Nuclear Fission

Physics of Nuclear Fission

In general, the neutron-induced fission reaction is the reaction in which the incident neutron enters the heavy target nucleus (fissionable nucleus), forming a compound nucleus that is excited to such a high energy level (E_excitation > E_critical) that the nucleus splits into two large fission fragments. A large amount of energy is released in the form of radiation and fragment kinetic energy. Moreover, and what is crucial, the fission process may produce 2, 3, or more free neutrons, and these neutrons can trigger further fission, and a chain reaction can occur. To understand the process of fission, we must understand processes that occur inside the nucleus to be fissioned. At first, the nuclear binding energy must be defined.

Uranium - 235 Fission Reaction

Uranium 235 is a fissile isotope, and its fission cross-section for thermal neutrons is about 585 barns (for 0.0253 eV neutron). For fast neutrons, its fission cross-section is on the order of barns. Most absorption reactions result in fission reaction, but a minority results in radiative capture forming ²³⁶U. The cross-section for radiative capture for thermal neutrons is about 99 barns (for 0.0253 eV neutron). Therefore about 15% of all absorption reactions result in radiative capture of neutrons. About 85% of all absorption reactions result in fission.

Uranium - 233 Fission Reaction

Uranium 233 is a very good fissile isotope, and its fission cross-section for thermal neutrons is about 531 barns (for 0.0253 eV neutron). For fast neutrons, its fission cross-section is on the order of barns. Most absorption reactions result in fission reaction, but a minority results in radiative capture forming ²³⁴U. The cross-section for radiative capture for thermal neutrons is about 45 barns (for 0.0253 eV neutron). Therefore about 6% of all absorption reactions result in radiative capture of neutrons. About 94% of all absorption reactions result in fission. The capture-to-fission ratio is much smaller than the other two major fissile fuels ²³⁵U and ²³⁹U.

Plutonium - 239 Fission Reaction

Plutonium 239 is a fissile isotope, and its fission cross-section for thermal neutrons is about 750 barns (for 0.025 eV neutron). For fast neutrons, its fission cross-section is on the order of barns. Most absorption reactions result in fission reaction, but a part of reactions result in radiative capture forming ²⁴⁰Pu. The cross-section for radiative capture for thermal neutrons is about 270 barns (for 0.025 eV neutron). Therefore about 27% of all absorption reactions result in radiative capture of incident neutron. About 73% of all absorption reactions result in fission.

Nuclear Binding Energy

At the nuclear level, the nuclear binding energy is required to disassemble (to overcome the strong nuclear force) a nucleus of an atom into its component parts (protons and neutrons). The protons and neutrons in an atomic nucleus are held together by the nuclear forces (strong force). The mass of a nucleus is always less than the sum of masses of the constituent protons and neutrons when separated. The difference is a measure of the nuclear binding energy (E_b) which holds the nucleus together. According to the Einstein relationship (E=m.c²) this binding energy is proportional to this mass difference, known as the mass defect.

During the nuclear splitting or nuclear fusion, some of the mass of the nucleus gets converted into vast amounts of energy. Thus this mass is removed from the total mass of the original particles, and the mass is missing in the resulting nucleus. The nuclear binding energies are enormous, and they are on the order of a million times greater than the electron binding energies of atoms.

For a nucleus with A (mass number) nucleons, the binding energy per nucleon E_b/A can be calculated. This calculated fraction is shown in the chart as a function of the mass number A. As can be seen, for low mass numbers, E_b/A increases rapidly and reaches a maximum of 8.8 MeV at approximately A=60. The nuclei with the highest binding energies that are most tightly bound belong to the “iron group” of isotopes (⁵⁶Fe, ⁵⁸Fe, ⁶²Ni). After that, the binding energy per nucleon decreases. A more stable configuration is obtained in the heavy nuclei (A>60) region when a heavy nucleus splits into two lighter nuclei. This is the origin of the fission process. It may seem that all the heavy nuclei may undergo fission or even spontaneous fission. In fact, for all nuclei with atomic number greater than about 60, fission occurs very rarely. For the fission process to take place, a sufficient amount of energy must be added to the nucleus, no matter how. The energetics and binding energies of a specific nucleus are well described by the Liquid Drop Model, which examines the global properties of nuclei.

Liquid Drop Model

One of the first models that could describe the behavior of the nuclear binding energies. Therefore nuclear masses were the mass formula of von Weizsaecker (also called the semi-empirical mass formula – SEMF) published in 1935 by German physicist Carl Friedrich von Weizsäcker. This theory is based on the liquid drop model proposed by George Gamow.

According to this model, the atomic nucleus behaves like the molecules in a drop of liquid. But in this nuclear scale, the fluid is made of nucleons (protons and neutrons), held together by the strong nuclear force. The liquid drop model of the nucleus considers that the nuclear forces on the nucleons on the surface are different from those on nucleons in the interior of the nucleus. Other attracting nucleons completely surround the interior nucleons. Here is the analogy with the forces that form a drop of liquid.

In the ground state, the nucleus is spherical. If sufficient kinetic or binding energy is added, this spherical nucleus may be distorted into a dumbbell shape and then maybe split into two fragments. Since these fragments are more stable, the splitting of such heavy nuclei must be accompanied by energy release. This model does not explain all the properties of the atomic nucleus but does explain the predicted nuclear binding energies.

The nuclear binding energy, as a function of the mass number A and the number of
protons Z, is based on the liquid drop model. Its formula can be written as:

This formula is called the Weizsaecker Formula (or the semi-empirical mass formula). The physical meaning of this equation can be discussed term by term.

Volume term

Volume term – a_V.A. The first two terms describe a spherical liquid drop of an incompressible fluid that contributes to the volume scaling with A and the surface, scaling with A^2/3. The first positive term a_V.A is known as the volume term, which is caused by attracting strong forces between the nucleons. The strong force has a very limited range, and a given nucleon may only interact with its direct neighbors. Therefore this term is proportional to A instead of A². The coefficient a_V is usually about ~ 16 MeV.

Surface term

Surface term – a_sf.A^2/3. The surface term is also based on the strong force. It is, in fact, a correction to the volume term. The point is that other particles do not completely surround particles at the surface of the nucleus. In the volume term, it is suggested that each nucleon interacts with a constant number of nucleons, independent of A. This assumption is nearly true for nucleons deep within the nucleus but causes an overestimation of the binding energy on the surface. By analogy with a liquid drop, this effect is indicated as the surface tension effect. If the volume of the nucleus is proportional to A, then the geometrical radius should be proportional to A^1/3. Therefore the surface term must be proportional to the surface area, i.e., proportional to A^2/3.

Coulomb term

Coulomb term – a_C.Z².A^-⅓. This term describes the Coulomb repulsion between the uniformly distributed protons and is proportional to the number of proton pairs Z²/R, whereby R is proportional to A^1/3. This effect lowers the binding energy because of the repulsion between charges of the equal sign.

Asymmetry term

Asymmetry term – a_A.(A-2Z)²/A. This term cannot be described as ‘classically’ as the first three. This effect is not based on any of the fundamental forces, and this effect is based only on the Pauli exclusion principle (no two fermions can occupy the same quantum state in an atom). The heavier nuclei contain more neutrons than protons. These extra neutrons are necessary for the stability of the heavier nuclei. They provide (via the attractive forces between the neutrons and protons) some compensation for the repulsion between the protons. On the other hand, if there are significantly more neutrons than protons in a nucleus, some of the neutrons will be higher in energy level in the nucleus. This is the basis for a correction factor, the so-called symmetry term.

Pairing term

Pairing term – δ(A,Z). The last term is the pairing term δ(A, Z). This term captures the effect of spin-coupling. Nuclei with an even number of protons and an even number of neutrons are (due to the Pauli exclusion principle) very stable thanks to the occurrence of “paired spin”. On the other hand, nuclei with an odd number of protons and neutrons are mostly unstable.

Table of Calculated Binding Energies

 

With the aid of the Weizsaecker formula, the binding energy can be calculated very well for nearly all isotopes. This formula provides a good fit for heavier nuclei. For light nuclei, especially for ⁴He, it provides a poor fit. The main reason is the formula does not consider the internal shell structure of the nucleus.

The coefficients a_V, a_S, a_C, a_A, and a_P must be known to calculate the binding energy. The coefficients have units of megaelectronvolts (MeV) and are calculated by fitting to experimentally measured masses of nuclei. They usually vary depending on the fitting methodology. According to ROHLF, J. W., Modern Physics from α to Z0, Wiley, 1994., the coefficients in the equation are following:

Using the Weizsaecker formula, also the mass of an atomic nucleus can be derived and is given by:

m = Z.m_p + N.m_n – E_b/c²

where m_p and m_n are the rest mass of a proton and a neutron, respectively, and E_b is the nuclear binding energy of the nucleus.

From the nuclear binding energy curve and from the table, it can be seen that, in the case of splitting a ²³⁵U nucleus into two parts, the binding energy of the fragments (A ≈ 120) together is more significant than that of the original ²³⁵U nucleus.
According to the Weizsaecker formula, the total energy released for such a reaction will be approximately 235 x (8.5 – 7.6) ≈ 200 MeV.

See also: Liquid Drop Model

Critical Energy – Threshold Energy for Fission

In principle, any nucleus, if brought into a sufficiently high excited state, can be split. For fission to occur, the excitation energy must be above a particular value for a certain nuclide. The minimum excitation energy required for fission to occur is known as the critical energy (E_crit) or threshold energy.

The critical energy depends on the nuclear structure and is significant for light nuclei with Z < 90. For heavier nuclei with Z > 90, the critical energy is about 4 to 6 MeV for A-even nuclei and generally is much lower for A-odd nuclei. It must be noted, some heavy nuclei (e.g.,, ²⁴⁰Pu or ²⁵²Cf) exhibit fission even in the ground state (without externally added excitation energy). This phenomenon is known as spontaneous fission. This process occurs without adding the critical energy by the quantum-mechanical process of quantum tunneling through the Coulomb barrier (similarly to alpha particles in the alpha decay). Spontaneous fission ensures sufficient neutron flux on source range detectors when the reactor is subcritical in long-term shutdown.

See also: Critical Energy – Threshold Energy for Fission.

The minimum excitation energy required for fission to occur is known as the critical energy (E_crit) or threshold energy.

This table shows critical energies compared to binding energies of the last neutron of a number of nuclei.

Energy Release from Fission

In general, nuclear fission results in the release of enormous quantities of energy. The amount of energy depends strongly on the nucleus fissioned and depends strongly on an incident neutron’s kinetic energy. To calculate the reactor’s power, it is necessary to accurately identify the individual components of this energy. At first, it is important to distinguish between the total energy released and the energy that can be recovered in a reactor.

The total energy released in fission can be calculated from binding energies of the initial target nucleus to be fissioned and binding energies of fission products. But not all the total energy can be recovered in a reactor. For example, about 10 MeV is released in the form of neutrinos (in fact antineutrinos). Since the neutrinos are weakly interacting (with extremely low cross-section of any interaction), they do not contribute to the energy that can be recovered in a reactor.

To understand this issue, we have first to investigate a typical fission reaction such as the one listed below.

Using this picture, we can identify and describe almost all the individual components of the total energy released during the fission reaction.

The total energy released in a reactor is about 210 MeV per ²³⁵U fission, distributed as shown in the table. The average recoverable energy per fission in a reactor is about 200 MeV (the total energy minus the energy of antineutrinos radiated away). This means that about 3.1⋅10¹⁰ fissions per second are required to produce a power of 1 W. Since 1 gram of any fissile material contains about 2.5 x 10²¹ nuclei, the fissioning of 1 gram of fissile material yields about one megawatt-day (MWd) of heat energy.

As can be seen from the description of the individual components of the total energy released during the fission reaction, there is a significant amount of energy generated outside the nuclear fuel (outside fuel rods). Especially the kinetic energy of prompt neutrons is primarily generated in the coolant (moderator). This phenomenon needs to be included in the nuclear calculations.

For LWR, it is generally accepted that about 2.5% of total energy is recovered in the moderator. This fraction of energy depends on the materials, their arrangement within the reactor, and thus on the reactor type.

Fission Fragments – Products of Nuclear Fission

Nuclear fission fragments are the fragments left after a nucleus fissions. Typically, when the uranium 235 nucleus undergoes fission, the nucleus splits into two smaller nuclei, along with a few neutrons and release of energy in the form of heat (kinetic energy of these fission fragments) and gamma rays. The average of the fragment mass is about 118, but very few fragments near that average are found. It is much more probable to break up into unequal fragments, and the most probable fragment masses are around mass 95 (Krypton) and 137 (Barium).

Most of these fission fragments are highly unstable (radioactive) and undergo further radioactive decays to stabilize themselves. Fission fragments interact strongly with the surrounding atoms or molecules traveling at high speed, causing them to ionize.

See also: Interaction of Heavy Charged Particles with Matter

Prompt and Delayed Neutrons

It is known the fission neutrons are of importance in any chain-reacting system. Neutrons trigger the nuclear fission of some nuclei (²³⁵U, ²³⁸U, or even ²³²Th). What is crucial the fission of such nuclei produces 2, 3, or more free neutrons.

But not all neutrons are released at the same time following fission. Even the nature of the creation of these neutrons is different. From this point of view, we usually divide the fission neutrons into two following groups:

• Prompt Neutrons. Prompt neutrons are emitted directly from fission, and they are emitted within a very short time of about 10^-14 seconds.
• Delayed Neutrons. Delayed neutrons are emitted by neutron-rich fission fragments that are called delayed neutron precursors. These precursors usually undergo beta decay, but a small fraction of them are excited enough to undergo neutron emission. The neutron is produced via this type of decay, which happens orders of magnitude later than the emission of the prompt neutrons, which plays a crucial role in the control of the reactor.

See also: Prompt Neutrons

See also: Delayed Neutrons

See also: Reactor control with and without delayed neutrons – Interactive chart.

Source: JANIS (Java-based Nuclear Data Information Software); The JEFF-3.1.1 Nuclear Data Library

Key Characteristics of Prompt Neutrons

• Prompt neutrons are emitted directly from fission, and they are emitted within a very short time of about 10^-14 seconds.

• Most of the neutrons produced in fission are prompt neutrons – about 99.9%.

• For example, fission of ²³⁵U by thermal neutron yields 2.43 neutrons, of which 2.42 neutrons are prompt neutrons, and 0.01585 neutrons are the delayed neutrons.

• The production of prompt neutrons slightly increases with incident neutron energy.

• Almost all prompt fission neutrons have energies between 0.1 MeV and 10 MeV.

• The mean neutron energy is about 2 MeV. The most probable neutron energy is about 0.7 MeV.

• In reactor design, the prompt neutron lifetime (PNL) belongs to key neutron-physical characteristics of the reactor core.

• Its value depends primarily on the type of the moderator and the energy of the neutrons causing fission.

• In an infinite reactor (without escape), prompt neutron lifetime is the sum of the slowing downtime and the diffusion time.

• In LWRs, the PNL increases with the fuel burnup.

• The typical prompt neutron lifetime in thermal reactors is on the order of 10^-4 seconds.

• The typical prompt neutron lifetime in fast reactors is on the order of 10^-7 seconds.

Key Characteristics of Delayed Neutrons

• The presence of delayed neutrons is perhaps the most important aspect of the fission process from reactor control.

• Delayed neutrons are emitted by neutron-rich fission fragments that are called delayed neutron precursors.

• These precursors usually undergo beta decay, but a small fraction of them are excited enough to undergo neutron emission.

• The emission of neutrons happens orders of magnitude later compared to the emission of the prompt neutrons.

• About 240 n-emitters are known between ⁸He and ²¹⁰Tl. About 75 of them are in the non-fission region.

• The grouping of precursors based on their half-lives is designed to simplify the kinetic calculations of the reactor.

• Therefore delayed neutrons are traditionally represented by six delayed neutron groups.

• Neutrons can also be produced in (γ, n) reactions (especially in reactors with heavy water moderator), and therefore, they are usually referred to as photoneutrons. Photoneutrons are usually treated no differently than regularly delayed neutrons in the kinetic calculations.

• The total yield of delayed neutrons per fission, v_d, depends on:
  • An isotope that is fissioned.
  • The energy of a neutron induces fission.

• Variation among individual group yields is much more significant than variation among group periods.

• In reactor kinetic calculations, it is convenient to use relative units usually referred to as delayed neutron fraction (DNF).

• At the steady-state condition of criticality, with k_eff = 1, the delayed neutron fraction is equal to the precursor yield fraction β.

• In LWRs, the β decreases with fuel burnup. This is due to isotopic changes in the fuel.

• Delayed neutrons have initial energy between 0.3 and 0.9 MeV with an average energy of 0.4 MeV.

• Depending on the type of the reactor, and their spectrum, the delayed neutrons may be more (in thermal reactors) or less effective than prompt neutrons (in fast reactors). To include this effect into the reactor kinetic calculations, the effectively delayed neutron fraction – β_eff must be defined.

• The effectively delayed neutron fraction is the product of the average delayed neutron fraction and the importance factor β_eff = β . I.

• The weighted delayed generation time is given by τ = ∑_iτ_i . β_i / β = 13.05 s, therefore the weighted decay constant λ = 1 / τ ≈ 0.08 s^-1.

• The mean generation time with delayed neutrons is about ~0.1 s, rather than ~10^-5 as in section Prompt Neutron Lifetime, where the delayed neutrons were omitted.

• Their presence completely changes the dynamic time response of a reactor to some reactivity change, making it controllable by control systems such as the control rods.

Capture-to-Fission Ratio

The probability that a neutron absorbed in a fissile nuclide causes a
fission is an important parameter of each fissile isotope. In terms of cross-sections, this probability is defined as:

σ_f / (σ_f + σ_γ) = 1 / (1 + σ_γ/σ_f) = 1 / (1 + α),

where α = σ_γ/σ_f is referred to as the capture-to-fission ratio. The capture-to-fission ratio may be used as an indicator of the “quality” of fissile isotopes. The lower C/F ratio simply means that an absorption reaction will result in fission rather than radiative capture. The ratio depends strongly on the incident neutron energy. In the fast neutron region, the C/F ratio decreases. It is determined by the steeper decrease in radiative capture cross-section (see chart).

For ²³⁵U and ²³³U, the thermal neutron capture-to-fission ratios are typically lower than fast neutrons (for mean energy of about 100 keV). It must be noted, the neutron flux of most fast reactors tends to peak around 200 keV, but the mean energy is between 100-200 keV depending on specific reactor design.

Further increase in neutron energy causes a decrease in the C/F ratio conversely. This is not the case with ²³⁹Pu. For 100 keV neutrons, the C/F ratio is lower than for thermal neutrons. A small capture-to-fission ratio is advantageous for the fissile isotopes (²³³U, ²³⁵U, and ²³⁹Pu) because neutrons captured onto them are lost.

Source: JANIS (Java-based Nuclear Data Information Software); The JEFF-3.1.1 Nuclear Data Library

Nuclear Fission Chain Reaction

The chain reaction can take place only in the proper multiplication environment and only under proper conditions. Obviously, suppose one neutron causes two further fissions. In that case, the number of neutrons in the multiplication system will increase in time, and the reactor power (reaction rate) will also increase in time. Moreover, this multiplication environment (nuclear reactor) behaves like an exponential system, which means the power increase is not linear but exponential. The non-fission neutron absorption in the system (e.g.,, insert control rods) must increase to stabilize such a multiplication environment.

On the other hand, if one neutron causes less than one further fission, the number of neutrons in the multiplication system will decrease in time, and the reactor power (reaction rate) will also decrease in time.  The non-fission neutron absorption in the system (e.g.,, to withdraw control rods) must be decreased to sustain the chain reaction.

There is always a competition for the fission neutrons in the multiplication environment. Some neutrons will cause further fission reaction, fuel materials or non-fuel materials will capture some, and some will leak out of the system.

The infinite and finite multiplication factors of a reactor must be defined to describe a multiplication system. The method of calculations of multiplication factors was developed in the early years of nuclear energy. It is only applicable to thermal reactors, where the bulk of fission reactions occurs at thermal energies. This method puts into context all the processes associated with the thermal reactors (e.g.,, neutron thermalization, neutron diffusion, or fast fission) because the most critical neutron-physical processes occur in energy regions that can be clearly separated from each other. In short, the calculation of the multiplication factor gives a good insight into the processes that occur in each thermal multiplying system.

The distinction between Fissionable, Fissile, and Fertile

In nuclear engineering, fissionable material (nuclide) is material that is capable of undergoing fission reaction after absorbing either thermal (slow or low energy) neutron or fast (high energy) neutron. Fissionable materials are a superset of fissile materials. Fissionable materials also include an isotope ²³⁸U that can be fissioned only with high energy (>1MeV) neutron. These materials are used to fuel thermal nuclear reactors because they can sustain a nuclear fission chain reaction.

Fissile materials undergo a fission reaction after absorption of the binding energy of the thermal neutron. They do not require additional kinetic energy for fission. If the neutron has higher kinetic energy, this energy will be transformed into the additional excitation energy of the compound nucleus. On the other hand, the binding energy released by the compound nucleus of (²³⁸U + n) after thermal neutron absorption is less than the critical energy, so the fission reaction cannot occur. The distinction is described in the following points.

• Fissile materials are a subset of fissionable materials.
• Fissionable material consists of isotopes that are capable of undergoing nuclear fission after capturing either fast neutron (high energy neutron – let say >1 MeV) or thermal neutron (low energy neutron – let say 0.025 eV).   Typical fissionable materials: ²³⁸U, ²⁴⁰Pu, but also ²³⁵U, ²³³U, ²³⁹Pu, ²⁴¹Pu
• Fissile material consists of fissionable isotopes capable of undergoing nuclear fission only after capturing a thermal neutron. Typical fissile materials: 235U, 233U, 239Pu, 241Pu. ²³⁸U is not a fissile isotope because a thermal neutron cannot fission 238U. ²³⁸U does not also meet alternative requirements to fissile materials. ²³⁸U cannot sustain a nuclear fission chain reaction because neutrons produced by the fission of ²³⁸U have lower energies than the original neutron (usually below the threshold energy of 1 MeV).
• Fertile material consists of isotopes that are not fissionable by thermal neutrons but can be converted into fissile isotopes (after neutron absorption and subsequent nuclear decay). Typical fertile materials: ²³⁸U, ²³²Th.

Fissile materials undergo a fission reaction after absorption of the binding energy of the thermal neutron. They do not require additional kinetic energy for fission. If the neutron has higher kinetic energy, this energy will be transformed into the additional excitation energy of the compound nucleus. On the other hand, the binding energy released by the compound nucleus of (²³⁸U + n) after absorption of the thermal neutron is less than the critical energy, so the fission reaction cannot occur. The distinction is described in the following points.

• Fissile materials are a subset of fissionable materials.
• Fissionable material consists of isotopes that are capable of undergoing nuclear fission after capturing either fast neutron (high energy neutron – let say >1 MeV) or thermal neutron (low energy neutron – let say 0.025 eV).   Typical fissionable materials: ²³⁸U, ²⁴⁰Pu, but also ²³⁵U, ²³³U, ²³⁹Pu, ²⁴¹Pu
• Fissile material consists of fissionable isotopes capable of undergoing nuclear fission only after capturing a thermal neutron. Typical fissile materials: 235U, 233U, 239Pu, 241Pu. ²³⁸U is not a fissile isotope because a thermal neutron cannot fission 238U. ²³⁸U does not also meet the alternative requirement to fissile materials. ²³⁸U cannot sustain a nuclear fission chain reaction because neutrons produced by the fission of ²³⁸U have lower energies than the original neutron (usually below the threshold energy of 1 MeV).
• Fertile material consists of isotopes that are not fissionable by thermal neutrons but can be converted into fissile isotopes (after neutron absorption and subsequent nuclear decay). Typical fertile materials: ²³⁸U, ²³²Th.

See also: Neutron cross-section.

Comparison of cross-sections

Source: JANIS (Java-based nuclear information software)  http://www.oecd-nea.org/janis/

Fissile / Fertile Material Cross-sections. Comparison of total fission cross-sections.
Uranium 235. Comparison of total fission cross-section and cross-section for radiative capture.

Uranium 238. Comparison of total fission cross-section and cross-section for radiative capture.