**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**.

In principle, **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 _{crit})** or

**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.**) exhibit fission even

^{240}Pu or^{252}Cf**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.

## Theory of Critical Energy

The amount of **excitation energy** required for fission to occur can be estimated from the magnitude of **the electrostatic potential barrier** and **the dissociation energy** of the fission.

In the figure, it is considered the potential energy of the fissioning nucleus as a function of the **distance r** between the **two separating lobes**. In order to deform the nucleus into a dumbbell shape, the sufficient kinetic or binding energy must be added to the system. This is due to the fact, nucleons attract one another and energy is required to increase the average distance between them. After the energy is added, the resulting nucleus is in an intermediate state with a** larger potential energy** than was the potential energy of original nucleus. The height of the potential barrier may be approximated given by:

**E _{c} = Z_{1}.Z_{2}.e^{2} / (4.π.ε_{0}.(R1+R2))**

where R1 and R2 are the respective nuclear radii and **ε _{0}** is the permittivity of the vacuum. If this point is reached, the two lobes of the dumbbell

**begin to separate**.

**The dissociation energy E _{d}** is equal to the difference between the binding energy of the compound nucleus to be fissioned and the sum of the binding energies of the fission fragments. The minimum activation energy

**E**that has to be added to a nucleus to cause fission reaction is thus

_{a}**E**.The minimum excitation energy required for fission to occur is known as the

_{c}– E_{d}**critical energy (E**or

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

**The activation energy E**of nuclides with mass numbers

_{a}**below about 230 is very large**, spontaneous fission of these nuclides does not occur. On the other hand nuclei with atomic numbers

**A > 260 have negative activation energies**, so that these nuclei must undergo a decay or they can undergo spontaneous fission. The following table shows

**critical energies**compared to

**binding energies of the last neutron**of a number of nuclei.

All fissioning nuclei **do not split in the same way**. Although the mass of the initial nucleus is well defined in a reaction, the masses of resulting fission fragments are not. This is the reason there is **no single Q-value**, but what is usually referred to as the fission Q-value is actually an average of Q-values over all ways of fission.This table shows critical energies compared to binding energies of the last neutron of a number of nuclei.For nuclei lighter than uranium the **critical energies** are considerably higher (e.g. **E _{c} ~ 20MeV for ^{208}Pb**). This is the reason only the heaviest nuclei are of importance in

**nuclear engineering**.

**The excitation energy** can be added to a nucleus by a neutron, but it is not the only way, **how to induce fission**. The excitation energy can be added also **by bombardment with photons** (**photofission**) or **charged particles**. In reactor engineering, the most attractive method of causing fission, however, is by forming a compound nucleus with the aid of a neutron.

In the case of** neutron-induced fission reactions,** an incident neutron provides additional energy to a target nucleus in the form of **the kinetic energy** and **the nuclear binding energy**. Neutrons have the principal advantage, they **do not need to overcome the coulomb forces** as in the case of charged particles.

It can be seen from the table that for fission of ** ^{238}U** or

**the neutron must have some additional kinetic energy (**

^{232}Th**negative BE**), while absorption of a neutron without kinetic energy can already cause fission of

_{n}– E_{crit}value**. For example, according to the table , the binding energy of the last neutron in**

^{235}U (or^{233}U,^{239}Pu)**is**

^{236}U**6.8 MeV**(

**target nucleus is**), while the critical energy is only

^{235}U**6.5 MeV**. Thus, when a thermal neutron is absorbed by

^{235}U, the

**compound nucleus**is produced at about

^{236}U**0.3 MeV above the critical energy**and the nucleus

**splits immediately**. Nuclei such as

^{235}U that lead to fission following the absorption of thermal neutron are called fissile nuclei.

For heavy nuclides with atomic number of **higher than 90**, most of fissile isotopes meet** the fissile rule**:

Fissile isotopes have **2 x Z – N = 43 ± 2** (example for **235U**: 2 x 92 – 143 = 41)

where** Z is number of protons** and **N is number of neutrons**.

In general, the heavy nuclei with an odd number of neutrons (^{235}U, ** ^{233}U, ^{239}Pu and ^{241}Pu**) can easily be split, because the neutron that is absorbed to form a compound nucleus with these nuclei is

**an ‘even’ neutron**, so that the

**binding energy due to the pairing effect is large**.

On the other hand, the heavy nuclei with an even number of neutrons (** ^{232}Th, ^{238}U, ^{240}Pu and ^{242}Pu**) have a

**threshold energy**(the kinetic energy) for fission by neutrons, because the absorbed neutron is an

**‘odd’ neutron**and this neutron makes relatively

**little binding energy available**. For these nuclides fission is thus

**a threshold reaction**.

For example, according to the table, the **binding energy of the last neutron in ^{239}U** is only

**5.5 MeV**(target nucleus is

^{238}U), while the critical energy is

**7.0 MeV**. Thus, when a thermal neutron is absorbed by

^{238}U,

**fission cannot occur**. In order to cause fission of

^{238}U, the incident neutron

**must have additional kinetic energy**. Nuclei such as

^{238}U that lead to fission following the absorption of fast neutron are called fissionable nuclei. It must be noted, also a

**when struck by a high energy neutron of**

^{208}Pb nucleus may be fissioned**about 20MeV**, but this nucleus is not ordinarily said to be fissionable.

When a nucleus is excited above the potential barrier, **it is not sure that fission will occur**. Most of absorption reactions result in fission reaction (**σ _{f} = 585 barns**), but a minority results in

**radiative capture**forming

**. The radiative capture is a reaction, in which the compound nucleus decays to its ground state by gamma emission.The cross-section for radiative capture for thermal neutrons is about**

^{236}U**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.

See also: Uranium 235 Fission

**See also: Plutonium 239 Fission**