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====== 15. Nuclear reactions ======


Chapter 15 from BASICS OF NUCLEAR PHYSICS AND OF RADIATION DETECTION AND MEASUREMENT - An open-access textbook for nuclear and radiochemistry students by Jukka Lehto

###
A nuclear reaction is an event in which a nucleus is targeted by a projectile ([[textbook:nrctextbook:chapter2#proton|proton]], [[textbook:nrctextbook:chapter2#deuterium|deuteron]], [[textbook:nrctextbook:chapter5#alpha_particle|alpha particle]], etc.) or gamma ray causing one of the following interactions with the [[textbook:nrctextbook:chapter2#nucleus|nucleus]]:
###

  * Transmutation or the formation of a new nucleus
  * [[textbook:nrctextbook:chapter15#inelastic_scattering| Inelastic Scattering]]
  * [[textbook:nrctextbook:chapter15#elastic|Elastic scattering]]

###
Since the specific aim of this communication is production of new nuclei, primarily radionuclides, in nuclear reactions, transmutation lead reactions will be the focus. Scattering is mentioned only briefly. Projectiles can be very heavy, even uranium nuclei, but this communication will concentrate on lighter ones ([[textbook:nrctextbook:chapter2#proton|p]], [[textbook:nrctextbook:chapter2#deuterium|d]], [[textbook:nrctextbook:chapter5#alpha_particle|α]], [[textbook:nrctextbook:chapter2#neutron|n]]).
###

In nature, nuclear reactions occur in the following sources:

  * The [[textbook:nrctextbook:chapter4#cosmic_radiation|cosmic]] [[textbook:nrctextbook:chapter2#proton|protons]] and [[textbook:nrctextbook:chapter5#alpha_particle|alpha particles]] of the upper parts of the atmosphere (as well as the [[textbook:nrctextbook:chapter2#neutron|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 [[textbook:nrctextbook:chapter4#radiocarbon|radiocarbon]],<sup>14</sup>C, from atmospheric nitrogen.
  * Fusion reactions occuring in stars.
  * A very small number of neutron activation reactions caused by neutrons generated in [[textbook:nrctextbook:chapter5#fission|spontaneous fission]] of uranium in bedrock and overburden.

###
The first artificial nuclear reaction was achieved by [[textbook:nrctextbook:chapter1#ernest_rutherford|Ernst Rutherford]] in 1919, when he targeted nitrogen gas with alpha radiation generated in <sup>214</sup>Po decay, which resulted in the following reaction:
###

$${}^{14}\mathrm{N} + {}^{4}\mathrm{He} \rightarrow {}^{17}\mathrm{O} + {}^{1}\mathrm{H}$$ ;;#
[XV.I]
;;#


{{anchor:target_nucleus}}
{{anchor:projectile_particle}}
###
The products of the reaction were oxygen, its <sup>17</sup>O [[textbook:nrctextbook:chapter2#isotope|isotope]] and [[textbook:nrctextbook:chapter2#proton|protons]]. The above reaction equation can also be presented in a shorter form:
###

$${}^{14}\text{N} ({}^{4}\text{He}, p) {}^{17}\text{O}$$
;#;
or
;#;

$${}^{14}\text{N} (\alpha, p) {}^{17}\text{O}$$  ;;#
[XV.II]
;;#


###
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/ray// and on the right an //emitting particle/ray//. The bombarded substance is called the target. The [[textbook:nrctextbook:chapter2#neutron|neutrons]] needed for nuclear reactions are primarily obtained by [[textbook:nrctextbook:chapter16#radionuclide_production_reactors|research reactors]], but also in neutron generators. The positive projectile particles are produced by [[textbook:nrctextbook:chapter16#radionuclide_production_accelerators|particle accelerators]] that are e.g. the van de Graaf accelerator, linear accelerator, [[textbook:nrctextbook:chapter11#cyclotrons|cyclotron]], and synchrotron.
###

### 
The first [[textbook:nrctextbook:chapter16#radionuclide_production_accelerators|particle accelerator]], which could achieve nuclear reactions, was developed in the early 1930s. The first nuclear reaction that created an artificial radionuclide was accomplished in 1934, when [[textbook:nrctextbook:chapter1#joliot_curie|Frederick Joliot]] and [[textbook:nrctextbook:chapter1#joliot_curie|Irene Curie]] used a [[textbook:nrctextbook:chapter16#radionuclide_production_accelerators|particle accelerator]] to bombard aluminum with [[textbook:nrctextbook:chapter5#alpha_particle|alpha particles]] and produced <sup>30</sup>P, which is a [[textbook:nrctextbook:chapter5#pure_positron_emitters| positron emitter]]. The reaction was, therefore, ${}^{27}\text{Al} (\alpha, p) {}^{30}\text{P}$.
###
{{anchor:nuclear_reaction_types}}
{{anchor:nuclear_reaction_models}}
{{anchor:liquid_drop_model_ii}}
===== 15.1. Nuclear reaction types and models =====

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In most nuclear reactions the [[textbook:nrctextbook:chapter15#target_nucleus|target nucleus]] absorbs the [[textbook:nrctextbook:chapter15#projectile_particle|projectile particles]] ([[textbook:nrctextbook:chapter2#proton|p]], [[textbook:nrctextbook:chapter2#deuterium|d]], [[textbook:nrctextbook:chapter5#alpha_particle|α]], [[textbook:nrctextbook:chapter2#neutron|n]]) and emits other particles in a fairly short period of time (10<sup>-14</sup>-10<sup>-18</sup> s). A [[textbook:nrctextbook:chapter3#liquid_drop_model|liquid drop model]] was developed to describe these types of reactions. The kinetic energy of the projectile particle and its merging with the [[textbook:nrctextbook:chapter2#nucleus|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 "evaporates" from the nucleus. The liquid drop model is supported, for example, by the fact that the reactions
###

$${}^{12}_{\phantom{1}6}\text{C} + {}^{144}_{\phantom{1}60}\text{Nd} \rightarrow {}^{150}_{\phantom{1}66}\text{Dy} + 6n$$

;#;
and
;#;


$${}^{20}_{10}\text{Ne} + {}^{136}_{\phantom{1}56}\text{Ba} \rightarrow {}^{150}_{\phantom{1}66}\text{Dy} + 6n$$ ;;#
[XV.III]
;;#


###
have roughly equal probability as a function of the [[textbook:nrctextbook:chapter15#projectile_particle|projectile]] energy (i.e. [[textbook:nrctextbook:chapter15#excitation_function|excitation function]], which is explained later). In this case, the explanation is that an excited intermediate [[textbook:nrctextbook:chapter2#nucleus|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 
origin.
###
{{anchor:spallation}}
{{anchor:fragmentation_reaction}}
###
The liquid drop model is not able to explain all of the observed reactions, in particular those with high [[textbook:nrctextbook:chapter15#projectile_particle|projectile]] energy. These reactions are, for example, //spallation reactions//, in which a large number, even many tens of [[textbook:nrctextbook:chapter2#nucleon|nucleons]], are emitted from the [[textbook:nrctextbook:chapter2#nucleus|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, 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.
###

###
[[textbook:nrctextbook:chapter5#5.1._fission|Fission]] is a reaction in which a heavy nucleus, e.g. <sup>235</sup>U, decays into two lighter elements. It is also explainable by the liquid drop model. Fusion is the opposite reaction to fission: lighter elements join together to form heavier nuclei.
###
{{anchor:elastic_scattering}}
{{anchor:elastic}}
{{anchor:inelastic_scattering}}
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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 [[textbook:nrctextbook:chapter2#nucleus|nucleus]] becomes excited and the kinetic energy of the emitting particle reduces.
###

{{anchor:coulomb_barrier}}
{{anchor:coulombs_law}}
{{anchor:coulombs_constant}}
===== 15.2. The Coulomb barrier =====

###
When a positively charged particle ([[textbook:nrctextbook:chapter2#proton|proton]], [[textbook:nrctextbook:chapter2#deuterium|deuteron]], [[textbook:nrctextbook:chapter5#alpha_particle|alpha particle]] etc.) collides with a [[textbook:nrctextbook:chapter2#nucleus|nucleus]], it is subjected to the positively charged protons of the [[textbook:nrctextbook:chapter15#target_nucleus|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//.  The Coulomb barrier is higher, the greater the target substance [[textbook:nrctextbook:chapter2#atomic_number|atomic number]] is. The Coulomb repulsion in also dependent on the bombarding particle charge: it is larger for alpha particles that have a charge of +2 than protons, with a charge of +1. According to the //Coulomb’s law// the repulsion force is
###

$$F_{\text{coul}} = k \cdot e \cdot Z_1 \cdot e \cdot Z_2 / x^2 = k \cdot e^2 \cdot Z_1 \cdot Z_2 / x^2$$ ;;#
[XV.IV]
;;#


###
where $\text{k}$  is //Coulomb’s constant// ($8.99 \times 10^9 \, \text{N} \, \text{m}^2 \, \text{C}^{-2}$), $\text{e}$ is the electron charge, $\text{Z}_1$ is the elemental number of the [[textbook:nrctextbook:chapter15#projectile_particle|projectile particle]], $\text{Z}_2$ is the corresponding value for the [[textbook:nrctextbook:chapter15#target_nucleus|target nucleus]], and $\text{x}$ is their distance. i.e. distance of their center points. So, this is the energy of projectile needed to cross the Coulomb barrier.  Figure XV.1 shows the dependence of the Coulomb barrier on the nuclear charge of both the projectile particle and the target nucleus.
###

{{:textbook:nrctextbook:height_of_coulomb_barrier_fig_15_1.png?400 |}}
Figure XV.1. The height of the Coulomb barrier for three projectile particles as a function of the target nucleus atomic number.

{{anchor:energetics_of_nuclear_reactions}}
{{anchor:exoergic_reaction}}
{{anchor:endoergic_reaction}}
{{anchor:treshold_energy}}
===== 15.3. Energetics of nuclear reactions =====

###
As in radioactive decay, the change in energy occurring in nuclear reactions can be calculated from the masses of the initial and resulting nuclei. The change in mass in a nuclear reaction $A(x,y)B$ is thus:
###

$$\Delta m = m_B + m_y - m_A - m_x$$ ;;#
[XV.V]
;;#

###
The reaction energy $Q \, (\text{MeV}) \, \text{is} \, -\Delta m \, (\text{amu}) \times 931.5 \, \text{MeV/amu}$. If the mass in the reaction decreases, it means that the energy is released, i.e., it is an //exoergic reaction//. On the other hand, if the mass increases, it is an //endoergic reaction//. For example, in the equation [XV.II] for the reaction ${}^{14}\text{N} (\alpha, p) {}^{17}\text{O}$, the reaction $\text{Q}$ is
###

$$Q = 931.5 \, \text{MeV/amu} \left( 16.999131 + 1.007825 - 14.003074 - 4.002603 \right) \, \text{amu}$$
$$= 931.5 \, \text{MeV/amu} \times -0.001279 \, \text{amu} = -1.19 \, \text{MeV}$$
;;#
[XV.VI]
;;#

###
The total mass increases in this reaction by 0.001279 [[textbook:nrctextbook:chapter3#amu|amu]] and thus this is an endoergic reaction, i.e. the kinetic energy of the [[textbook:nrctextbook:chapter15#projectile_particle|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 
//threshold energy// ($\text{E}_{th}$) of the reaction.  From the energy and momentum conservation laws one can derive threshold energy value by:
###
{{anchor:eq_157}}
$$E_{\text{th}} = \frac{m_x + m_A}{m_A} \times (-Q)$$ ;;#
[XV.VII]
;;#

###
The threshold energy of the reaction ${}^{14}\text{N} (\alpha, p) {}^{17}\text{O}$ is thus $-(-1.19 \, \text{MeV}) \left( \frac{4+14}{14} \right) = 1.53 \, \text{MeV}$, being 0.34 MeV larger than the reaction energy.
###

###
To achieve an exoergic reaction, the projectile particle must have enough energy to cross the Coulomb barrier. In exoergic reactions the kinetic energy of the emitting particle is, however, not the same as the reaction energy since also in this case the resulting nucleus gets part of the released energy as recoil energy.
###

###
In addition to particle emission the nucleus also often emits [[textbook:nrctextbook:chapter5#gamma|gamma rays]]. For example, in the reaction
###
{{anchor:eq_158}}
$${}^{184}_{\phantom{1}74}\text{W} + \alpha \rightarrow {}^{186}_{\phantom{1}76}\text{Os} + 2n$$ ;;#
[XV.VIII]
;;#

###
the intermediate [[textbook:nrctextbook:chapter2#nucleus|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 radiation.
###

{{anchor:cross_section}}
===== 15.4. Cross sections =====

###
The cross section describes of the probability of a nuclear reaction occurrence. In other words, it tells us how large a fraction of [[textbook:nrctextbook:chapter15#projectile_particle|bombarding particles]] brings about a nuclear reaction. The cross section is derived in this section.
###

###
Let’s expose a [[textbook:nrctextbook:chapter15#target_nucleus|target]] with $\text{N}$ number of [[textbook:nrctextbook:chapter2#nucleon|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/m<sup>2</sup>×s<sup>1</sup>) 
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 ($m^2$):
###
{{anchor:eq_159}}
$$- d\phi = \phi_0 \times N \times \sigma \times dx$$  ;;#
[XV.IX]
;;#

###
where $\sigma$ is the probability of events or the cross section. When the target is so thick ($x$) that the particle flux decreases significantly, its value can be calculated with the [[textbook:nrctextbook:chapter15#eq_1510|equation XV.X]], which is obtained by integrating the [[textbook:nrctextbook:chapter15#eq_159|equation XV.IX]]. with respect to thickness. 
###
{{anchor:eq_1510}}
$$\phi = \phi_0 \times e^{-\sigma \times N \times x}$$ ;;#
[XV.X]
;;#

{{anchor:barn}}
###
The unit of the cross section derived from the [[textbook:nrctextbook:chapter15#eq_157|equation XV.VII]] is unit area. Because this has a very small value, //barn// (b = 10<sup>-28</sup> m<sup>2</sup>, approximately a unit cross section of a nucleus) is used instead m<sup>2</sup>. If we neglect Coulombic interactions and nuclear forces the cross section should approximately be 
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.
###

###
The decrease in particle flux in the [[textbook:nrctextbook:chapter15#target_nucleus|target]] does not yet explicitly describe the number of a specific nuclear reaction. Several nuclear reactions can occur simultaneously in the target, for example a reaction can lead to the emission of one neutron and a reaction with emission of two neutrons simultaneously. When all simultaneous reactions are considered, the cross section is called the total 
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.
###
{{anchor:nuclear_reaction_kinetics}}
===== 15.5. Nuclear reaction kinetics =====

###
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}}
$$\frac{dN_B}{dt} = \sigma \times \phi \times N_A$$ ;;#
[XV.XI]
;;#


###
where $\text{N}_A$ and $\text{N}_B$ are the numbers of target and product atoms, respectively. If the product nuclide B is radioactive, it decays at the same time by a factor $-\lambda N_B$, where $\lambda$ is the [[textbook:nrctextbook:chapter6#decay_constant|decay constant]] of nuclide B. In this case, the total rate of growth for the product nucleus is
###
{{anchor:eq_1512}}
$$\frac{dN_B}{dt} = \sigma \times \phi \times N_A - \lambda N_B$$ ;;#
[XV.XII]
;;#

###
When the irradiation time is //t//, the number of product [[textbook:nrctextbook:chapter2#nucleus|nuclei]] is calculated at the end of irradiation by the [[textbook:nrctextbook:chapter15#eq_1513|formula XV.XIII]], which is obtained by integrating the [[textbook:nrctextbook:chapter15#eq_1512|formula XV.XII]] in the time interval $0 - t$ assuming that at the start of irradiation $N_B = 0$.
###
{{anchor:eq_1513}}
$$N_B = \frac{\sigma \times \phi \times N_A}{\lambda} \times (1 - e^{-\lambda t})$$ ;;#
[XV.XIII]
;;#


###
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}}

$$A_B = \sigma \times \phi \times N_A \left(1 - e^{-\lambda t}\right)$$  ;;#
[XV.XIV]
;;#

###
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}}
$$A_B = \frac{m \times I \times 6.023 \times 10^{23} \times \sigma \times \phi}{M} \times \left(1 - e^{-\ln 2 \times \frac{t}{t_{1/2}}}\right)$$ ;;#
[XV.XV]
;;#

###
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.
###

{{: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.


\\ 
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\\ 
\\ 
\\ 
###
Finally, accounting for the continuing decay of the radionuclide after irradiation, the [[textbook:nrctextbook:chapter6#activity|activity]] of the produced [[textbook:nrctextbook:chapter4|radionuclide]] can be calculated at the time point t* after the end of the irradiation by the formula
###
\\ 

{{anchor:eq_1516}}
$$A_B = \frac{m \times I \times 6.023 \times 10^{23} \times \sigma \times \phi}{M} \times \left(1 - 2^{-\frac{t}{t_{1/2}}}\right) \times 2^{-\frac{t*}{t_{1/2}}}$$ ;;#
[XV.XVI]
;;#

{{anchor:excitation_function}}
===== 15.6. Excitation function =====

###
The probability of nuclear reactions is also dependent on the [[textbook:nrctextbook:chapter15#projectile_particle|projectile particle]] energy. Presenting the [[textbook:nrctextbook:chapter15#cross_section|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 <sup>54</sup>Fe irradiation with [[textbook:nrctextbook:chapter5#alpha_particle|alpha particles]]. As seen, the predominant reactions at low projectile energy lead to emission of a single particle ($n$ and $p$). As the projectile energy increases also two particle emissions occur and at even higher energies also three particle emissions. The reactions in the figure are:
###
{{ :textbook:nrctextbook:excitation_function_in_irradiation_of_54fe_fig_15_3.png?400|}}



  * $^{54}\text{Fe}(\alpha,p)^{57}\text{Co}$
  * $^{54}\text{Fe}(\alpha,n)^{57}\text{Ni}$
  * $^{54}\text{Fe}(\alpha,pn)^{56}\text{Co}$
  * $^{54}\text{Fe}(\alpha,2n)^{56}\text{Ni}$
  * $^{54}\text{Fe}(\alpha,2pn)^{55}\text{Fe}$
  * $^{54}\text{Fe}(\alpha,p2n)^{55}\text{Co}$

\\ 
\\ 

;;#
Figure XV.3. The excitation function in irradiation of 54Fe with alpha particles (data from Houck and Miller, Physical Review 123(1961)231.
;;#

\\ 
{{anchor:photonuclear_reactions}}
===== 15.7. Photonuclear reactions =====

###
[[textbook:nrctextbook:chapter5#gamma|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 [[textbook:nrctextbook:chapter15#treshold_energy|threshold energy]] from [[textbook:nrctextbook:chapter5#gamma|gamma rays]] to overcome the [[textbook:nrctextbook:chapter3#binding_energy|binding energies]] of [[textbook:nrctextbook:chapter2#proton|protons]] and [[textbook:nrctextbook:chapter2#neutron|neutrons]]. The threshold energies of protons are higher than those of neutrons, because their removal from the [[textbook:nrctextbook:chapter2#nucleus|nucleus]] also 
requires energy to cross the [[textbook:nrctextbook:chapter15#coulomb_barrier|Coulomb barrier]]. Threshold energies are usually at least 5 MeV, so most of the gamma rays generated in [[textbook:nrctextbook:chapter5|radioactive decay]] do not cause nuclear reactions. Some nuclei, such as <sup>2</sup>H, <sup>9</sup>Be, and <sup>13</sup>C, have lower threshold energies, however, and for example the <sup>9</sup>Be($\gamma, n$)<sup>8</sup>Be reaction can already occur at gamma energies of 1.67 MeV.
###

{{anchor:neutron_induced_nuclear_reactions}}
===== 15.8. Neutron induced nuclear reactions =====


{{ :textbook:nrctextbook:excitation_function_of_nuclear_reactions_by_neutrons_fig_15_4.png?400|}}
###
Since [[textbook:nrctextbook:chapter2#neutron|neutrons]] have no charge they are not affected by the repulsion caused by the positively charged [[textbook:nrctextbook:chapter2#nucleus|nuclei]]. Therefore, neutron induced reactions do not have a [[textbook:nrctextbook:chapter15#treshold_energy|threshold energy]]. On the contrary, reactions are achievable even with very little energy. In fact, the reaction probability ([[textbook:nrctextbook:chapter15#cross_section|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.).
###
;;#
{{anchor:figure_154}}
Figure XV.4. The excitation function of nuclear reactions induced by neutrons (neutron capture) in a <sup>113</sup>Cd target http://thorea.wikia.com/wiki/Thermal,_Epithermal_and_Fast_Neutron_Spectra).
;;#

{{anchor:thermal_neutron}}
{{anchor:neutron_capture}}
###
For slowest [[textbook:nrctextbook:chapter2#neutron|neutrons]], called //thermal neutrons// having energies of 0.005-0.1 eV, the neutron capture [[textbook:nrctextbook:chapter15#cross_section|cross sections]] are as high as 10<sup>5</sup> [[textbook:nrctextbook:chapter15#barn|b]]. The ability of slow neutrons to cause nuclear reactions is due to their high wavelength of about 0.1 nm, while the wavelengths of faster neutrons (<0.1 MeV) is a thousand times smaller. Therefore, the probability of slower neutrons hitting the [[textbook:nrctextbook:chapter2#nucleus|nuclei]] is greater.
###

###
Slow neutrons often cause a capture reaction, that is, the neutrons are absorbed into the [[textbook:nrctextbook:chapter2#nucleus|nucleus]] and [[textbook:nrctextbook:chapter5#gamma|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.
###

###
As shown in [[textbook:nrctextbook:chapter15#figure_154|Figure XV.4]] the [[textbook:nrctextbook:chapter15#excitation_function|excitation function]] of neutrons at intermediate energies has many peaks. These are resonances, in which the excitation energy produced by the [[textbook:nrctextbook:chapter15#projectile_particle|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.
###

###
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:nrctextbook:chapter15#coulomb_barrier|Coulomb barrier]], these reactions always have a [[textbook:nrctextbook:chapter15#treshold_energy|threshold energy]].
###

###
Neutrons also cause fission reactions, which will be discussed in the next section.
###
{{anchor:induced_fission}}
===== 15.9. Induced fission =====

###
[[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
###

$${}^{235}\mathrm{U} + \mathrm{n} \rightarrow ({}^{236}\mathrm{U})^* \rightarrow {}^{143}\mathrm{Ba} + {}^{89}\mathrm{Kr} + 3{}^{1}\mathrm{n} + 200\mathrm{MeV}$$ ;;#
[XV.XVI]
;;#

###
where $(^{236}\text{U})^*$ is the excited intermediate uranium nucleus generated by neutron absorption, which quickly breaks down.
###

{{:textbook:nrctextbook:induced_fission_fig_15_5.png |}} Figure XV.5. Induced fission of a heavy nucleus into two lighter nuclei 
(http://chemwiki.ucdavis.edu/Physical_Chemistry/Nuclear_Chemistry/Nuclear_Reactions).
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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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{{:textbook:nrctextbook:heavy_nucleus_fission_fig_15_6.png?200 |}}
Figure XV.6. Heavy nucleus fission after neutron capture.
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The fission process can be explained as follows. Two opposing forces exist in the nucleus. The short-range nuclear forces which are smallest when the nucleus is spherical. While the longer range electrical forces, i.e. proton repulsion, aims to tear apart and deform the nucleus. Therefore, this is why, for example, the uranium nucleus is slightly elliptical, not perfectly spherical.  When a neutron 
absorbs into the uranium nucleus it causes an increase in the ellipticity and disintegration of the nucleus if the excitation energy is sufficiently large (Figure XV.5).
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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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Table XV.1. The distribution of the 200 MeV total energy generated by a <sup>235</sup>U fission event.

^Kinetic energy of fission products^| 165 MeV|
^Kinetic energy of neutrons^| 5 MeV|
^Gamma energy released immediately at fission^| 7 MeV|
^Beta particle energy of fission products^| 7 MeV| 
^Gamma ray energy of fission products^| 6 MeV| 
^Energy generated by neutrinos in beta decay^| 10 MeV|

{{anchor:distribution_of_fission_products}}
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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 [[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 [[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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{{:textbook:nrctextbook:yields_of_fission_products_fig_15_6.png |}}
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}}
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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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$${}^{137}_{\phantom{1}52}\mathrm{Te} \rightarrow 3.5\mathrm{s} \rightarrow {}^{137}_{\phantom{1}53}\mathrm{I} \rightarrow 24.5\mathrm{s} \rightarrow {}^{137}_{\phantom{1}54}\mathrm{Xe} \rightarrow 3.8\ \mathrm{min} \rightarrow {}^{137}_{\phantom{1}55}\mathrm{Cs} \rightarrow 30\alpha \rightarrow {}^{137}_{\phantom{1}56}\mathrm{Ba} \ (\text{stable})$$

in which the n/p ratio decreases correspondingly as:

$$\text{n/p: } {}^{137}_{\phantom{1}52}\mathrm{Te} \ (1.63) \rightarrow {}^{137}_{\phantom{1}53}\mathrm{I} \ (1.58) \rightarrow {}^{137}_{\phantom{1}54}\mathrm{Xe} \ (1.54) \rightarrow {}^{137}_{\phantom{1}55}\mathrm{Cs} \ (1.49) \rightarrow {}^{137}_{\phantom{1}56}\mathrm{Ba} \ (1.45)$$


{{anchor:delayed_neutron}}

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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.
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{{:textbook:nrctextbook:cross_section_of_235u_fission_fig_15_8.png?400 |}}

{{anchor:fissionable}}
{{anchor:fissile}}
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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.
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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}}
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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.
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