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textbook:nrctextbook:chapter15 [2025-04-28 15:13]
Merja Herzig 		textbook:nrctextbook:chapter15 [2025-05-05 11:10] (current)
Merja Herzig
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 When [[textbook:nrctextbook:chapter2#nucleus|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:nrctextbook:chapter2#nucleus|nucleus]] B is produced by irradiating a nucleus A in a nuclear reaction, the equation for the growth rate of the resulting nucleus is:
 ### ###
 +{{anchor:eq_1511}}
 $$\frac{dN_B}{dt} = \sigma \times \phi \times N_A$$ ;;# $$\frac{dN_B}{dt} = \sigma \times \phi \times N_A$$ ;;#
 [XV.XI] [XV.XI]
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 When producing [[textbook:nrctextbook:chapter4|radionuclides]], the [[textbook:nrctextbook:chapter6#activity|activity]] of the nuclide is of more interest than the number of nuclei. Since $A = \lambda \times N_B$, we can replace $N_B$ with $\frac{A}{\lambda}$ in the formula XV.XIII to get: When producing [[textbook:nrctextbook:chapter4|radionuclides]], the [[textbook:nrctextbook:chapter6#activity|activity]] of the nuclide is of more interest than the number of nuclei. Since $A = \lambda \times N_B$, we can replace $N_B$ with $\frac{A}{\lambda}$ in the formula XV.XIII to get:
 ### ###
 +{{anchor:eq_1514}}
  
 $$A_B = \sigma \times \phi \times N_A \left(1 - e^{-\lambda t}\right)$$  ;;# $$A_B = \sigma \times \phi \times N_A \left(1 - e^{-\lambda t}\right)$$  ;;#
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 The mass m is used instead of the [[textbook:nrctextbook:chapter15#target_nucleus|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:nrctextbook:chapter15#target_nucleus|target nuclei]] number $N_A$ and the half-life t½ is used instead of the decay constant $\lambda$, in which case the formula becomes
 ### ###
 +{{anchor:eq_1515}}
 $$A_B = \frac{m \times I \times 6.023 \times 10^{23} \times \sigma \times \phi}{M} \times \left(1 - e^{-\ln 2 \times \frac{t}{t_{1/2}}}\right)$$ ;;# $$A_B = \frac{m \times I \times 6.023 \times 10^{23} \times \sigma \times \phi}{M} \times \left(1 - e^{-\ln 2 \times \frac{t}{t_{1/2}}}\right)$$ ;;#
 [XV.XV] [XV.XV]
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 where $m$ is the mass of the target element, $I$ the target nuclide’s [[textbook:nrctextbook:chapter2#isotope|isotopic abundance]] of the element, 6.023·10<sup>23</sup> is the [[textbook:nrctextbook:chapter3#avogadro_number|Avogadro´s number]], and $M$ is the molar mass of the element. where $m$ is the mass of the target element, $I$ the target nuclide’s [[textbook:nrctextbook:chapter2#isotope|isotopic abundance]] of the element, 6.023·10<sup>23</sup> is the [[textbook:nrctextbook:chapter3#avogadro_number|Avogadro´s number]], and $M$ is the molar mass of the element.
 ### ###
 +{{anchor:figure_152}}
 ### ###
 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.
 ### ###
 +
 {{:textbook:nrctextbook:relative_amount_of_radionuclide_in_target_as_function_of_irradiation_time_fig_15_2.png?400 |}} {{:textbook:nrctextbook:relative_amount_of_radionuclide_in_target_as_function_of_irradiation_time_fig_15_2.png?400 |}}
 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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 \\  \\ 
  
 +{{anchor:eq_1516}}
 $$A_B = \frac{m \times I \times 6.023 \times 10^{23} \times \sigma \times \phi}{M} \times \left(1 - 2^{-\frac{t}{t_{1/2}}}\right) \times 2^{-\frac{t*}{t_{1/2}}}$$ ;;# $$A_B = \frac{m \times I \times 6.023 \times 10^{23} \times \sigma \times \phi}{M} \times \left(1 - 2^{-\frac{t}{t_{1/2}}}\right) \times 2^{-\frac{t*}{t_{1/2}}}$$ ;;#
 [XV.XVI] [XV.XVI]
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 ### ###
-[[textbook:nrctextbook:chapter5|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:nrctextbook:chapter5|Chapter V]] dealt with [[textbook:nrctextbook:chapter5#fission|spontaneous fission]] and found that the reason for it is that a [[textbook:nrctextbook:chapter2#nucleus|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 [[textbook:nrctextbook:chapter2#neutron|neutrons]], for example
 ### ###
  
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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 onlypertains to heavy nuclei, induced fission can also be achieved with medium heavy nuclei, even lanthanides, as long as the projectile particle energy is high enough, at least 50 MeV.+Despite fission usually being caused by a [[textbook:nrctextbook:chapter2#neutron|neutron]], it can be produced by other particles, such as [[textbook:nrctextbook:chapter2#proton|protons]], [[textbook:nrctextbook:chapter2#deuterium|deuterons]] and [[textbook:nrctextbook:chapter5#alpha_particle|alpha particles]] (and even by the [[textbook:nrctextbook:chapter5#gamma|gamma rays]]) that have enough energy to cross the [[textbook:nrctextbook:chapter15#coulomb_barrier|Coulomb barrier]] and introduce enough excitation energy via their kinetic energy. The requisite excitation energy of the intermediate [[textbook:nrctextbook:chapter2#nucleus|nucleus]] is 4-6 MeV. While [[textbook:nrctextbook:chapter5#fission|spontaneous fission]] only pertains to heavy nuclei, induced fission can also be achieved with medium heavy nuclei, even lanthanides, as long as the [[textbook:nrctextbook:chapter15#projectile_particle|projectile particle]] energy is high enough, at least 50 MeV.
 ### ###
  
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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/nucleon, than heavy nuclei in which it is about 7 MeV/nucleon (see Figure II.2). For each nucleon an energy of almost 1 MeV is released and in an individual fission event of a heavy nucleus a total of 200 MeV of energy is released, the distribution of which is shown in Table XV.I+Fission releases a large amount of energy, because medium heavy nuclei have a stronger [[textbook:nrctextbook:chapter3#binding_energy|binding energy]] to nucleons, about 8 MeV/nucleon, than heavy nuclei in which it is about 7 MeV/nucleon (see Figure II.2). For each nucleon an energy of almost 1 MeV is released and in an individual fission event of a heavy nucleus a total of 200 MeV of energy is released, the distribution of which is shown in Table XV.I
 ### ###
  
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 ^Energy generated by neutrinos in beta decay^| 10 MeV| ^Energy generated by neutrinos in beta decay^| 10 MeV|
  
 +{{anchor:distribution_of_fission_products}}
 ### ###
-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 <sup>235</sup>U, <sup>239</sup>Pu, and <sup>241</sup>Pu. Events, in which fission products are of equal size (symmetric fission) are extremely rare, occurring in only 0.05-0.01% of the cases. Distribution of uranium fission products has two peaks, at a mass numbers of 90-100 and of 130-140. The largest fission yields at these mass numbers are about 6-7%. The fission nuclides <sup>90</sup>Sr and <sup>137</sup>Cs belong to this category: they constitute the major part of the activity of fission products within the next few hundred years. The half-life of both is relatively long, about 30 years. Plutonium also has a fission product peak at the same upper mass number, but the lower mass  +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:nrctextbook:chapter15#thermal_neutron|thermal neutron]] induced fission of three nuclides <sup>235</sup>U, <sup>239</sup>Pu, and <sup>241</sup>Pu. Events, in which fission products are of equal size (symmetric fission) are extremely rare, occurring in only 0.05-0.01% of the cases. Distribution of [[textbook:nrctextbook:chapter4#uranium|uranium]] fission products has two peaks, at a mass numbers of 90-100 and of 130-140. The largest fission yields at these [[textbook:nrctextbook:chapter2#mass_number|mass numbers]] are about 6-7%. The fission nuclides <sup>90</sup>Sr and <sup>137</sup>Cs belong to this category: they constitute the major part of the [[textbook:nrctextbook:chapter6#activity|activity]] of fission products within the next few hundred years. The [[textbook:nrctextbook:chapter6#half_life|half-life]] of both is relatively long, about 30 years. [[textbook:nrctextbook:chapter1#plutonium|Plutonium]] also has a fission product peak at the same upper mass number, but the lower mass  
-number peak is transferred to a higher range, 95-105.  When going further into heavier fissioning nuclides this lower mass number peak range moves closer to the upper range and the valley between them narrows and becomes shallower, increasing the likelihood of symmetric fission. The probability of symmetric fission also increases when the projectile particle energy grows (Figure XV.7b): in the fission of <sup>235</sup>U induced by 14 MeV neutrons already one out of a hundred results in a symmetric fission. When the neutron energy is raised to 100 MeV, the valley between the peaks disappears.+number peak is transferred to a higher range, 95-105.  When going further into heavier fissioning nuclides this lower mass number peak range moves closer to the upper range and the valley between them narrows and becomes shallower, increasing the likelihood of symmetric fission. The probability of symmetric fission also increases when the [[textbook:nrctextbook:chapter15#projectile_particle|projectile particle]] energy grows (Figure XV.7b): in the fission of <sup>235</sup>U induced by 14 MeV neutrons already one out of a hundred results in a symmetric fission. When the [[textbook:nrctextbook:chapter2#neutron|neutron]] energy is raised to 100 MeV, the valley between the peaks disappears.
 ### ###
  
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 Figure XV.7. Yields of fission products (%) as a function of their mass number: a) thermal neutron induced fission of <sup>233</sup>U, <sup>235</sup>U, <sup>239</sup>Pu and <sup>241</sup>Pu (https://en.wikipedia.org/wiki/Fission_product_yield#/media/File:ThermalFissionYield.svg) b): <sup>235</sup>U fission with thermal and 14 MeV neutrons (http://www.tpub.com/doenuclearphys/nuclearphysics29.htm). Figure XV.7. Yields of fission products (%) as a function of their mass number: a) thermal neutron induced fission of <sup>233</sup>U, <sup>235</sup>U, <sup>239</sup>Pu and <sup>241</sup>Pu (https://en.wikipedia.org/wiki/Fission_product_yield#/media/File:ThermalFissionYield.svg) b): <sup>235</sup>U fission with thermal and 14 MeV neutrons (http://www.tpub.com/doenuclearphys/nuclearphysics29.htm).
  
 +{{anchor:daughter_nuclides_fission}}
 ### ###
-In a fission event 2-3 neutrons, prompt neutrons, form at disintegration moment. The daughter nuclides formed in fission are always radioactive, because fissioning heavy elements have a greater neutron to proton ratio than lighter elements. Thus, the fissioning nuclides, even after emitting 2-3 neutrons, have too many neutrons and they decay via β-decay to correct the unstable neutron/proton ratio. Decay occurs in several stages forming a decay chain. The neutron to proton ratio of <sup>235</sup>U is 1.55 and is roughly the same with the primary fission product nuclides. For example, the stable barium isotopes, however, have a much lower neutron to proton ratio of 1.32-1.46. An example of this type of beta decay chain in which the neutron/proton ratio decreases is:+In a fission event 2-3 [[textbook:nrctextbook:chapter2#neutron|neutrons]], prompt neutrons, form at disintegration moment. The //daughter nuclides// formed in fission are always radioactive, because fissioning heavy elements have a greater [[textbook:nrctextbook:chapter3#neutron_to_proton_ratio|neutron to proton ratio]] than lighter elements. Thus, the fissioning nuclides, even after emitting 2-3 neutrons, have too many neutrons and they decay via [[textbook:nrctextbook:chapter5#beta|β-decay]] to correct the unstable neutron/proton ratio. Decay occurs in several stages forming a decay chain. The neutron to proton ratio of <sup>235</sup>U is 1.55 and is roughly the same with the primary fission product nuclides. For example, the stable barium [[textbook:nrctextbook:chapter2#isotope|isotopes]], however, have a much lower neutron to proton ratio of 1.32-1.46. An example of this type of [[textbook:nrctextbook:chapter6#beta_decay_chains|beta decay chain]] in which the neutron/proton ratio decreases is:
 ### ###
  
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 $$\text{n/p: } {}^{137}_{\phantom{1}52}\mathrm{Te} \ (1.63) \rightarrow {}^{137}_{\phantom{1}53}\mathrm{I} \ (1.58) \rightarrow {}^{137}_{\phantom{1}54}\mathrm{Xe} \ (1.54) \rightarrow {}^{137}_{\phantom{1}55}\mathrm{Cs} \ (1.49) \rightarrow {}^{137}_{\phantom{1}56}\mathrm{Ba} \ (1.45)$$ $$\text{n/p: } {}^{137}_{\phantom{1}52}\mathrm{Te} \ (1.63) \rightarrow {}^{137}_{\phantom{1}53}\mathrm{I} \ (1.58) \rightarrow {}^{137}_{\phantom{1}54}\mathrm{Xe} \ (1.54) \rightarrow {}^{137}_{\phantom{1}55}\mathrm{Cs} \ (1.49) \rightarrow {}^{137}_{\phantom{1}56}\mathrm{Ba} \ (1.45)$$
 +
 +
 +{{anchor:delayed_neutron}}
  
 ### ###
-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 <sup>235</sup>U fission.+As shown, when going towards stable nuclides from the primary fission nuclides the [[textbook:nrctextbook:chapter6#half_life|half-lives]] lengthen, reflecting the increase in stability. In some [[textbook:nrctextbook:chapter5#5.3._beta_decay_processes|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 <sup>235</sup>U fission.
 ### ###
 {{:textbook:nrctextbook:cross_section_of_235u_fission_fig_15_8.png?400 |}} {{:textbook:nrctextbook:cross_section_of_235u_fission_fig_15_8.png?400 |}}
 +
 +{{anchor:fissionable}}
 +{{anchor:fissile}}
 ### ###
-The nuclides, in which a fission reaction is possible, are called fissionable, i.e. eligible for fission. Nuclides able to undergo fission induced by thermal neutron are called fissile. Of these the most important are <sup>235</sup>U and <sup>239</sup>Pu, which play an important role as nuclear reactor fuel and nuclear weapons material. Of these, <sup>235</sup>U is the only naturally occurring fissile material. Neutron  +The nuclides, in which a fission reaction is possible, are called //fissionable//, i.e. eligible for fission. Nuclides able to undergo fission induced by [[textbook:nrctextbook:chapter15#thermal_neutron|thermal neutron]] are called //fissile//. Of these the most important are <sup>235</sup>U and <sup>239</sup>Pu, which play an important role as [[textbook:nrctextbook:chapter4#nuclear_power_production|nuclear reactor fuel]] and nuclear weapons material. Of these, <sup>235</sup>U is the only naturally occurring fissile material. Neutron bombardment of <sup>238</sup>U produce <sup>239</sup>Pu by [[textbook:nrctextbook:chapter15#neutron_capture|neutron capture]] and [[textbook:nrctextbook:chapter5#beta|beta decay]]. A characteristics of uranium, and plutonium, isotopes is that the [[textbook:nrctextbook:chapter2#isotope|isotopes]](<sup>233</sup>U, <sup>235</sup>U), with an [[textbook:nrctextbook:chapter5#odd_even_problem|odd mass number]] are fissile, but ones with an [[textbook:nrctextbook:chapter5#odd_even_problem|even mass number]] (<sup>234</sup>U, <sup>238</sup>U) only undergo fission induced by high energy neutrons. This is because when all the protons in the uranium nucleus (Z = 92) are paired, the uranium isotopes with an odd mass number have unpaired neutrons. [[textbook:nrctextbook:chapter3#binding_energy|Binding energy]] released in the pair formation of the absorbed neutron with the unpaired neutron is enough to induce fission. Instead, with uranium isotopes with an even mass number, not enough binding energy is released to induce fission since the absorbed neutron remains unpaired. In this case, to induce fission, kinetic energy of the fast neutrons is needed. [[textbook:nrctextbook:chapter15#cross_section|Cross sections]] of induced fission of <sup>235</sup>U and <sup>238</sup>U are seen in Figure XV.8.
-bombardment of <sup>238</sup>U produce <sup>239</sup>Pu by neutron capture and beta decay. A characteristics of uranium, and plutonium, isotopes is that the isotopes (<sup>233</sup>U, <sup>235</sup>U), with an odd mass number are fissile, but ones with an even mass number (<sup>234</sup>U, <sup>238</sup>U) only undergo fission induced by high energy neutrons. This is because when all the protons in the uranium nucleus (Z = 92) are paired, the uranium isotopes with an odd mass number have unpaired neutrons. Binding energy released in the pair formation of the absorbed neutron with the unpaired neutron is enough to induce fission. Instead, with uranium isotopes with an even mass number, not enough binding energy is released to induce fission since the absorbed neutron remains unpaired. In this case, to induce fission, kinetic  +
-energy of the fast neutrons is needed. Cross sections of induced fission of <sup>235</sup>U and <sup>238</sup>U are seen in Figure XV.8.+
 ### ###
  
 Figure XV.8. Cross section of neutron induced fission of <sup>235</sup>U and <sup>238</sup>U (D.T.Hughes and R.B. Schwartz, Neutron Cross Sections, Brookhaven National Laboratory Report 325, 2<sup>nd</sup> Edition, 1958). Figure XV.8. Cross section of neutron induced fission of <sup>235</sup>U and <sup>238</sup>U (D.T.Hughes and R.B. Schwartz, Neutron Cross Sections, Brookhaven National Laboratory Report 325, 2<sup>nd</sup> Edition, 1958).
  
 +{{anchor:critical_mass}}
 ### ###
-In order for fission events to continue spontaneously, become a chain reaction, in fissile material there has to be a sufficient amount of material. If there is too little material the neutrons escape without causing new fission. At least one neutron generated by a fission event must cause at least one new fission in order to create a chain reaction. The chain reaction is controlled, if only one neutron causes one new fission. If more than one neutron induces fission, it is uncontrolled fission,  +In order for fission events to continue spontaneously, become a chain reaction, in [[textbook:nrctextbook:chapter15#fissile|fissile]] material there has to be a sufficient amount of material. If there is too little material the neutrons escape without causing new fission. At least one neutron generated by a fission event must cause at least one new fission in order to create a chain reaction. The chain reaction is controlled, if only one neutron causes one new fission. If more than one neutron induces fission, it is uncontrolled fission, i.e. a bomb. The minimum mass of a spherical fissile material at which fission chain reaction occurs is called the //critical mass//. It is 52 kg for <sup>235</sup>U and only 16 kg for <sup>239</sup>Pu.
-i.e. a bomb. The minimum mass of a spherical fissile material at which fission chain reaction occurs is called the critical mass. It is 52 kg for <sup>235</sup>U and only 16 kg for <sup>239</sup>Pu.+
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textbook/nrctextbook/chapter15.1745846018.txt.gz · Last modified: 2025-04-28 15:13 by Merja Herzig

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