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textbook:nrctextbook:chapter15 [2025-01-23 23:30] Merja Herzig |
textbook:nrctextbook:chapter15 [2025-05-05 11:10] (current) Merja Herzig |
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| - | A nuclear reaction is an event in which a nucleus is targeted by a projectile (proton, deuteron, alpha particle, etc.) or gamma ray causing one of the following interactions with the nucleus: | + | A nuclear reaction is an event in which a nucleus is targeted by a projectile ([[textbook: |
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| * Transmutation or the formation of a new nucleus | * Transmutation or the formation of a new nucleus | ||
| - | * Inelastic | + | * [[textbook: |
| - | * Elastic scattering | + | * [[textbook: |
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| - | Since the specific aim of this communication is production of new nuclei, primarily radionuclides, | + | Since the specific aim of this communication is production of new nuclei, primarily radionuclides, |
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| In nature, nuclear reactions occur in the following sources: | In nature, nuclear reactions occur in the following sources: | ||
| - | * The cosmic protons and alpha particles of the upper parts of the atmosphere (as well as the neutrons generated by these nuclear reactions) cause nuclear reactions upon hitting with the gas molecules of the atmosphere. The most important of these is the emergence of the < | + | * The [[textbook: |
| * Fusion reactions occuring in stars. | * Fusion reactions occuring in stars. | ||
| - | * A very small number of neutron activation reactions caused by neutrons generated in spontaneous fission of uranium in bedrock and overburden. | + | * A very small number of neutron activation reactions caused by neutrons generated in [[textbook: |
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| - | The first artificial nuclear reaction was achieved by [[textbook: | + | The first artificial nuclear reaction was achieved by [[textbook: |
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| - | The products of the reaction were oxygen, its < | + | The products of the reaction were oxygen, its < |
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| - | where the left side of the parentheses is the target nucleus and on the right is the result nucleus. Inside the parentheses on the left is a projectile particle/ | + | where the left side of the parentheses is the //target nucleus// and on the right is the //result nucleus//. Inside the parentheses on the left is a //projectile particle/ |
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| - | The first particle accelerator, | + | The first [[textbook: |
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| ===== 15.1. Nuclear reaction types and models ===== | ===== 15.1. Nuclear reaction types and models ===== | ||
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| - | In most nuclear reactions the target nucleus absorbs the projectile particles (n, p, d, α) and emits other particles in a fairly short period of time (10-14-10-18 s). A liquid drop model was developed to describe these types of reactions. The kinetic energy of the projectile particle and its merging with the nucleus generates bonding energy that spreads evenly as the nuclear excitation energy. Much like a liquid droplet, the particles at some point in the nucleus have such an ample energy that it | + | In most nuclear reactions the [[textbook: |
| - | " | + | |
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| - | have roughly equal probability as a function of the projectile energy (i.e. excitation function, which is explained later). In this case, the explanation is that an excited intermediate nucleus or compound nucleus ${}^{156}_{\phantom{1}66}\text{Dy}$ is generated in both cases, which breaks down in the same way, regardless of its | + | have roughly equal probability as a function of the [[textbook: |
| origin. | origin. | ||
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| - | The liquid drop model is not able to explain all of the observed reactions, in particular those with high projectile energy. These reactions are, for example, spallation reactions, in which a large number, even many tens of nucleons, are emitted from the nucleus and the resulting nucleus is of a clearly lighter element. Such reactions are also the fragmentation reactions, in which instead of one several lighter nuclei develop. These reactions are depicted by the direct interaction model, | + | The liquid drop model is not able to explain all of the observed reactions, in particular those with high [[textbook: |
| - | according to which the intermediate nucleus does not have time to form, in other words the excitation energy has no time to spread throughout the nucleus and instead breaks down immediately as a direct effect of the projectile particles. | + | |
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| [[textbook: | [[textbook: | ||
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| - | Scattering is characteristic of neutron interactions with target nuclei. In scattering the projectile particle and emitting particle are identical. If there is no change in the energy of the nucleus during scattering, it is called elastic. In inelastic scattering the nucleus becomes excited and the kinetic energy of the emitting particle reduces. | + | Scattering is characteristic of neutron interactions with target nuclei. In scattering the projectile particle and emitting particle are identical. If there is no change in the energy of the nucleus during scattering, it is called |
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| ===== 15.2. The Coulomb barrier ===== | ===== 15.2. The Coulomb barrier ===== | ||
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| - | When a positively charged particle (proton, deuteron, alpha particle etc.) collides with a nucleus, it is subjected to the positively charged protons of the target nucleus causing a repellent force. In order to penetrate into the nucleus the particle must have enough kinetic energy to overcome this repulsion, in other words cross the Coulomb barrier. | + | When a positively charged particle ([[textbook: |
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| - | where $\text{k}$ | + | where $\text{k}$ |
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| Figure XV.1. The height of the Coulomb barrier for three projectile particles as a function of the target nucleus atomic number. | Figure XV.1. The height of the Coulomb barrier for three projectile particles as a function of the target nucleus atomic number. | ||
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| ===== 15.3. Energetics of nuclear reactions ===== | ===== 15.3. Energetics of nuclear reactions ===== | ||
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| - | The reaction energy $Q \, (\text{MeV}) \, \text{is} \, -\Delta m \, (\text{amu}) \times 931.5 \, \text{MeV/ | + | The reaction energy $Q \, (\text{MeV}) \, \text{is} \, -\Delta m \, (\text{amu}) \times 931.5 \, \text{MeV/ |
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| - | The total mass increases in this reaction by 0.001279 amu and thus this is an endoergic reaction, i.e. the kinetic energy of the projectile particle must import the required energy (1.19 MeV) to the target nucleus. In addition to this energy, the projectile particle must have enough energy to also encompass the kinetic energy of the emitting particle and the recoil energy of the resulting nucleus. The smallest possible projectile particle energy able to cause an endoergic reaction is called the | + | The total mass increases in this reaction by 0.001279 |
| - | threshold energy ($\text{E}_{th}$) of the reaction. | + | //threshold energy// ($\text{E}_{th}$) of the reaction. |
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| $$E_{\text{th}} = \frac{m_x + m_A}{m_A} \times (-Q)$$ ;;# | $$E_{\text{th}} = \frac{m_x + m_A}{m_A} \times (-Q)$$ ;;# | ||
| [XV.VII] | [XV.VII] | ||
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| - | In addition to particle emission the nucleus also often emits gamma rays. For example, in the reaction | + | In addition to particle emission the nucleus also often emits [[textbook: |
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| $${}^{184}_{\phantom{1}74}\text{W} + \alpha \rightarrow {}^{186}_{\phantom{1}76}\text{Os} + 2n$$ ;;# | $${}^{184}_{\phantom{1}74}\text{W} + \alpha \rightarrow {}^{186}_{\phantom{1}76}\text{Os} + 2n$$ ;;# | ||
| [XV.VIII] | [XV.VIII] | ||
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| - | the intermediate nucleus ${}^{188}_{\phantom{1}76}\text{Os}$ is generated, with an excitation energy of 20 MeV. The release of each neutron reduces the binding energy by 6 MeV and their kinetic energy of 3 MeV, i.e. a total of 18 MeV of excitation energy is removed with their emission. The remaining 2 MeV is not enough to overcome the binding energy of a third neutron, but this portion departs the nucleus as gamma | + | the intermediate |
| - | radiation. | + | |
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| ===== 15.4. Cross sections ===== | ===== 15.4. Cross sections ===== | ||
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| - | The cross section describes of the probability of a nuclear reaction occurrence. In other words, it tells us how large a fraction of bombarding particles brings about a nuclear reaction. The cross section is derived in this section. | + | The cross section describes of the probability of a nuclear reaction occurrence. In other words, it tells us how large a fraction of [[textbook: |
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| - | Let’s expose a target with $\text{N}$ number of nuclei per unit volume ($m^3$) and $\text{dx}$ in thickness ($m$) with a coherent particle flux. When the target is so thin that the particle flux density $\phi_o$ (particles/ | + | Let’s expose a [[textbook: |
| does not essentially change, the decrease in particle flux density equals with the number of collisions leading to nuclear reactions in a unit of time per unit area ($m^2$): | does not essentially change, the decrease in particle flux density equals with the number of collisions leading to nuclear reactions in a unit of time per unit area ($m^2$): | ||
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| $$- d\phi = \phi_0 \times N \times \sigma \times dx$$ ;;# | $$- d\phi = \phi_0 \times N \times \sigma \times dx$$ ;;# | ||
| [XV.IX] | [XV.IX] | ||
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| - | where $\sigma$ is the probability of events or the cross section. When the target is so thick ($x$) that the particle flux decreases significantly, | + | where $\sigma$ is the probability of events or the cross section. When the target is so thick ($x$) that the particle flux decreases significantly, |
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| $$\phi = \phi_0 \times e^{-\sigma \times N \times x}$$ ;;# | $$\phi = \phi_0 \times e^{-\sigma \times N \times x}$$ ;;# | ||
| [XV.X] | [XV.X] | ||
| ;;# | ;;# | ||
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| - | The unit of the cross section derived from the equation XV.VII is unit area. Because this has a very small value, barn (b = 10< | + | The unit of the cross section derived from the [[textbook: |
| comparable with the size of a nucleus, which in fact applies to many neutron-induced reactions, i.e. cross sections are close to 1 barn. However, due to action of the repulsive and attractive forces from nuclear and Coulombic interactions the cross sections vary several orders of magnitude, both above and below 1 barn. | comparable with the size of a nucleus, which in fact applies to many neutron-induced reactions, i.e. cross sections are close to 1 barn. However, due to action of the repulsive and attractive forces from nuclear and Coulombic interactions the cross sections vary several orders of magnitude, both above and below 1 barn. | ||
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| - | The decrease in particle flux in the target does not yet explicitly describe the number of a pecific | + | The decrease in particle flux in the [[textbook: |
| cross section while when individual reactions are considered separately it is called a partial cross section in which case the total cross section is the sum of all the partial cross sections of simultaneous nuclear reactions. | cross section while when individual reactions are considered separately it is called a partial cross section in which case the total cross section is the sum of all the partial cross sections of simultaneous nuclear reactions. | ||
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| ===== 15.5. Nuclear reaction kinetics ===== | ===== 15.5. Nuclear reaction kinetics ===== | ||
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| - | When nucleus B is produced by irradiating a nucleus A in a nuclear reaction, the equation for the growth rate of the resulting nucleus is: | + | When [[textbook: |
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| $$\frac{dN_B}{dt} = \sigma \times \phi \times N_A$$ ;;# | $$\frac{dN_B}{dt} = \sigma \times \phi \times N_A$$ ;;# | ||
| [XV.XI] | [XV.XI] | ||
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| - | where $\text{N}_A$ and $\text{N}_B$ are the numbers of target and product atoms, respectively. If the product nuclide B is radioactive, | + | where $\text{N}_A$ and $\text{N}_B$ are the numbers of target and product atoms, respectively. If the product nuclide B is radioactive, |
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| $$\frac{dN_B}{dt} = \sigma \times \phi \times N_A - \lambda N_B$$ ;;# | $$\frac{dN_B}{dt} = \sigma \times \phi \times N_A - \lambda N_B$$ ;;# | ||
| [XV.XII] | [XV.XII] | ||
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| - | When the irradiation time is t, the number of product nuclei is calculated at the end of irradiation by the formula XV.XIII, which is obtained by integrating the formula XV.XII in the time interval $0 - t$ assuming that at the start of irradiation $N_B = 0$. | + | When the irradiation time is //t//, the number of product |
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| $$N_B = \frac{\sigma \times \phi \times N_A}{\lambda} \times (1 - e^{-\lambda t})$$ ;;# | $$N_B = \frac{\sigma \times \phi \times N_A}{\lambda} \times (1 - e^{-\lambda t})$$ ;;# | ||
| [XV.XIII] | [XV.XIII] | ||
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| - | When producing radionuclides, | + | When producing |
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| $$A_B = \sigma \times \phi \times N_A \left(1 - e^{-\lambda t}\right)$$ | $$A_B = \sigma \times \phi \times N_A \left(1 - e^{-\lambda t}\right)$$ | ||
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| - | The mass m is used instead of the target nuclei number $N_A$ and the half-life t½ is used instead of the decay constant $\lambda$, in which case the formula becomes | + | The mass m is used instead of the [[textbook: |
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| $$A_B = \frac{m \times I \times 6.023 \times 10^{23} \times \sigma \times \phi}{M} \times \left(1 - e^{-\ln 2 \times \frac{t}{t_{1/ | $$A_B = \frac{m \times I \times 6.023 \times 10^{23} \times \sigma \times \phi}{M} \times \left(1 - e^{-\ln 2 \times \frac{t}{t_{1/ | ||
| [XV.XV] | [XV.XV] | ||
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| - | where $m$ is the mass of the target element, $I$ the target nuclide’s isotopic abundance of the element, 6.023·10< | + | where $m$ is the mass of the target element, $I$ the target nuclide’s |
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| Figure XV.2. shows the relative amount of nuclide produced in the target as a function of irradiation time. Time here is the irradiation time divided by the nuclide’s half-life, i.e. it is the number of half-lives. As seen, 50% of the maximum obtainable activity (saturation activity) is produced during one half-life, 75% during two half-lives, and about 99% during ten. | Figure XV.2. shows the relative amount of nuclide produced in the target as a function of irradiation time. Time here is the irradiation time divided by the nuclide’s half-life, i.e. it is the number of half-lives. As seen, 50% of the maximum obtainable activity (saturation activity) is produced during one half-life, 75% during two half-lives, and about 99% during ten. | ||
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| {{: | {{: | ||
| Figure XV.2. The relative amount of a radionuclide in the target as a function of irradiation time up to ten half-lives of the product nuclide and the decay of the product nuclide after irradiation. | Figure XV.2. The relative amount of a radionuclide in the target as a function of irradiation time up to ten half-lives of the product nuclide and the decay of the product nuclide after irradiation. | ||
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| \\ | \\ | ||
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| - | Finally, accounting for the continuing decay of the radionuclide after irradiation, | + | Finally, accounting for the continuing decay of the radionuclide after irradiation, |
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| \\ | \\ | ||
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| $$A_B = \frac{m \times I \times 6.023 \times 10^{23} \times \sigma \times \phi}{M} \times \left(1 - 2^{-\frac{t}{t_{1/ | $$A_B = \frac{m \times I \times 6.023 \times 10^{23} \times \sigma \times \phi}{M} \times \left(1 - 2^{-\frac{t}{t_{1/ | ||
| [XV.XVI] | [XV.XVI] | ||
| ;;# | ;;# | ||
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| ===== 15.6. Excitation function ===== | ===== 15.6. Excitation function ===== | ||
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| - | The probability of nuclear reactions is also dependent on the projectile particle energy. Presenting the cross sections of all individual reactions as a function of the projectile energy is called an excitation function. The Figure XV.3 demonstrates the excitation function of < | + | The probability of nuclear reactions is also dependent on the [[textbook: |
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| {{ : | {{ : | ||
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| \\ | \\ | ||
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| ===== 15.7. Photonuclear reactions ===== | ===== 15.7. Photonuclear reactions ===== | ||
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| - | Gamma radiation can cause nuclear reactions resulting in particle emissions, e.g. ($\gamma, n$) and ($\gamma, p$). These type of reactions are called photonuclear reaction. These reactions require a certain threshold energy from gamma rays to overcome the binding energies of protons and neutrons. The threshold energies of protons are higher than those of neutrons, because their removal from the nucleus also | + | [[textbook: |
| - | requires energy to cross the Coulomb barrier. Threshold energies are usually at least 5 MeV, so most of the gamma rays generated in radioactive decay do not cause nuclear reactions. Some nuclei, such as < | + | requires energy to cross the [[textbook: |
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| ===== 15.8. Neutron induced nuclear reactions ===== | ===== 15.8. Neutron induced nuclear reactions ===== | ||
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| {{ : | {{ : | ||
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| - | Since neutrons have no charge they are not affected by the repulsion caused by the positively charged nuclei. Therefore, neutron induced reactions do not have a threshold energy. On the contrary, reactions are achievable even with very little energy. In fact, the reaction probability (cross section) for neutrons with energy less than 1 MeV is higher the lower the energy: the cross section is inversely proportional to the kinetic energy of the neutron (Figure XV.4.). | + | Since [[textbook: |
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| ;;# | ;;# | ||
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| Figure XV.4. The excitation function of nuclear reactions induced by neutrons (neutron capture) in a < | Figure XV.4. The excitation function of nuclear reactions induced by neutrons (neutron capture) in a < | ||
| ;;# | ;;# | ||
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| - | For slowest neutrons, called thermal neutrons having energies of 0.005-0.1 eV, the neutron capture cross sections are as high as 10< | + | For slowest |
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| - | Slow neutrons often cause a capture reaction, that is, the neutrons are absorbed into the nucleus and gamma rays are emitted. These gamma rays get kinetic energy from the binding energy of absorbed neutrons. Typically, the energies of the gamma rays are high, between 5-8 MeV. | + | Slow neutrons often cause a capture reaction, that is, the neutrons are absorbed into the [[textbook: |
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| - | As shown in Figure XV. 4 the excitation function of neutrons at intermediate energies has many peaks. These are resonances, in which the excitation energy produced by the projectile particle nuclei are equivalent to the excitation levels of the nucleus. When the projectile energy value rises above 1 MeV the resonance states overlap and the excitation function levels. | + | As shown in [[textbook: |
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| - | Neutrons with a higher projectile energy can also produce reactions leading to alpha and proton emissions. Since their departure from the nucleus requires crossing the Coulomb barrier, these reactions always have a threshold energy. | + | Neutrons with a higher projectile energy can also produce reactions leading to alpha and proton emissions. Since their departure from the nucleus requires crossing the [[textbook: |
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| Neutrons also cause fission reactions, which will be discussed in the next section. | Neutrons also cause fission reactions, which will be discussed in the next section. | ||
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| ===== 15.9. Induced fission ===== | ===== 15.9. Induced fission ===== | ||
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| - | Chapter V dealt with spontaneous fission, and found that the reason for it is that a nucleus is too heavy and that it occurs only in the heaviest nuclei. In many ways, induced nuclear fission is similar to spontaneous fission. In both cases the nucleus disintegrates into two lighter nuclei, but not spontaneously in case of induced fission but through the excitation energy of external particles, typically neutrons, for example | + | [[textbook: |
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| - | Despite fission usually being caused by a neutron, it can be produced by other particles, such as protons, deuterons and alpha particles (and even by the gamma rays) that have enough energy to cross the Coulomb barrier and introduce enough excitation energy via their kinetic energy. The requisite excitation energy of the intermediate nucleus is 4-6 MeV. While spontaneous fission | + | Despite fission usually being caused by a [[textbook: |
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| - | Fission releases a large amount of energy, because medium heavy nuclei have a stronger binding energy to nucleons, about 8 MeV/ | + | Fission releases a large amount of energy, because medium heavy nuclei have a stronger |
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| ^Energy generated by neutrinos in beta decay^| 10 MeV| | ^Energy generated by neutrinos in beta decay^| 10 MeV| | ||
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| - | In conventional fission types the fission products generated are mostly of a different size (asymmetric fission). Figure XV.7a shows the distribution of fission products of the thermal neutron induced fission of three nuclides < | + | In conventional fission types the fission products generated are mostly of a different size (asymmetric fission). Figure XV.7a shows the distribution of fission products of the [[textbook: |
| - | number peak is transferred to a higher range, 95-105. | + | number peak is transferred to a higher range, 95-105. |
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| Figure XV.7. Yields of fission products (%) as a function of their mass number: a) thermal neutron induced fission of < | Figure XV.7. Yields of fission products (%) as a function of their mass number: a) thermal neutron induced fission of < | ||
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| - | In a fission event 2-3 neutrons, prompt neutrons, form at disintegration moment. The daughter nuclides formed in fission are always radioactive, | + | In a fission event 2-3 [[textbook: |
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| $$\text{n/ | $$\text{n/ | ||
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| - | As shown, when going towards stable nuclides from the primary fission nuclides the half-lives lengthen, reflecting the increase in stability. In some beta decay events, neutrons, called delayed neutrons, are also emitted. They are only a small fraction of the prompt neutrons, e.g. 0.02% in < | + | As shown, when going towards stable nuclides from the primary fission nuclides the [[textbook: |
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| {{: | {{: | ||
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| - | The nuclides, in which a fission reaction is possible, are called fissionable, | + | The nuclides, in which a fission reaction is possible, are called |
| - | bombardment of < | + | |
| - | energy of the fast neutrons is needed. Cross sections of induced fission of < | + | |
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| Figure XV.8. Cross section of neutron induced fission of < | Figure XV.8. Cross section of neutron induced fission of < | ||
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| + | In order for fission events to continue spontaneously, | ||
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textbook/nrctextbook/chapter15.1737671431.txt.gz · Last modified: 2025-01-23 23:30 by Merja Herzig