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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 and decay are observed. With emission of an 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 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

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  • Radioactive Decay


    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 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 particle into the same nuclide (Table 1), thus closing the branching. In both branches the sequence of decay alternates: either decay is followed by - decay or - decay is followed by decay.

    TABLE 1. Thorium decay series: A=4n

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

    Th-232 1.405x1010 y 4.01

    Ra-228 (MsTh1) 5.75 y - 0.04

    Ac-228 (MsTh2) 6.13 h - 2.11

    Th-228 (RdTh) 1.91 y 5.42

    Ra-224 (ThX) 3.66 d 5.69

    Rn-220 (Tn) 55.6 s 6.29

    Po-216 (ThA) 0.15 s 6.78

    Pb-212 (ThB) 10.64 h - 0.57

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

    Po-212 (ThC) 3.0x10-7 s 8.79

    Tl-208 (ThC) 3.05 min - 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 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------- N=

    A B x E+ +

    E Mc2 MA Mb Mx+( )[ ]c2= =

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  • Radioactive Decay


    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 , 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 , we write from Eq(4) at time t = t1/2:


    B + x





    excited state ofnuclide A

    N Noet=

    Engineering Aspects of Food Irradiation

  • Radioactive Decay

    (EQ 5)

    Taking the natural log of both sides gives:

    (EQ 6)

    and therefore:

    (EQ 7)


    (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

  • Radioactive Decay


    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 (Ci) = 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------- N 2ln

    t1 2--------N= = =

    A Aoet=

    mNMNAv--------- AM

    NAv------------ AM

    NAv 2ln-----------------t1 2= = =

    AsAm---- s[ ]=

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  • 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 by the half-lives t1/2:

    nuclide 1 nuclide 2 nuclide 3



    --------- 2N2 1N1 2N2= =


    --------- 2N2 1N0

    1e1t+ + 0=


    2 1-----------------N01 e

    1t e2t( ) N02e

    2 t+=


    2 1-----------------N01 e

    1t e2 t( )=


    2 1-----------------N1 1 e

    1 2( ) t( )=

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  • Radioactive Decay


    (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 daughte


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