# Difference between revisions of "Nucleus Recoil-Energy in Neutron Capture Reactions"

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− | A nucleus which captures a thermal neutron must, since the momentum is conserved, receive a recoil energy. Immediately after capturing a neutron, the nucleus will emit γ quantas to get rid of the excess energy liberated when the neutron is bound to the nucleus. This also result in a certain amount of recoil energy on the nucleus. <br> | + | A nucleus which captures a thermal neutron must, since the momentum is conserved, receive a recoil energy. Immediately after capturing a neutron, the nucleus will emit γ quantas to get rid of the excess energy liberated when the neutron is bound to the nucleus (usually refered to as "promt γ to distinguis it from the "normal" γs which is emitted after the nucleus has disintegrated). This also result in a certain amount of recoil energy on the nucleus. <br> |

==== Recoil energy from n-capture<br> ==== | ==== Recoil energy from n-capture<br> ==== | ||

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The conservation of momentum demands that<br> | The conservation of momentum demands that<br> | ||

− | <math>\overrightarrow{P}_n + \overrightarrow{P}_T = \overrightarrow{P}_{T+n} = \overrightarrow{P}_R</math> | + | <math>\overrightarrow{P}_n + \overrightarrow{P}_T = \overrightarrow{P}_{T+n} = \overrightarrow{P}_R</math> |

− | where P denotes the momentum, index n denots the neutron, index T the target nucleus, and index R the recoil. | + | where P denotes the momentum, index ''n'' denots the neutron, index ''T'' the target nucleus, and index ''R'' the recoil. |

− | The kinetic energy, E<sub>K</sub>, is given by: | + | The general relationship between kinetic energy, ''E<sub>K</sub>'', and momentum ''p'' is given by: |

− | = | + | <math>E_K = \frac{\overrightarrow{p}^2}{2m}</math> |

− | + | The mass of the neutron is ''1'' (atomic mass unit). the mass of the target nucleus is ''A''. The new nucleus will therefore have mass ''A+1''. Then | |

+ | |||

+ | <math>E_{K,R} = \frac{\overrightarrow{P}^2_R}{2(A+1)} | ||

+ | = \frac{\overrightarrow{P}^2_n m_n}{2 m_n (A+1)} | ||

+ | =\frac{E_{K,n} m_n}{A+1} | ||

+ | =\frac{E_{K,n}}{A+1}</math> | ||

+ | |||

+ | (remember that the momemtun of the target nucleus initially is 0.) | ||

+ | |||

+ | ==== Recoil energy from γ emission<br> ==== | ||

+ | |||

+ | For emission of the mass-less quantas we have the following relationship: | ||

+ | |||

+ | <math>\overrightarrow{P}_R = \overrightarrow{P}_\gamma</math> | ||

+ | |||

+ | and | ||

+ | |||

+ | <math>P_\gamma = \frac{E_\gamma}{c}</math> | ||

+ | |||

+ | In this case the nucleus has mass ''A+1'', then | ||

+ | |||

+ | <math>E_{K,R}=\frac{\overrightarrow{P}^2_R}{2(A+1)} | ||

+ | = \frac{\overrightarrow{P}^2_\gamma c^2}{2(A+1) c^2} | ||

+ | = \frac{E^2_\gamma (MeV)}{2(A+1) 931.5 MeV}</math> | ||

+ | |||

+ | ==== The iodine case<br> ==== | ||

+ | |||

+ | For iodine, A = 127. Thermal neutrons have E<sub>K,n</sub> = 0.025 eV. E<sub>γ</sub> will be around 3 MeV. We then get that for neutron capture the recoil energy is<br> | ||

+ | |||

+ | <math>E_{K,R}= \frac{0.025 eV}{128} = 0.2 \; meV</math> | ||

+ | |||

+ | and for the prompt γ-emission it is<br> | ||

+ | |||

+ | <math>E_{K,R} = \frac{9 \cdot 10^6 eV}{2 \cdot 128 \cdot 931.5 \; MeV} = 38 \; eV</math> | ||

+ | |||

+ | Since chemical binding-energies typically are between 0.1 to 1 eV, the recoil from γ emission is large enough to break chemical bounds. However, the recoil from n capture is not. <br> |

## Latest revision as of 14:53, 14 November 2012

A nucleus which captures a thermal neutron must, since the momentum is conserved, receive a recoil energy. Immediately after capturing a neutron, the nucleus will emit γ quantas to get rid of the excess energy liberated when the neutron is bound to the nucleus (usually refered to as "promt γ to distinguis it from the "normal" γs which is emitted after the nucleus has disintegrated). This also result in a certain amount of recoil energy on the nucleus.

#### Recoil energy from n-capture

The conservation of momentum demands that

where P denotes the momentum, index *n* denots the neutron, index *T* the target nucleus, and index *R* the recoil.

The general relationship between kinetic energy, *E _{K}*, and momentum

*p*is given by:

The mass of the neutron is *1* (atomic mass unit). the mass of the target nucleus is *A*. The new nucleus will therefore have mass *A+1*. Then

(remember that the momemtun of the target nucleus initially is 0.)

#### Recoil energy from γ emission

For emission of the mass-less quantas we have the following relationship:

and

In this case the nucleus has mass *A+1*, then

#### The iodine case

For iodine, A = 127. Thermal neutrons have E_{K,n} = 0.025 eV. E_{γ} will be around 3 MeV. We then get that for neutron capture the recoil energy is

and for the prompt γ-emission it is

Since chemical binding-energies typically are between 0.1 to 1 eV, the recoil from γ emission is large enough to break chemical bounds. However, the recoil from n capture is not.