Radiation Physics for Medical Physiscists - E.B. Podgorsak

340 8 Radioactivity

8.13.2 Internal Conversion Factor

In any nuclear de-excitation both the γ ray emission and the internal conversion electron emission are possible. The two nuclear processes are competing with one another and are governed essentially by the same selection rules. Thus, similar to the situation in atomic competing processes represented by the emission of characteristic (fluorescent) photons and emission of Auger electrons that are governed by the fluorescent yield, the internal conversion factor governs the two nuclear processes: emission of gamma photons and ejection of conversion electrons. However, in contrast to the fluorescent yield ω (see Sect. 3.1.2) that is defined as the number of characteristic photons emitted per vacancy in a given atomic shell, the total internal conversion factor αIC is defined as

In addition to the total internal conversion factor αIC one can define partial internal conversion factors according to the shell from which the electron was ejected, i.e.,

where αIC(i) represents the partial internal conversion factors. Further distinction is possible when one accounts for subshell electrons.

The total internal conversion factors αIC are defined with respect to Nγ so that αIC can assume values greater or smaller than 1, in contrast to fluorescent yield ω that is always between 0 and 1.

Since the K-shell electrons of all atomic electrons are the closest to the nucleus, most often the conversion electrons originate from the K atomic shell. The vacancy in the K shell, of course, is filled by a higher shell electron and the associated emission of characteristic photon or Auger electron, as discussed in Sect. 3.1.

An example for both the emission of γ photons and emission of conversion electrons is given in Fig. 8.22 with the β− decay scheme for cesium-137 decaying into barium-137. Two channels are available for β− decay of cesium-137:

1.94.6% of disintegrations land in a barium-137 isomeric state (barium137m) that has a half-life of 2.552 min and de-excitation energy of 662 keV.

2.5.4% of disintegrations land directly in the barium-137 ground state.

The de-excitation energy of 0.662 MeV is emitted either in the form of a 662 keV gamma photon or a conversion electron of kinetic energy 662 keV.

As shown in Fig. 8.22, for 100 disintegrations of cesium-137, 94.6 transitions land in barium-137m; of these 85 result in γ photons; 7.8 in K conversion electrons and 1.8 in higher shell conversion electrons. The internal conversion factor αIC is (7.8 + 1.8)/85 = 0.113.

8.14 Spontaneous Fission

In addition to disintegrating through α and β decay processes, nuclei with very large atomic mass numbers A may also disintegrate by splitting into two nearly equal fission fragments an concurrently emit 2 to 4 neutrons. This decay process is called spontaneous fission (SF) and is accompanied by liberation of a significant amount of energy. It was discovered in 1940 by Russian physicists Georgij N. Flerov and Konstantin A. Petrˇzak who noticed that uranium-238, in addition to α decay, may undergo the process of spontaneous fission.

Spontaneous fission follows the same process as nuclear fission, except that it is not self-sustaining, since it does not generate the neutron fluence rate required to sustain a “chain reaction”. In practice, SF is only energetically feasible for nuclides with atomic masses above 230 u or with Z2/A ≥ 235 where Z is the atomic number and A the atomic mass number of the radionuclide. SF can thus occur in thorium, protactinium, uranium and transuranic elements.

Transuranic (or transuranium) elements are elements with atomic numbers Z greater than that of uranium (Z = 92). All transuranic elements have more protons than uranium and are radioactive, decaying through β decay, α decay, or spontaneous fission. Generally, the transuranic elements are man-made and synthesized in nuclear reactions in a process referred to as nucleosynthesis. The nucleosynthesis reactions are generally produced in particle accelerators or nuclear reactors; however, neptunium (Z = 93) and plutonium (Z = 94) are also produced naturally in minute quantities, following the spontaneous fission decay of uranium-238. The spontaneous fission neutrons emitted by U-238 can be captured by other U-238 nuclei thereby producing U-239 which is unstable and decays through β− decay with a halflife of 23.5 m into neptunium-239 which in turn decays through β− decay with a half-life 2.35 d into plutonium-239, as shown in (8.201):

SF is a competing process to α decay; the higher is A above uranium-238, the more prominent is the spontaneous fission in comparison with the α decay

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and the shorter is the half-life for spontaneous fission. For the heaviest nuclei, SF becomes the predominant mode of radioactive decay suggesting that SF is a limiting factor in how high in atomic number Z and atomic mass number A one can go in producing new elements.

• In uranium-238 the half-life for SF is 1016 y, while the half-life for α decay is 4.5 × 109 y. The probability for SF in uranium-238 is thus about 2 × 106 times lower than the probability for α decay.

•Fermium-256 has a half-life for SF of about 3 hours making the SF in fermium-256 about 10 times more probable than α decay.

•Another interesting example is californium-256 which decays essentially 100% of the time with SF and has a half-life of 12.3 m.

•For practical purposes, the most important radionuclide undergoing the SF decay is the transuranic californium-252 (Cf-252), used in industry and medicine as a very e cient source of fast neutrons (see Sect. 6.6.4). Californium-252 decays through α decay into curium-248 with a half-life of 2.65 y; however, about 3% of Cf-252 decays occur through SF producing

on the average 3.8 neutrons per fission decay. The neutron production rate of Cf-252 is thus equal to 2.35 × 106 (µg · s)−1.

8.15 Proton Emission Decay

Proton-rich nuclides normally approach stability through β+ decay or α decay. However, in the extreme case of a very large proton excess a nucleus may also move toward stability through emission of one or even two protons. Proton emission is thus a competing process to β+ and α decay and is, similarly to α decay, an example of particle tunneling through the nuclear barrier potential.

Proton emission decay is much less common than are the β+ and α decay and is not observed in naturally occurring radionuclides. In this type of decay the atomic number Z decreases by 1 and so does the atomic mass number A:

• When a proton is ejected from a radionuclide P, the parent nucleus P sheds an orbital electron from its outermost shell to become a neutral

daughter atom A−1D.

Z−1

•The energetic proton slows down in moving through the absorber medium and captures an electron from its surroundings to become a neutral hydrogen atom 11H.

•Since N , the number of neutrons does not change in proton emission decay, the parent P and daughter D are isotones.

•For lighter, very proton-rich nuclides with an odd number of protons Z, proton emission decay is likely (see example in Sect. 8.16.2).

•For lighter, very proton-rich nuclides (A ≈ 50) with an even number of protons Z, a simultaneous two-proton emission may occur in situations where a sequential emission of two independent protons is energetically not possible (see example in Sect. 8.16.3).

8.15.1 Decay Energy in Proton Emission Decay

The decay energy Qp released in proton emission decay appears as kinetic energy shared between the emitted proton and the daughter nucleus and is expressed as follows:

Qp = {M(P) − [M(D) + M(H)]}c2 = {M (P) − [M (D) + mp]}c2 , (8.203)

where M(P), M(D), and M(11H) are the atomic rest masses of the parent, daughter and hydrogen atom, respectively, and M (P), M (D) and mp are nuclear rest masses of the parent, daughter and hydrogen nucleus (proton), respectively.

The total number of protons as well as the total number of neutrons does not change in the proton emission decay. Therefore, Qp may also be expressed in terms of binding energies of the parent and daughter nucleus as follows:

where

EB(D) is the total binding energy of the daughter D nucleus EB(P) is the total binding energy of the parent P nucleus.

The nuclear binding energy is defined in Eq. (1.13). For proton emission decay to be feasible, Qp must be positive and this implies that the total binding energy of the daughter nucleus EB(D) must exceed the total binding energy of the parent nucleus EB(P); that is, EB(D) > EB(P), or else that the rest mass of the parent nucleus must exceed the combined rest masses of the daughter nucleus and the proton, that is, M (P) > M (D) + mp.

Two products are released in proton emission decay: a proton and the daughter product. For a decay of the parent nucleus at rest this implies that the proton and the daughter will acquire momenta p equal in magnitude but opposite in direction. The kinetic energy of the proton is (EK)P = p2/2mp and of the daughter nucleus it is (EK)D = p2/2M (D).

The total decay energy Qp must be positive for the proton emission decay and can be written as the sum of the kinetic energies of the two decay

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From (8.205) we determine the emitted proton kinetic energy (EK)p as

The kinetic energy of the recoil daughter (EK)D, on the other hand, is given as follows

The decay energy Q2p released in two-proton emission decay appears as kinetic energy shared among the three emitted particles (two protons and the daughter nucleus) and may be calculated simply from the di erence in binding energies EB between the daughter D and the parent P nucleus

where M stands for the atomic rest masses, M for nuclear rest masses and mp for the proton rest mass.

8.15.2 Example of Proton Emission Decay

An example of proton emission decay is the decay of lithium-5 into helium-4 with a half-life of 10−21 s. The decay is schematically written as follows

and the decay energy may be calculated from (8.203) or (8.204). The required atomic and nuclear data are given as follows

M(53Li)c2 = 5.012541u × 931.5 MeV/u = 4669.18 MeV

M(42He)c2 = 4.002603u × 931.5 MeV/u = 3728.43 MeV

M(11H)c2 = 1.007825u × 931.5 MeV/u = 938.79 MeV

EB(53Li) = 26.330674 MeV

EB(42He) = 28.295673 MeV

We first notice that M(53Li) > M(42He)+M(21H) and that EB(42He) > EB(53Li). This leads to the conclusion that the proton emission decay is possible. Next we use (8.203) and (8.204) to calculate the decay energy Qp and get 1.96 MeV from both equations. Equations (8.206) and (8.207) give 1.57 MeV and 0.39 MeV for the kinetic energies of the ejected proton and the recoil helium-4 atom, respectively.

8.15.3 Example of Two-Proton Emission Decay

An example of two-proton emission decay is the decay of iron-45 (a highly proton rich radionuclide with Z = 26 and N = 19) which decays with a simultaneous emission of two protons at a half-life of 0.35 µs into chromium43 (a proton-rich radionuclide with Z = 24 and N = 19). The decay is schematically written as follows:

and the decay energy Q2p may be calculated from (8.208) or (8.209).

At first glance one could expect the iron-45 radionuclide to decay by a single proton emission into manganese-44; however, a closer inspection shows that the one-proton decay would produce negative decay energy Qp from (8.203) and (8.204) and thus is not energetically feasible.

The atomic and nuclear data for radionuclides 4526Fe, 4425Mn, and 4324Cr are given as follows:

M(4526Fe)c2 = 45.014564u × 931.5 MeV/u = 41931.07 MeV

M(4425Mn)c2 = 44.006870u × 931.5 MeV/u = 40992.40 MeV

M(4324Cr)c2 = 42.997711u × 931.5 MeV/u = 40052.37 MeV

M(11H)c2 = 1.007825u × 931.5 MeV/u = 938.79 MeV

EB(4526Fe) = 329.306 MeV

EB(4425Mn) = 329.180 MeV

EB(4324Cr) = 330.426 MeV .

Inspection of (8.204) shows that one-proton emission decay of 4526Fe into 4425Mn is not possible, since it results in a negative Qp. On the other hand, (8.208) results in positive decay energy Q2p for a two-proton decay of 4526Fe into its isotone 4324Cr. The decay energy Q2p calculated from (8.208) and (8.209) then amounts to 1.12 MeV for the two-proton decay of 4526Fe into 4324Cr.

8.16 Neutron Emission Decay

Neutron emission from a neutron-rich nucleus is a competing process to β− decay but is much less common then the β− decay and is not observed in naturally occurring radionuclides. In contrast to spontaneous fission which also produces neutrons, in neutron emission decay the atomic number Z remains the same but the atomic mass number A decreases by 1. Both the parent nucleus P and the daughter nucleus D are thus isotopes of the same nuclear species. The neutron emission decay relationship is written as follows:

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8.16.1 Decay Energy in Neutron Emission Decay

The decay energy Qn released in neutron emission decay appears as kinetic energy shared between the emitted neutron and the daughter nucleus and is expressed as follows:

Qn = {M(P) − [M(D) + mn]}c2 = {M (P) − [M (D) + mn]}c2 , (8.213)

where M(P) and M(D) are atomic masses of the parent and daughter atom, respectively; M (P) and M (D) are the nuclear masses of the parent and daughter respectively, and mn is the neutron rest mass.

The total number of protons Z as well as the total number of neutrons N does not change in the neutron emission decay. Therefore, Qn may also be expressed in terms of binding energies of the parent and daughter nucleus as follows

where

EB(D) is the total binding energy of the daughter D nucleus EB(P) is the total binding energy of the parent P nucleus.

For the neutron emission decay to be feasible, Qn must be positive and this implies that the total binding energy of the daughter nucleus EB(D) must exceed the total binding energy of the parent nucleus EB(P); that is, EB(D) > EB(P), or else that the rest mass of the parent nucleus M (P) must exceed the combined rest masses of the daughter nucleus and the neutron; that is, M (P) > M (D) + mn.

Two products are released in neutron emission decay: a neutron and the daughter product. For a decay of the parent nucleus at rest this implies that the neutron and the daughter will acquire momenta p equal in magnitude but opposite in direction. The kinetic energy of the neutron is (EK)n = p2/2mn and of the daughter nucleus the kinetic energy is (EK)D = p2/2M (D).

The total decay energy Qn must be positive for the neutron emission decay and is expressed as follows:

From (8.212) we determine the emitted neutron kinetic energy (EK)n as

The kinetic energy of the recoil daughter (EK)D, on the other hand, is given as follows:

8.16.2 Example of Neutron Emission Decay

An example of neutron emission decay is the decay of helium-5 into helium-4 with a half-life of 8 × 10−22 s. The decay is schematically written as follows:

and the decay energy may be calculated from (8.210) or (8.211). The required atomic and nuclear data are as follows:

M(52He)c2 = 5.012221u × 931.5 MeV/u = 4668.88 MeV

M(42He)c2 = 4.002603u × 931.5 MeV/u = 3728.43 MeV

mnc2 = 1.008665u × 931.5 MeV/u

EB(52He) = 27.405673 MeV

EB(42He) = 28.295673 MeV .

We first notice that M(P) > M(D) + mn and EB(42He) > EB(52He) and conclude that neutron emission decay is possible. Next we use (8.210) and (8.211) to calculate the decay energy Qn and get 0.89 MeV from both equations. Equations (8.212) and (8.213) give 0.71 MeV and 0.18 MeV for the kinetic energies of the ejected neutron and recoil helium-4 atom, respectively.

8.17 Chart of the Nuclides

All known nuclides are uniquely characterized by their number of protons Z (atomic number) and their number of neutrons N = A − Z where A is the number of nucleons (atomic mass number). The most pertinent information on the 275 known stable nuclides and over 3000 known radioactive nuclides (radionuclides) is commonly summarized in the Chart of the Nuclides in such a way that it is relatively easy to follow the atomic transitions resulting from the various radioactive decay modes used by radionuclides to attain more stable configurations. Usually the ordinate of the chart represents Z and the abscissa represents N with each nuclide represented by a unique square (pixel) that is placed onto the chart according to the N and Z value of the nuclide.

The chart of the nuclides is also referred to as the Segr` chart in honor of Emilio Segr` who was first to suggest the arrangement in the 1930s. Similarly

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Fig. 8.27. Chart of the Nuclides also known as the Segr` Chart. Each known stable and radioactive nuclide is characterized by its unique combination of the number of protons Z and number of neutrons N, and assigned a pixel in a chart displaying Z on the ordinate axis and N on the abscissa axis. The stable nuclides are shown by dark pixel squares, radioactive nuclides by light pixel squares. The plot of stable nuclides forms a “curve of stability”, neutron-rich radionuclides are below the curve of stability and proton-rich radionuclides are above the curve of stability. The magic numbers for neutrons and protons are also indicated

to the Periodic Table of Elements introduced by Mendeleyev in the 1870s to represent conveniently the periodicity in chemical behavior of elements with increasing atomic number Z, Segr`e’s chart of the nuclides presents an orderly formulation of all nuclear species (stable and radioactive) against both Z and N and, in addition, indicates the possible decay paths for radionuclides.

In addition to Z and N for a given nuclide the Segr` Chart usually provides other data, such as:

•For stable nuclides the atomic mass number A; the nuclear mass in u; and the natural abundance.

•For radionuclides the atomic mass number A, nuclear mass in u, radioactive half-life, and mode of decay.

A schematic representation of the Segr` Chart is given in Fig. 8.27 for the currently known stable and radioactive nuclides ranging in number of protons Z from 1 to 118 and in number of neutrons N from 0 to 292. The magic numbers (see Sect. 1.15.2) for protons and neutrons are shown on the chart; the stable nuclides are shown with black squares, the radionuclides with light squares. For each element the rows in the Segr` Chart give a list of isotopes (Z = const), the vertical columns give a list of isotones (N = const).