attraction and

ﬂuctuations in particle density caused condensations of matter with the formation of

protostars. When in the center of a protostar, the density becomes sufﬁciently high

and the fusion cycle starts.

The ﬁrst reaction of the cycle occurs through weak interaction, that is,

p C p !

H C e

C 

Q D 0:42 MeV: (14.48)

The cross-section for this low energy process is extremely small,   10

55

this is why the stars burn very slowly. The reaction in which the formed deuteron

interacts with a proton occur via strong interaction and has a large cross-section:

p C

H !

He C QD 5:49 MeV: (14.49)

Thus, the produced deuterons are immediately transformed. When the density of

He nuclei is high enough, the reaction

He C

He !

He C p CpQD 12:86 MeV (14.50)

starts and one helium nucleus plus two protons are formed; the latter are available

to start other cycles. Further reactions of the cycle, particularly those producing

neutrinos, are shown in Fig. 12.11. The proton–proton cycle can be summarized as

450 14 Fundamental Aspects of Nucleon Interactions

4p !

He C 2e

C 2

Q D 24:7 MeV: (14.51)

To this value of Q, 2 MeV must be added due to the annihilation of the two

positrons. Neutrinos do not energetically contribute to star equilibrium because they

have a low interaction probability and immediately escape. Since each neutrino has

an average energy

E  0:3 MeV which is subtracted, the total energy balance

of the cycle is about 26 MeV. The measurement of solar neutrino ﬂux (Sect. 12.7

and Exercise 14.7) provided important information not only on the neutrino physics

properties, but also on how the fusion process occurs in the Sun and on the properties

of our star (Standard Solar Model [B89]).

Due to many other processes, the energy produced by fusion propagates to the

surface of the Sun, the photosphere. The diffusion time of photons from the stellar

core to the photosphere (taking into account the -p cross-section) is of the order of

–10

years.

The Sun has converted hydrogen to helium for roughly 5  10

years. The value

of solar mass and the emitted power indicate that it will continue for about as many

years. A star with roughly the solar mass, when hydrogen is exhausted, tends to

contract and to increase its density; this happens because the radiation produced

by the fusion reactions is no longer able to balance the gravitational pressure. With

contraction, the gravitational energy is converted into kinetic energy of the nuclei:

the temperature increases and further fusion reactions may be ignited.

The critical point is the carbon formation. In a star composed mainly of

nuclei,

Be is continuously formed.

Be has a mass which is slightly larger than

twice the

He mass, that is,

He C

He !

Be I Q D0:09 MeV. Once formed,

it splits again into two

He nuclei. When the

He density is extremely high, a

fusion reaction forming carbon nuclei in an excited state occurs with a resonant

cross-section:

He C

Be !



. The excited state C



immediately decays to

the ground state. The carbon abundance in the universe is relatively high and it may

also be present in stars that have not exhausted the proton cycle. In the presence of

protons, the nucleus

C actsasacatalyst for another cycle, similar to the proton–

proton cycle, that produces energy transforming protons into helium nuclei: the

CNO cycle.

Reaction Q (MeV)

p C

C !

N C  1.94

C !

C Ce

C 

1.20

p C

C !

N C  7.55

p C

N !

O C 7.29

O !

N C e

C 

1.94

p C

N !

C C

He 4.96

14.10 Nuclear Fusion in Astrophysical Environments 451

Finally, one has 4p !

He C 2e

C 3 C 2

with a total energy balance of

about 26MeV. The

C nucleus is strongly bound and is the starting point for the

formation of heavier nuclei through fusion; for example,

Reaction Q (MeV) E

(MeV)

He C

C !

O C  7.16 3.6

He C

O !

Ne C  4.73 4.5

He C

Ne !

Mg C 9.31 5.4

Note that the Coulomb potential barrier E

increases with the atomic number.

The probability of each reaction is inversely proportional to E

and the relative

nuclear abundance decreases with increasing A.

When helium is exhausted as a fuel, the star tends to contract again (if massive

enough), increasing density and temperature. It can start a new fusion cycle

using carbon and oxygen as fuel. The potential barrier is respectively 9.0 and

14.6 MeV; to sustain the following fusion reactions, temperatures T  10

Kare

required.

Reaction Q (MeV)

C C

C !

Ne C ˛ 4.62

C C

C !

Na C p 2.24

C C

C !

Mg Cn 2.61

C C

C !

Ne C  13.93

O C

O !

Si C ˛ 9.59

O C

O !

P C p 7.68

O C

O !

S Cn 1.46

O C

O !

S C 16.54

In the above reactions, in addition to the heavy nuclei, photons, protons, neutrons

and ˛ particles are also produced. These particles have energies high enough to pro-

duce other heavy nuclei from

Si and

S nuclei. These reactions are energetically

favored when compared to fusion reactions, for example,

Si C

Si !

NiC,

that require an even higher temperature.

14.10.2 Formation of Elements Heavier than Fe in Massive Stars

Nucleosynthesis from nuclear fusion proceeds until the formation of nuclei with

A  60. The heavier elements up to iron are only synthesized in massive stars

with M>8M

. Near the end of its life, the star begins to convert the silicon

core into iron and nickel (this phase lasts a few days). Once the star core is

452 14 Fundamental Aspects of Nucleon Interactions

made primarily of iron, further compression of the core does not ignite nuclear

fusion anymore; the star is unable to thermodynamically support its outer envelope

made of concentric shells of silicon, sulfur, oxygen, neon, carbon, helium and

hydrogen (in order from the innermost to the outermost part). As the surrounding

matter falls inward under gravity, the temperature of the core is so high that the

iron disintegrates with a large absorption of energy (core cooling) into helium

nuclei, neutrons and protons. As can be seen in Fig. 14.3, the transformation

of iron nuclei into heavier or lighter elements is endothermic, and energy must

be provided. Now, the electron capture on protons becomes favored and elec-

tron neutrinos are produced as the core gets neutronized (a process known as

neutronization).

The core cooling deprives the star of the energy power that allowed the balance

with the gravitational pull of falling matter. The rest of the star, without the support

of the radiation, collapses, compressing the star nucleus. When the core reaches

densities of above 10

g/cm

, neutrinos are trapped in the so-called neutrinosphere.

The collapse continues until 3–4 times the density of ordinary nuclei is reached,

after which the inner core rebounds sending a shock-wave across the outer stellar

shells. This shock-wave loses energy heating the matter that it crosses. Particle–

antiparticle annihilation (mainly e



) is a source of neutrino–antineutrino pairs of

all ﬂavors. In this way, 99% of the star binding energy is released in the form of

neutrinos in a few tens of seconds.

The shock wave generates local compressions that trigger new interactions. The

formation of nuclei heavier than iron (from silver, platinum, gold, mercury up to

uranium) is induced by neutron capture and ˇ



decay:

n C

X !

AC1

X C 

AC1

X !

AC1

ZC1

Y C e



C 

: (14.52)

Charged particles such as p or ˛ cannot be easily captured due to the additional

Coulomb barrier. The neutron capture can occur when free neutrons are available,

just immediately after the supernova explosion. In particular, nickel and cobalt are

produced; the ﬂuorescence due to their radioactive decay is the primary cause of

the visible light observed by astronomers in the ﬁrst weeks after the supernova

explosion.

The relative abundance of heavy nuclei depends on the lifetime 

for the neutron

capture and on the lifetime 

for the ˇ decay:

•If

 

, the nucleus

AC1

X formed by neutron capture has enough time to

undergo a ˇ



decay; the nucleus tends to stabilize along the stability valley.

Since the probability of neutron capture 1=

is small, the reactions occur at a

low rate and are classiﬁed as s-processes (s Dslow).

•If



, the newly formed nucleus does not decay immediately; after sub-

sequent neutron captures, the isotopes move away from the stability valley

X !

AC1

X !

AC2

X ! :::/. The number of neutrons increases and thus

the instability for ˇ



decay. Since the probability of neutron capture is large and

the reactions occur with a high rate, these are called r-processes (r Drapid).