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CHAPTER 1 Radioactive Decay & Decay Modes

Decay Series

The terms ‘radioactive transmutation” and radioactive decay” are synonymous. Many radionuclides were found after the discovery of radioactivity in 1896. Their atomic mass and mass numbers were determined later, after the concept of isotopes had been established.

The great variety of radionuclides present in thorium are listed in Table 1. Whereas thorium has only one isotope with a very long half-life (Th-232, uranium has two (U-238 and U-235), giving rise to one decay series for Th and two for U. To distin-guish the two decay series of U, they were named after long lived members: the uranium-radium series and the actinium series. The uranium-radium series includes the most important radium isotope (Ra-226) and the actinium series the most important actinium isotope (Ac-227).

In all decay series, only α and β− decay are observed. With emission of an α particle (He-4) the mass number decreases by 4 units, and the atomic number by 2 units (A’ = A - 4; Z’ =

Z - 2). With emission β− particle the mass number does not change, but the atomic number increases by 1 unit (A’ = A; Z’ = Z + 1). By application of the displacement laws it can easily be deduced that all members of a certain decay series may differ from each other in their mass numbers only by multiples of 4 units. The mass number of Th-232 is 323, which can be written 4n (n=58). By variation of n, all possible mass numbers of the members of decay

Engineering Aspects of Food Irradiation 1

Radioactive Decay

2

series of Th-232 (thorium family) are obtained. Thus, A = 4n is a common label for the tho-rium family. For uranium-radium family the label is A = 4n + 2, and for actinium family A = 4n + 3. The radioactive decay series with A = 4n + 1 is for the neptunium family (Np-237) created artificially by nuclear reactions.

The genetic correlations of the radionuclides withis families are often characterized by the terms “mother” and “daughter”. Thus Th-232 is the mother nuclide of all members of the thorium family, U-238 of the uranium family, Ra-226 of the mother nuclide of Rn-222, and so forth.

The final members of the decay series are stable nuclides: Pb-208 at the end of the thorium family, Pb-206 at the end of the uranium-radium family, Pb-207 at the end of the actinium family, and Bi-209 at the end of the neptunium family. In all four decay series one or more branching are observed. For instance Bi-212 decays with a certain probability by emission of α particle into Tl-208, and with another probability by emission of an electron into Po-212. Tl-208 decays by emission of an electron into Pb-208, and Po-212 by emission of an α particle into the same nuclide (Table 1), thus closing the branching. In both branches the sequence of decay alternates: either α decay is followed

by β- decay or β- decay is followed by α decay.

TABLE 1. Thorium decay series: A=4n

Nuclide Half-life Decay modeMaximum energy of the radiation [MeV]

Th-232 1.405x1010 y α 4.01

Ra-228 (MsTh1) 5.75 y β- 0.04

Ac-228 (MsTh2) 6.13 h β- 2.11

Th-228 (RdTh) 1.91 y α 5.42

Ra-224 (ThX) 3.66 d α 5.69

Rn-220 (Tn) 55.6 s α 6.29

Po-216 (ThA) 0.15 s α 6.78

Pb-212 (ThB) 10.64 h β- 0.57

Bi-212 (ThC) 60.6 min α,β- α: 6.09; β: 2.25

Po-212 (ThC’) 3.0x10-7 s α 8.79

Tl-208 (ThC”) 3.05 min β- 1.80

Pb-208 (ThD) stable

Engineering Aspects of Food Irradiation

Radioactive Decay

Law & Energy of Radioactive Decay

Radioactive decay follows the laws of statistics. If a sufficiently great number of radioactive atoms are observed for a sufficient long time, the law of radioactive decay is found to be:

(EQ 1)

where N is the number of atoms of a certain radionuclide, -dN/dt is the disintegra-tion rate, and λ the disintegration or decay constant (1/s). The negative signal is needed because N decreases as the time t increases. Equation (1) is the measure of the probability of radioactive decay. The law of radioactive decay describes the kinetics of the reaction:

(EQ 2)

where A denotes the radioactive mother nuclide, B the daughter nuclide, x the parti-cle emitted and ∆E the energy set free by the decay process, which is also called the Q-value. This equation represents a first-order reaction and in the present case a mononuclear reaction.

Radioactive decay is only possible if ∆E > 0. ∆E can be determined by comparison

of the masses. According to the relation by Einstein (E = mc2):

(EQ 3)

By calculation of ∆E it can be decided whether a decay process is possible or not.

Even if ∆E > 0, the question of the probability of a radioactive decay process is still open. It can only be answered if the energy barrier is known. The energetics of radioactive decay are plotted schematically in Figure 1. The energies of the mother nuclide and the products of the mononuclear reaction differ by ∆E. But the nuclide A has to surmount an energy barrier with the threshold energy Es. The nuclide may occupy discrete energy levels above ground level. However, only if its excitation energy is high enough can decay occur. The energy barrier must either be sur-mounted or crossed by quantum mechanical tunnelling.

dNdt-------– λN=

A B x ∆E+ +→

∆E ∆Mc2

MA Mb Mx+( )–[ ]c2= =


Radioactive Decay

4

Figure 1: Energy barrier of radioactive decay.

The law governing radioactive decay (Eq(1)) is analogous to that of first-order chemical kinetics. The excited state on top of the energy barrier corresponds to the activated complex, and Es is equivalent to the activation energy.

Integration of Eq(1) gives:

(EQ 4)

where No is the number of radioactive atoms at time = 0. Instead of the decay con-stant λ, the half-life t1/2 is frequently used. This is the time after which half the radioactive atoms have decayed: To find t1/2 in terms of λ, we write from Eq(4) at time t = t1/2:

A

B + x

Es

∆E

ener

gy

excited state ofnuclide A

N Noeλt–=


Radioactive Decay

(EQ 5)

Taking the natural log of both sides gives:

(EQ 6)

and therefore:

(EQ 7)

and

(EQ 8)

This equation shows that the number of radioactive atoms has decreased to one-half after one half-life, to 1/128 (less than 1%) after 7 half-lives, and to 1/1024 (about 0.1%) after 10 half-lives. If the time t is small compared with the half-life of the radionuclide (t<<t1/2), the following approximation can be used:

(EQ 9)

(EQ 10)

The average lifetime τ is obtained by usual calculation of an average value:

(EQ 11)

From Eq(4) it follows that after the average lifetime t the number of radioactive atoms has decreased from No to No/e (τ = t1/2/(ln(2)).

12--- e

λt1 2⁄–=

λt1 2⁄–12--- ln 2ln–= =

t1 2⁄2ln

λ-------- 0.693

λ-------------= =

N No12---

tt1 2⁄---------

=

eλ t– 1 λt–

λt( )2

2------------ …–+=

1 2ln( ) tt1 2⁄-------- –

2ln( )2

2--------------- t

t1 2⁄-------- 2

…–+=

τ 1No

------ N td

0

∞

∫ eλt–

td

0

∞

∫ 1λ---= = =


Radioactive Decay

6

Generally, the half-life of a radionuclide does not depend on pressure, temperature, state of matter or chemical bonding. However, in some special cases in which low-energy transitions occur, these parameters have been found to have a small influ-ence.

The activity A of a radionuclide is given by its disintegration rate:

(EQ 12)

The dimensions is s-1, and the unit is called bacquerel (Bq): 1 Bq = 1 s-1. An older unit still used is curie (Ci). It is related to the activity of 1 g of Ra-226 and is

defined as 1 Ci = 3.700x1010 s-1 = 37 GBq. Smaller units are 1 millicurie (nCi) = 37 MBq, 1 microcurie (µCi) = 37 kBq, 1 nanocurie (nCi) = 37 Bq, 1 picocurie (pCi) = 0.37 Bq.

As the activity A is proportional to the number N of radioactive atoms, the exponen-tial law, Eq(4), holds also the activity:

(EQ 13)

The mass m of the radioactive atoms can be calculated from their number N and their activity A:

(EQ 14)

The ratio of the activity to the total mass m of the element (the sum of radioactive and stable isotopes) is called the specific activity A:

(EQ 15)

Radioactive Equilibria

Genetic relations between radionuclides, as in the decay series, can be written in the form:

AdNdt------- λN

2lnt1 2⁄--------N= = =

A Aoeλt–=

mNMNAv

--------- AMNAvλ------------ AM

NAv 2ln-----------------t1 2⁄= = =

AsAm---- s[ ]=


Radioactive Decay

(EQ 16)

Nuclide 1 is transformed by radioactive decay into nuclide 2, and the latter into nuclide 3. Nuclide 1 is the mother nuclide of nuclide 2, and nuclide 2 the daughter of nuclide 1. At any instant, the rate of production of nuclide 2 is given by the decay rate of nuclide 1 diminished by the decay rate of nuclide 2:

(EQ 17)

Which the decay rate of nuclei 1 it follows:

(EQ 18)

where N10 is the number of atoms of nuclide 1 at time 0. The solution of the first-

order differential Eq(18) is:

(EQ 19)

where N20 is the number of atoms of nuclide 2 at time 0. If nuclides 1 and 2 are sep-

arated quantitatively at t =0, the situation becomes simpler and two fractions are obtained. In the fraction containing nuclide 2, this nuclide is not produced any more by any decay of nuclide 1, and for the fraction containing nuclide 1 it follows with

N20 = 0:

(EQ 20)

Rearranging gives:

(EQ 21)

or, after substitution of the decay constants λ by the half-lives t1/2:

nuclide 1 nuclide 2 nuclide 3→→

dN2

dt---------

dN1

dt---------– λ2N2– λ1N1 λ2N2–= =

dN2

dt--------- λ2N2 λ1N

01e

λ1t–+ + 0=

N2

λ1

λ2 λ1–-----------------N

01 e

λ1t–e

λ2t––( ) N

02e

λ2 t–+=

N2

λ1

λ2 λ1–-----------------N0

1 eλ1t–

eλ2 t–

–( )=

N2

λ1

λ2 λ1–-----------------N1 1 e

λ1 λ2–( ) t––( )=


Radioactive Decay

8

(EQ 22)

The term in the exponent of 1/2 in Eq(22) may be written to show the influence of the ratio of the half-lives t1/2(1)/t1/2(2):

(EQ 23)

The time necessary to attain radioactive equilibrium depends on the half-life of the daughter nuclide as well as on the ratio of the half-lives. This is seen in Figure 2. After a sufficiently long time, the exponential function in Eq(21) becomes zero and radioactive equilibrium is established:

(EQ 24)

In the radioactive equilibrium, the ratio N2/N1, the ratio of the masses and the ratio

of the activities are constant. This is not an equilibrium in the sense used in thermo-dynamics and chemical kinetics, because it is not reversible, and it does not repre-sent a stationary state.

Four cases can be distinguished:

• the half-life of the mother nuclide is much longer that of the daughter nuclide, t1/2 (1) >> t1/2 (2).

• the half-life of the mother nuclide is longer than that of the daughter nuclide, but the decay of the mother nuclide cannot be neglected, t1/2 (1) > t1/2 (2)

• the half-life of the mother nuclide is shorter than that of the daughter nuclide: t1/2 (1) < t1/2 (2)

• the half-lives of the mother nuclide and the daughter nuclide are similar: t1/2 (1)

= t1/2 (2)

Secular radioactive equilibrium

In secular radioactive equilibrium (t1/2 (1) > > t1/2 (2)), Eq(20) reduces to:

N2

t1 2⁄ 2( ) t1 2⁄ 1( )⁄1 t1 2⁄– t1 2⁄ 1( )⁄--------------------------------------N1 1

12---

t 2( ) t 1( )–

–=

tt1 2⁄ 2( )----------------- t

t1 2⁄ 1( )-----------------– 1

t1 2⁄ 2( )t1 2⁄ 1( )-----------------–

tt1 2⁄ 2( )-----------------=

N2

λ1

λ2 λ1–-----------------N1

t1 2⁄ 2( ) t1 2⁄ 1( )⁄1 t1 2⁄– t1 2⁄ 1( )⁄--------------------------------------N1= =


Radioactive Decay

Figure 2: Attainment of radioactive equilibrium as a function of t/t1/2(2) for different ratios of the half-lives of the mother and daughter nuclides.

(EQ 25)

Assuming that mother and daughter nuclide are separated from each other at time t = 0, the growth of the daughter nuclide in the fraction of the mother nuclide and the decay of the daughter nuclide in the separated fraction are plotted in Figure 3.

0

1.0

0.6

0.4

0.2

0.8

11 5432 6 109870

)2(/ 2/1tt

)2()1(

)2(1

1

12

1

2 2/12/1

2/1

2

11

t

t

t

t

N

NY

−

−=−=

λλλ

10)2(/)1( 2/12/1 =tt

2)2(/)1( 2/12/1 =tt

5.1)2(/)1( 2/12/1 =tt

Y

N2

λ1

λ2

-----N1 1 eλ2t–

–( )=


Radioactive Decay

10

Figure 3: Decay of the daughter nuclide and its formation from the mother nuclide in the case of secular equilibrium as a function of t/t1/2 (2).

After t >> t1/2 (2) (in practice, after about 10 half-lives of nuclide 2), radioactive

equilibrium is established and the following holds:

(EQ 26)

(EQ 27)

The activities of the mother nuclide and of all the nuclides emerging from it by nuclear transformation or a sequence of nuclear transformation are the same, pro-vided that secular radioactive equilibrium is established.

Secular radioactive equilibrium has several practical applications:

0

1.0

0.6

0.4

0.2

0.8

21 3 540

)2(/ 2/1tt

Act

ivit

y, A

6 7

decay

formation

N2

N1

------λ1

λ2

-----t1 2⁄ 2( )t1 2⁄ 1( )-----------------= =

A1 A2=


Radioactive Decay

• Determination of the long half-life of a mother nuclide by measuring the mass ratio of daughter and mother nuclides, provided that the half-life of the daughter nuclide is known. Examples are the determination of the half-lives of Ra-226 and U-238 which cannot be obtained directly by measuring theory radioactive decay because of the long half-lives. The half-life of Ra-226 is obtained by measuring the absolute activity of the daughter nuclide Rn-222 in radioactive equilibrium with Ra-226, and its half-life. From the activity and the half-life, the number of radioactive atoms of Rn-222 is calculated by use of Eq.(12), and the half-life of Ra-226 from Eq.(26). The half-life of U-238 is determined by measuring the mass ratio of Ra-226 and U-238 in a uranium mineral. With the known half-life of Ra-236, that of U-238 is calculated by applying Eq.(26).

• Calculation of the mass ratios of radionuclides that are in secular radioactive equilibrium. From the half-lives, the masses of all radionuclides of the natural decay series in radioactive equilibrium with the long-lived mother nuclides can be calculated using Eq.(14).

• Calculation of the mass of a mother nuclide from the measured activity of a daughter nuclide. For example, the amount of U-238 in a sample can be deter-mined by measuring the activity of Th-234 or Pa-234m. The latter emits high

energy β- radiation and can therefore be measured easily. The Mass of U-238 is obtained by application of Eq.(14) with A1=A2:

(EQ 28)

where m1 is the mass and M1 the nuclide mass of the long-lived mother nuclide, and NAv Avogadro’s number.

• Finally, the previous application can be reversed inasmuch as a sample of U or U3O8 can be weighed to provide a source of known activity of Pa-234m. The

radiation of U-238 is filtered from the high-energy β- radiation of Pa-234m by covering the sample with thin aluminum foil. From Eq.(14) it follows with

A1=A2 that 1 mg of U-238 is a radiation source emitting 740 β- particles from

Pa-234m per minute. Such sample may be used as a β- standard.

Transient radioactive equilibrium

The attainments of a transient radioactive equilibrium is shown in Figure 4 for t1/2(1)/t1/2(2)=5. Now t1/2(2) alone does not regulate the attainment of the radioac-

m1

M1

MAv

---------A2

2ln-------- t1 2⁄ 1( )=


Radioactive Decay

12

tive equilibrium; its influence is modified by a factor containing the ratio t1/2(1)/t1/2(2) as already explained before.

Figure 4: Transient equilibrium: activities of mother and daughter nuclide as a function of t/ t1/2(2) (t1/2(1)/t1/2(2)=5).

After attainment of radioactive equilibrium, Eq.(22) is valid. Introducing the half-lives, this equation becomes:

(EQ 29)

Whereas is secular radioactive equilibrium the activities of the mother and the daughter nuclide are the same, in transient radioactive equilibrium the daughter activity is always higher:

)2(/ 2/1tt

Act

ivit

y, A

10

1

100

A2 growing in (1)

A2 separated from(1)

Total activity A = A1 + A2

A1

0 105

N2

N1

------t1 2⁄ 2( )

t1 2⁄ 1 ) t1 2⁄ 2( )–--------------------------------------=


Radioactive Decay

(EQ 30)

The possibilities of application of transient radioactive equilibrium are similar to those explained for secular radioactive equilibrium. Instead of Eq.(28), the follow-ing equation holds:

(EQ 31)

Decay Modes

Unstable, radioactive nuclei may be transformed by emission of nucleons (α decay and emission of protons and neutrons) or by emission of electrons or positrons

(β- and β+,respectively). To the emission of a positron, the unstable nucleus may capture an electron of the electron shell of the atom (symbol ε).

In most cases the emission of nucleons, electrons and positrons leads to an excited state of the new nucleus, which gives off its excitation energy in the form of one or several photons (γ -rays). This de-excitation occurs most frequently within about

10-13 s after the preceding α and β decay, but in some cases the transition to the ground state is ‘forbidden’ resulting in a metastable isomeric state that decays inde-pendently of the way it was formed.

Alpha nuclei is observed for heavy nuclei with atomic numbers Z > 83 and for some groups of nuclei far away from the line of β stability. Radionuclides with very long half-lives are mainly α emitters. Proton emission has been found for nuclei with high excess of protons far away from the line of β stability and more fre-

quently as a two-stage process after β+decay (β delayed proton emission).

With increasing atomic numbers spontaneous fission begins to compete with α decay and prevails for some radionuclides with Z > 96. However, due to high fis-sion barrier, α decay is still the dominating mode of decay for many heavy nuclides with Z > 105.

Details of decay of radionuclides are recorded in the form of decay schemes, in which the energy levels are plotted and the half-lives, the nuclear spins, the parity and the transitions are indicated. Nuclei with higher atomic numbers are pout to the

A1

A2

-----λ1N1

λ2N2

------------ 1λ1

λ2

-----– 1t1 2⁄ 2( )t1 2⁄ 1( )-----------------–= = =

m1

M1

MAv

---------A2

2ln-------- t1 2⁄ 1( ) t1 2⁄ 2( )–[ ]=


Radioactive Decay

14

right, and energies are given in MeV. For example, see the decay scheme of U-238 in Figure 5.

Figure 5: Decay scheme of U-238 (energies of excited states, α decay and γ transition in MeV, 0 for ground states; nuclear spin and parity are indicated).

Alpha Decay

Almost all naturally occurring alpha emitters are heavy elements with Z > 83. For

example, nuclei are emitted by α decay, and so the atomic number decreases

by two units and the mass number by four units (first displacement law of Soddy and Fajans):

)1047.4( 9238 yUO ×

α (4.04)0.23%

α (4.15)23%

α (4.20)77%

0.16

0.0496

4+

2+

0+ 0

γ (0.1105)

γ (0.0496)

)1.24(234 dTh

0+

4

2He


Radioactive Decay

(EQ 32)

The energy ∆E of α decay can be calculated by means of the Einstein formula

:

(EQ 33)

where m1, m2, and mα are the masses of the mother nucleus, the daughter nucleus and the particle, respectively. Introducing the masses of the nuclides (nucleus plus electrons), M = m + Zme, gives:

(EQ 34)

where M1, M2, and Mα are the nuclide masses of the mother nuclide, the daughter

nuclide and the α particle, respectively. By application of Eq(34) it is found that all nuclides with mass numbers A > 140 are unstable with respect to α decay. The rea-son that the binding energy of an α particle in the nucleus is relatively small is the high binding energy of the four nucleons in the α particle. However, as long as ∆E is small, α decay is not observed due to the energy barrier which has to be sur-mounted by the α particle. Therefore, nuclides with A > 140 are energetically unstable, but kinetically more or less stable with respect to α decay.

All α particles originating from a certain decay process are monoenergetic, i.e. they have the same energy. The energy of decay process is split into two parts, the kinetic energy of the α particle, Eα, and the kinetic energy of the recoiling nucleus, EN:

(EQ 35)

From the law of conservation of momentum it follows that:

(EQ 36)

where mα and mN are the masses and vα and vN are the velocities of the α particle

and the nucleus, respectively, so Eq(35) becomes:

ZA

Z 2–( )A 4– 4

2He

2 ++→

U α( )234Th238

∆E ∆mc2=

∆E m1 m2– mα–( )c2=

∆E MA MB– Mα–( )c2=

∆E Eα EN+=

mαva mNvN=


Radioactive Decay

16

(EQ 37)

Because the mass of heavy nuclei is appreciably higher than that of an α particle (mΝ >> mα), Eα is only 2% smaller than ∆E.

Beta Decay

Nuclides with excess of neutrons experience β− decay. In the nucleus a neutron is converted into a proton, an electron and an electron antineutrino:

(EQ 38)

The atomic number increases by one unit, whereas the mass number does not change (second displacement law of Soddy and Fajans). The energy of the decay process can again be calculated by comparison of the masses according to Einstein:

(EQ 39)

where m1, m2, and me are the masses of the mother nucleus, the daughter nucleus and the electron, respectively. The mass of the antineutrino is neglected, because it

is extremely small (< 2. 10-7 u). Inserting the masses of the nuclides (nucleus plus electrons), M = m + Zme, gives:

(EQ 40)

(EQ 41)

Nuclides with an excess of protons exhibit β+ decay. A proton in the nucleus is con-verted into a neutron, a positron and an electron neutrino:

∆E Eα 1mα

mN

-------+ =

1

0n

1

1p

0

1–e– 0

0ve+ +→

ZA

Z 1–( )→

C β–( )14 14

N

∆E m1 m2– me–( )c2=

∆E M1 Z1me M2– Z1 1+( )me me–+–( )c2=

∆E M1 M2–( )c2=


Radioactive Decay

(EQ 42)

The atomic number decreases by one unit, and the mass number remains

unchanged. As in the case of β− decay, the energy of the decay process is obtained by Eq.(39). But because Z2 = Z1, it follows that:

(EQ 43)

(EQ 44)

This means that β+ decay can occur only if M1 is at least two electrons masses higher than M2:

(EQ 45)

In contrast to α particles, β particles do not have a distinct energy, but they show a continuous energy distribution (Fig.6). The energy of the emitted electrons varies between zero and the maximum energy Emax, whereas the mean energy of the elec-

trons is only about on-third of the Emax. In addition to the electron another particle, the neutrino (correctly speaking and electron antineutrino), is emitted (neutrino hypothesis), which carries away the missing energy:

(EQ 46)

where Ee is the energy of the electron and Ev the energy of the electron neutrino or antineutrino, respectively. The neutrino has no charge, an extremely small mass (mv < 1/1000 me), the spin 1/2h/2π, and it obeys the Fermi-Dirac statistics. These

properties were postulated to fulfil the conservation laws.

Taking into account the formation of electron neutrinos in addition to emission of

electrons and positrons, respectively, the following equations are valid for β+ and

β− decay:

1

1p

1

0n

0

1e

0

0ve+ +→

ZA

Z 1–( )→

C β( )11 11B

∆E M1 Z1me M2– Z1 1–( )me me–+–( )c2=

∆E M1 M2– 2me–( )c2=

M1 M2 2me+>

Emax Ee Ev+=


Radioactive Decay

18

(EQ 47)

(EQ 48)

Electron capture is described by the following equation:

(EQ 49)

The energy ∆E, given by Eq.(39), is split up:

(EQ 50)

where Ee is the energy of the electron or positron, Ev the energy of the electron neu-

trino or antineutrino, respectively, and EN the recoil energy of the nucleus. As the mass of the electron is very small compared with the mass of a nucleus, it follows that:

(EQ 51)

Figure 6: Shape of typical beta-particle energy spectrum.

β– …1

0n nucleus( )

1

1p nucleus( )

0

1–e– 0

0ve+ +→

β + …1

1p nucleus( )

1

0n nucleus( )

0

1–e

0

0ve+ +→

ε…1

0p nucleus( )

1

1–e

–shell( )

1

0n nucleus( )→

0

0ve+ +

∆E Ee Ev EN+ +=

∆E Emax≈ Ee Ev+=

Rel

ativ

e #

of β

part

icle

s

Energy [keV]

Emax


Radioactive Decay

For nuclei with mN > 5 u, the difference between ∆E and Emax is < 0.01%. In the

case of electron capture, Ee in Eq.(50) is given by the binding energy of the electron in the electron shell which is very small compared to ∆E, and the neutrino receives the whole energy of the decay process:

(EQ 52)

The decay scheme of Cu-64 is plotted in Figure 7: 39.6% of Cu-64 show β− decay

to the ground state of Zn-64, 19.3% β+ decay to the ground state of Ni-64, 50.5% electron capture (ε) and transition to the ground state of Ni-64 and 0.6% electron capture and transition to the excited state of Ni-64.

Figure 7: Decay scheme of Cu-64.

Gamma Transition

If a nucleus changes from an excited state to the ground state or another excited state of lower energy, γ-ray are emitted. As an example, the decay scheme of Au-198 is plotted in Figure 8. With 98.7% probability, Au-198 changes into the first excited state of the daughter nuclide Hg-198 (0.412 MeV above ground level), with

∆E Ev≈

)8.12(64 hCu

%6.0)33.0( =∆Eε2+

0+

1.346

0+

)578.0(−β39.6%

Zn64

)655.0(+β19.3%

)677.1( =∆Eε40.5%

Ni64

0

)346.1(γ


Radioactive Decay

20

1.3% probability into the second excited state (1,087 MeV above ground level), and 0.025% to the ground level of Hg-198. Accordingly, the γ transitions are observed: the second excited state changes with 20% probability directly to the ground state and with 80% probability to the first excited state at 0.412 MeV, resulting in the following intensities relative to the total β activity: g(1.087), 1.3x0.2=0.26%; γ(0.676), 1.3x0.8=1.04%; γ(0.412), 98.7+1.3x0.8=99.74%.

Figure 8: Decay scheme of Au-138.

One or more gamma photons can be emitted from the excited states of daughter nuclei following radioactive decay. Transitions that result in gamma emission leave Z and A unchanged and are called isomeric; nuclides in the initial and final states are called isomers.

All γ−rays emitted by a certain nucleus are monoenergetic, i.e. they have well-defined energies. Because the recoil energies transmitted to the nuclei by emission of the γ−ray photons are very small compared with the energy of the γ−rays, the lat-ter are practically equal to the excitation energies or differences in the excitation energies of the nuclei:

(EQ 53)

)696.2(198 dAu

0+

)29.0(−β1.3%

2+

2+)962.0(−β

)371.1(−β

98.7%

0.025%

2-0

Hg198

1.087

0.412

0

)087.1(γ)676.0(γ

)412.0(γ

∆Eγ ∆E=


Radioactive Decay

Gamma spectrometry is therefore the most important tool for studying the proper-ties of atomic nuclei.

Generally, the lifetime of the excited states is very small, of the order of 10-16 to 10-13s; the γ−radiation is emitted immediately after a preceding α and β decay. However, if immediate γ transitions are ‘forbidden’, because of high differences of the nuclear spins of the excited state and the ground state in combination with the laws of conservation of nuclear momentum and of parity, a metastable state or nuclear isomer results which decays with its own half-life. The transition from the metastable isomeric state into the ground state is called isomeric transition (IT). Isomeric transition is free from accompanying α or β radiation, and some nuclear isomers are of great practical importance as pure γ emitters.

Instead of emitting a γ-ray photon, the excited nucleus may transmit its excitation energy to an electron of the atomic shell, preferable a K electron, a process called internal conversion (IC). The probability of this alternative increases with increas-ing atomic number and with decreasing excitation energy. The conversion electron

(symbol e-) is emitted instead of a γ-ray photon and its energy is:

(EQ 54)

where Eγ is the energy of the γ-ray photon and EB the energy of the electron. In con-trast to β particles, conversion electrons are monoenergetic. Internal conversion is followed by emission of characteristics X rays, as in the case of electron capture (previous section).

Proton Decay

With an increasing excess of protons, on the left-hand side of the line of β stability, the binding energy of the last proton decreases markedly, and a region is expected in which this binding energy approaches to zero and proton emission from the ground state becomes energetically possible. However, as in the case of α decay, the protons leaving the nucleus have to pass an energy barrier by tunneling; this gives these nuclei that are unstable with respect to proton decay a certain lifetime.

Proton activity was observed for the first time for Tm-147 (t1/2 = 0.56 s) and Lu-151 (t1/2 = 85 ms). Both nulcides emit monoenergetic protons 0f 1.06 and 1.23

MeV, respectively, by transmutation of the ground state of the mother nuclide into the ground state of the daughter nuclide:

Ee Eγ EB–=


Radioactive Decay

22

(EQ 55)

(EQ 56)

Thus, besides α decay, β decay and γ transition, a fourth type of decay is known.

More frequently, p emission occurs after β+decay in a two-stage process: β+decay leads to an excited state of the daughter nuclide, and from this excited state the pro-

ton can easily surmount the energy barrier. This two-stage process is called β+

delayed proton emission. It is observed for several β+ emitters from C-9 to Ti-41 with N = Z-3, with half-lives in the range of 1 ms to 0.5 s. Simultaneous emission of two protons has been observed for a few proton-rich nuclides, e.g., Ne-16 (t1/2 =

10-20 s).

Spontaneous Fission

Spontaneous fission (sf) is another mode of radioactive decay, which is observed only for high numbers of A. For U-238 the ratio of probability of spontaneous fis-

sion to that of α decay is 1:106. It increases with the atomic number Z and the num-ber of neutrons in the nucleus. For Fm-256 the probability of spontaneous fission relative to the total probability of decay is already 92%.

Spontaneous fission can be described as:

(EQ 57)

(1) (2)

where v is the number of neutrons and ∆E the energy set free by the fission process. The resulting nuclei (1) and (2) have, in general, have different mass numbers A and atomic numbers Z. Because of the high neutron excess of heavy fissioning nuclei, the fission products (1) and (2) are found in the chart of nuclides on the neu-tron-rich side of the line of β stability, as illustrated in Fig.(9).

(a) the nucleus oscillates between a more spherical and more ellipsoidal shape. By further distortion and constriction near the center of the ellipsoid the nucleus attains the shape of a dumbbell, in which at least one part has magic number of protons and electrons

Tm147

Er p1

+146

→

Lu151

Yb p1

+150

→

ZA

Z'A'

Z Z'–( )A A'– ν– νn ∆E+ + +→


Radioactive Decay

Figure 9: The steps of spontaneous fission

(b) the nucleus splits into two parts. If this split takes place at A, two parts of nearly equal mass, but different excitation energy, are formed (symmetric fusion). Most probable is the fission at B, by which two parts of different mass, but similar excita-tion energy, are formed (symmetric fission). Fission at C leads to two parts of every different mass and different excitation energy. The Coulomb repulsion energy, which has a much greater range than the nuclear forces, drives both products apart, and the fission products attain high kinetic energies.

(c) the highly excited fission products emit neutrons (prompt neutrons) and photons (propmpt photons), and sometimes also charged particles. Up to this stage, the pro-

cesses take place within about 10-15 s.

AZ

A1Z1

A2Z2

A B C

A3Z3

A4Z4

A5Z5

A6Z6

A7Z7

A8Z8

γ

γ

γ

γ

n

n

β−

β−

(a)

(b)

(c)

(d)

(e)


Radioactive Decay

24

(d) the fission products change by one or several β− transformations and emission of γ-ray photons into stable products. In the case of high excitation energies, emis-sion of further neutrons (delay neutrons) may be observed.


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