**Nuclear fission**is a nuclear reaction in which the nucleus of an atom

**splits**into smaller parts (lighter nuclei). The fission process often produces free neutrons and photons (in the form of gamma rays), and releases a

**large amount of energy**. In nuclear physics, nuclear fission is

**either a nuclear reaction**

**or a radioactive decay process**. The case of decay process is called

**spontaneous fission**and it is very rare process. In this section,

**the neutron-induced nuclear fission**, the process of the greatest practical importance in reactor physics, will be discussed.

When they finally published the results in 1939, they came to the attention of Lise Meitner, an Austrian-born physicist who had worked with Hahn on his nuclear experiments. She was the first to realize that Hahn’s barium and other lighter products from the neutron bombardment experiments were coming from the fission of U-235. Meitner and Frisch carried out further experiments which showed that the U-235 fission can release large amounts of energy both as electromagnetic radiation and as kinetic energy of the fragments (heating the bulk material where fission takes place). They realized that this made possible a chain reaction with an unprecedented energy yield.

## Basics of Nuclear Fission

**undergo fission**on their own spontaneously, but only certain nuclei, like uranium-235, uranium-233 and plutonium-239, can

**sustain a fission chain reaction**. This is because these nuclei release neutrons when they break apart, and these neutrons can induce fission of other nuclei. Free neutrons released by each fission play very important role as a trigger of the reaction.

## Chain reaction

**A nuclear chain reaction** occurs when one single nuclear reaction causes an average of **one or more** subsequent nuclear reactions, thus leading to the possibility of a self-propagating series of these reactions. The “**one or more**” is the key parameter of reactor physics. To raise or lower the power, the amount of reactions must be changed (using the control rods) so that the number of neutrons present (and hence the rate of power generation) is either reduced or increased.

**Nuclear fission**is the main process generating nuclear energy.- Most of the energy (~85%) is released in the form of kinetic energy of the splitted parts.
**Neutrons**trigger the**nuclear fission**.- The fission process produces free neutrons (2 or 3).
- The chain reaction means if the reaction induces
**one or more**reactions. - The probability that fission will occur depends on incident neutron energy.
- Therefore the moderator is used to slow down neutrons (to increase the probability of fission)
- For reactors using light water as moderator, enriched uranium fuel is required.
- Control rods contains material which absorb neutrons (boron, cadmium, …)
- Withdrawal of the rods increases the parameter
**one or more**(multiplication factor), thus increase the power. - Insertion of the rods decreases the parameter
**one or more**(multiplication factor), thus decrease the power. - The multiplication factor is influenced also by other parameters such as temperature, fuel burnup and reactor poisoning.

- The fission takes place in the
**nuclear reactor core**. - The reactor core consists of nuclear fuel, control rods, moderator (coolant) and neutron reflector.

** **

## Youtube animation

## Principles 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 take place. In order 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 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 of absorption reactions result in fission reaction, but a minority results in radiative capture forming ^{236}U. 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 neutron. About 85% of all absorption reactions result in fission.

Uranium 233 is a very good fissile isotope and its fission cross-sectionfor 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 of absorption reactions result in fission reaction, but a minority results in radiative capture forming ^{234}U. 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 neutron. About 94% of all absorption reactions result in fission. The capture-to-fission ratio is much smaller than the other two major fissile fuels ^{235}U and ^{239}U.

Plutonium 239 is a fissile isotope and its fission cross-section forthermal neutrons is about **750 barns** (for 0.025 eV neutron). **For fast neutrons** its fission cross-section is **on the order of barns**. Most of absorption reactions result in fission reaction, but a part of reactions result in radiative capture forming ^{240}Pu. 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

**A binding energy**is generally the energy required to

**disassemble**a whole system into separate parts. It is known the sum of separate parts has typically a higher potential energy than a bound system, therefore the

**bound system is more stable**. A creation of bound system is often accompanied by subsequent energy release. We usually distinguish the binding energy according to these levels:

- Atomic level
- Molecular level
- Nuclear level

**nuclear binding energy**is the energy 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**which holds the nucleus together. According to the Einstein relationship (E=m.c

_{b})^{2}) this binding energy is proportional to this mass difference and it is known as

**the mass defect**.

During the **nuclear splitting** or nuclear fusion, some of the mass of the nucleus gets **converted** into huge amounts of energy and 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**, 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 them mass number A. As can be seen, for low mass numbers

**E**increases rapidly and reaches a maximum of

_{b}/A**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 (

^{56}Fe,

^{58}Fe,

^{62}Ni).

**After that, the binding energy per nucleon decreases**. In the heavy nuclei (A>60) region, a more stable configuration is obtained, 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. In order to fission process to take place, a

**sufficient amount of energy must be added to the nucleus**and no matter how. The energetics and binding energies of certain nucleus are well described by

**the Liquid Drop Model**, which examines the global properties of nuclei.

**mass defect**of the

**3000MW**reactor core after one year of operation.

_{th}It is known the average recoverable energy per fission is about **200 MeV**, being the total energy minus the energy of the energy of antineutrinos that are radiated away.

The **reaction rate** per entire **3000MW _{th}** reactor core is about

**9.33×10**.

^{19}fissions / second**The overall energy release** in the units of joules is:

200×10^{6} (eV) x 1.602×10^{-19} (J/eV) x 9.33×10^{19} (s^{-1}) x 31.5×10^{6} (seconds in year) = **9.4×10 ^{16} J/year**

The mass defect is calculated as:

Δm = ΔE/c^{2}

**Δm** = 9.4×10^{16} / (2.9979 x 10^{8})^{2} = **1.046 kg**

That means in a typical **3000MWth** reactor core about 1 kilogram of matter is **converted** into pure energy.

Note that, a typical annual uranium load for a **3000MWth **reactor core is about **20 tonnes** of **enriched uranium **(i.e. about**22.7 tonnes of UO _{2}**). Entire reactor core may contain about 80 tonnes of enriched uranium.

### Mass defect directly from E=mc^{2}

The mass defect can be calculated directly from the Einstein relationship (**E = mc ^{2}**) as:

Δm = ΔE/c^{2}

Δm = 3000×10^{6} (W = J/s) x 31.5×10^{6} (seconds in year) / (2.9979 x 10^{8})^{2 }= **1,051 kg**

## Liquid Drop Model

**von Weizsaecker**(also called

**the semi-empirical mass formula – SEMF**), that was 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), which are held together by **the strong nuclear force**. The liquid drop model of the nucleus takes into account the fact that the nuclear forces on the nucleons on the surface are different from those on nucleons in the interior of the nucleus. The **interior nucleons are completely surrounded** by other attracting nucleons. Here is the analogy with the forces that form a drop of liquid.

In the ground state the nucleus is **spherical**. If the sufficient kinetic or binding energy is added, this spherical nucleus may be distorted into a **dumbbell shape** and then may be splitted into **two fragments**. Since these fragments are a more stable configuration, 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 based on **the liquid drop model** 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 – a**The first two terms describe a spherical liquid drop of an incompressible fluid with a contribution from the volume scaling with A and from the surface, scaling with A

_{V}.A.^{2/3}. The first positive term

**a**is known as the volume term and it is caused by the attracting strong forces between the nucleons. The

_{V}.A**strong force**has a

**very limited range**and a given nucleon may only interact with its

**direct neighbours**. Therefore this term is proportional to A, instead of A

^{2}. The coefficient

**a**is usually about ~ 16 MeV.

_{V}**Surface term – a**The surface term is also based on the strong force, it is, in fact, a

_{sf}.A^{2/3}.**correction to the volume term**. The point is that particles at the surface of the nucleus are not completely surrounded by other particles. In the volume term, it is suggested that each nucleon interacts with a constant number of nucleons, independent of A. This assumption is very 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}and therefore the surface term must be proportional to the surface area i.e. proportional to A

^{2/3}.

**Coulomb term – a**This term describes the Coulomb repulsion between the uniformly distributed protons and is proportional to the number of proton pairs

_{C}.Z^{2}.A^{-⅓}.**Z**, whereby R is proportional to A

^{2}/R^{1/3}.

**This effect lowers the binding energy**because of the repulsion between charges of equal sign.

**Asymmetry term – a**This term cannot be described as ‘classically’ as the first three. This effect is not based on any of the fundamental forces, this effect is based only on

_{A}.(A-2Z)^{2}/A.**the Pauli exclusion principle**(no two fermions can occupy exactly the same quantum state in an atom). The heavier nuclei contain more neutrons than protons. These extra neutrons are necessary for 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 – 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 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.

**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

^{4}He, it provides a poor fit. The main reason is the formula does not consider the internal shell structure of the nucleus.

In order to calculate the binding energy, the coefficients a_{V}, a_{S}, a_{C}, a_{A} and a_{P} must be known. 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}/c^{2}**

where **m _{p}** and

**m**are the rest mass of a proton and a neutron, respectively, and

_{n}**E**is the nuclear binding energy of the nucleus.

_{b}From the nuclear binding energy curve and from the table it can be seen that, in the case of splitting a ^{235}U nucleus into two parts, the binding energy of the fragments (A ≈ 120) together is larger than that of the original ^{235}U nucleus.

According to the Weizsaecker formula, the total energy released for such reaction will be approximately **235 x (8.5 – 7.6) ≈ 200 MeV.**

See also: Liquid Drop Model

## Critical Energy – Threshold Energy for Fission

**any nucleus**, if brought into sufficiently

**high excited state**,

**can be splitted**. For fission to occur, the excitation energy must be

**above a particular value**for certain nuclide. The minimum excitation energy required for fission to occur is known as

**the critical energy (E**or

_{crit})**threshold energy**.

The critical energy depends on the nuclear structure and is quite large 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 (**eg. ^{240}Pu or ^{252}Cf**) exhibit fission even

**in the ground state**(without externally added excitation energy). This phenomena is known as

**the spontaneous fission**. This process occur without the addition of the critical energy by the quantum-mechanical process of

**quantum tunneling**through the Coulomb barrier (similarly like alpha particles in the alpha decay).

**The spontaneous fission**contributes to ensure sufficient neutron flux on source range detectors when reactor is

**subcritical**in long term shutdown.

## Energy Release from Fission

**nuclear fission**results in the release of

**enormous quantities of energy**. The amount of energy

**depends strongly on**the nucleus to be fissioned and also depends strongly on the kinetic energy of an incident neutron. In order to calculate the power of a reactor, it is necessary to be able precisely 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 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.

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

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

^{235}U have masses of about 94 and 139.

**The largest part of the energy** produced during fission (about 80 % or about 170 MeV or about 27 picojoules) appears as **kinetic energy of the fission fragments**. The fission fragments interact strongly (intensely) with the surrounding atoms or molecules traveling at high speed, **causing them to ionize**. Creation of ion pairs requires energy, which is lost from the kinetic energy of the charged fission fragment causing it to decelerate. The positive ions and free electrons created by the passage of the charged fission fragment will then reunite, releasing energy** in the form of heat** (e.g. vibrational energy or rotational energy of atoms).

The range of these massive, highly charged particles in the fuel is **of the order of micrometers**, so that the recoil energy is effectively deposited as heat at the point of fission. This is the principle how fission fragments heat up fuel in the reactor core.

See also: Interaction of Heavy Charged Particles with Matter

**Prompt neutrons**are emitted directly from fission and they are emitted within very short time of

**about 10**. Usually more than

^{-14}second**99 percent**of the fission neutrons are the

**prompt neutrons**, but the exact fraction is dependent on the nuclide to be fissioned and is also dependent on an incident neutron energy (usually increases with energy).

For example a fission of ^{235}U by thermal neutron yields **2.43 neutrons**, of which **2.42 neutrons are the prompt neutrons** and 0.01585 neutrons (**0.01585/2.43=0.0065=ß**) are the delayed neutrons. 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**.

Most of this energy is deposited in the coolant (moderator), because the water have the highest macroscopic slowing down power (MSDP) of the materials that are in a reactor core (PWR). The range of neutrons in a reactor depends strongly on certain reactor type, in the case of PWRs it is usually of the order of centimeters.

**prompt gamma rays**are associated.

**Most of prompt gamma rays are emitted after prompt neutrons.**The fission reaction releases approximately

**~7 MeV**in prompt gamma rays.

The gamma rays are well attenuated by** high-density** and **high Z materials**. In a reactor core the largest share of the energy will be deposited **in the fuel **containing uranium dioxide, but a significant share of the energy will be deposited also in the fuel cladding and in the coolant (moderator).

The range of gamma rays in a reactor vary according to the initial energy of the gamma ray. It can be stated the most of gammas in a reactor have range from 10cm-1m.

**About 6 MeV**of fission energy is in the form of

**kinetic energy of electrons**(beta particles). The fission fragments are

**neutron-rich nuclei**and therefore they usually undergo

**beta decay**in order to stabilize itself. Beta particles deposit their energy essentially

**in the fuel element**, within about 1 mm of the fission fragment.

**negative beta decay**. In a nuclear reactor occurs especially the β− decay, because the common feature of the fission fragments is an

**excess of neutrons**. The existence of emission of antineutrinos and their

**extremely low cross-section**for any interaction leads to very interesting phenomenon. Roughly

**about 5% of released energy**per one fission is

**radiated away**from reactor in the form of antineutrinos.

For a typical nuclear reactor with a thermal power of **3000 MWth** (~1000MWe of electrical power), the total power produced is in fact higher, approximately **3150 MW**, of which 150 MW is radiated away into space as antineutrino radiation. This amount of energy is **forever lost**, because antineutrinos are able to penetrate all reactor materials without any interaction.

In fact, a common statement in physics texts is that **the mean free path** of a neutrino is approximately **a light-year of lead**. Moreover, a neutrino of moderate energy can easily penetrate **a thousand light-years of lead** (according to the J. B. Griffiths).

**neutron-rich**and

**very unstable nuclei.**These nuclei undergo

**many beta decays**in order to stabilize itself. Gamma rays usually

**accompany**the beta decay. Their energy is transferred as heat to the surrounding material similarly as the energy carried by

**prompt γ-rays**.

**radiative capture**followed by

**gamma ray emission**, producing on average

**about 7 MeV per fission**in the form of energetic gamma rays. Their energy is transferred as heat to the surrounding material similarly as the energy carried by prompt γ-rays.

**The total energy released**in a reactor is

**about 210 MeV**per

^{235}U fission, distributed as shown in the table. In a reactor,

**the average recoverable energy**per fission is

**about 200 MeV**, being the total energy minus the energy of the energy of antineutrinos that are radiated away. This means that

**about 3.1⋅10**are required to produce a power of

^{10}fissions per second**1 W**. Since

**1 gram**of any fissile material contains about

**2.5 x 10**, the fissioning of 1 gram of fissile material yields

^{21}nuclei**about 1 megawatt-day (MWd)**of heat energy.

As can be seen from the description of the individual components of the total energy energy released during the fission reaction, there is **significant amount of energy generated outside the nuclear fuel** (outside fuel rods). Especially the kinetic energy of prompt neutrons is largely generated** in the coolant (moderator)**. This phenomena 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 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 the 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 itself. 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

^{235}U,

^{238}U or even

^{232}Th). 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 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**very short time of about 10**.^{-14}second**Delayed Neutrons.**Delayed neutrons are emitted by**neutron rich fission fragments**that are called**the delayed neutron precursors**. These precursors usually undergo beta decay but a small fraction of them are excited enough to undergo**neutron emission**. The fact the neutron is produced via this type of decay and this happens**orders of magnitude later**compared to the emission of the prompt neutrons, plays an extremely important 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

## Key Characteristics of Prompt Neutrons

- Prompt neutrons are emitted
**directly from fission**and they are emitted within very short time of about**10**.^{-14}second

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

- For example a fission of
^{235}U 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 increase 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 reactor core.

- Its value depends especially on the type of the moderator and on the energy of the neutrons causing fission.

- In an infinite reactor (without escape) prompt neutron lifetime is the sum of the
**slowing down time 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**second.^{-4}

- The typical prompt neutron lifetime in fast reactors is on the order of
**10**second.^{-7}

## Key Characteristics of Delayed Neutrons

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

- Delayed neutrons are emitted by neutron rich fission fragments that are called the
**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 neutron happens orders
**of magnitude later**compared to the emission of the prompt neutrons.

- About
**240 n-emitters**are known between^{8}He and^{210}Tl, about 75 of them are in the non-fission region.

- In order to simplify reactor kinetic calculations it is suggested
**to group together the precursors**based on their half-lives.

- Therefore delayed neutrons are traditionally represented by
**six delayed neutron groups**.

- Neutrons can be produced also 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 regular delayed neutrons in the kinetic calculations.

- The total yield of delayed neutrons per fission, v
_{d}, depends on:- Isotope, that is fissioned.
- Energy of a neutron that induces fission.

- Variation among individual group yields is much greater 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). In order to include this effect into the reactor kinetic calculations**the effective delayed neutron fraction – β**must be defined._{eff}

- The effective 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 = ∑**, therefore the weighted decay constant_{i}τ_{i}. β_{i}/ β = 13.05 s**λ = 1 / τ ≈ 0.08 s**.^{-1}

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

- 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

fission is very 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 “quality” of fissile isotopes. The lower

**C/F ratio**simply means that an absorption reaction will result in the fission rather than in the radiative capture. The ratio depends strongly on the incident neutron energy. In the fast neutron region, C/F ratio decreases. It is determined by the steeper decrease in radiative capture cross-section (see chart).

For ^{235}U and ^{233}U the thermal neutron **capture-to-fission ratios** are typically lower than those for 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 certain reactor design.

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

## Nuclear Fission Chain Reaction

**The chain reaction**can take place only in the

**proper**

**multiplication environment**and only under

**proper conditions**. It is obvious, if one neutron causes two further fissions, the number of neutrons in the multiplication system will increase in time and the reactor power (reaction rate) will also increase in time. In order to stabilize such multiplication environment, it is necessary to increase the non-fission neutron absorption in the system (e.g. to

**insert control rods**). Moreover, this multiplication environment (nuclear reactor) behaves like the exponential system, that means the power increase is not linear, but it is

**exponential**.

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. In order to **sustain the chain reaction**, it is necessary to decrease the non-fission neutron absorption in the system (e.g. to **withdraw control rods**).

In fact, there is always a** competition** for the fission neutrons in the multiplication environment, some neutrons will cause further **fission reaction**, some will be **captured** by fuel materials or non-fuel materials and some will** leak out** of the system.

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

## Distinction between Fissionable, Fissile and Fertile

**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

**that can be fissioned only with high energy (>1MeV) neutron. These materials are used to fuel thermal nuclear reactors, because they are capable of sustaining a nuclear fission chain reaction.**

^{238}UFissile materials undergoes fission reaction after absorption of the binding energy of thermal neutron. They do not require additional kinetic energy for fission. If the neutron has higher kinetic energy, this energy will be transformed into additional excitation energy of the compound nucleus. On the other hand, the binding energy released by compound nucleus of **( ^{238}U + n)** after absorption of 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**consist 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:,^{238}U, but also^{240}Pu,^{235}U,^{233}U,^{239}Pu^{241}Pu**Fissile material**consist of fissionable isotopes that are capable of undergoing nuclear fission**only after capturing a thermal neutron**.is not fissile isotope, because^{238}Ucannot be fissioned by thermal neutron.^{238}Udoes not meet also alternative requirement to fissile materials.^{238}Uis not capable of sustaining a nuclear fission chain reaction, because neutrons produced by fission of^{238}Uhave lower energies than original neutron (usually below the threshold energy of 1 MeV). Typical fissile materials:^{238}U,^{235}U,^{233}U,^{239}Pu.^{241}Pu**Fertile material**consist 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:,^{238}U.^{232}Th

**(**after absorption of thermal neutron is less than the critical energy, so the fission reaction

^{238}U + n)**cannot occur**. The distinction is described in the following points.

**Fissile materials are a subset of fissionable materials**.**Fissionable material**consist 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:,^{238}U, but also^{240}Pu,^{235}U,^{233}U,^{239}Pu^{241}Pu**Fissile material**consist of fissionable isotopes that are capable of undergoing nuclear fission**only after capturing a thermal neutron**.is not fissile isotope, because^{238}Ucannot be fissioned by thermal neutron.^{238}Udoes not meet also alternative requirement to fissile materials.^{238}Uis not capable of sustaining a nuclear fission chain reaction, because neutrons produced by fission of^{238}Uhave lower energies than original neutron (usually below the threshold energy of 1 MeV). Typical fissile materials:^{238}U,^{235}U,^{233}U,^{239}Pu.^{241}Pu**Fertile material**consist 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:,^{238}U.^{232}Th

See also: Neutron cross-section

**Comparison of cross-sections**

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