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

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(Created page with "A nucleus which captures a thermal neutron must, since the momentum is conserved, receive a recoil energy. This will also happen when a gamma quantum is emitted.")
 
 
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A nucleus which captures a thermal neutron must, since the momentum is conserved, receive a recoil energy. This will also happen when a gamma quantum is emitted.
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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 (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.&nbsp;<br>
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==== Recoil energy from n-capture<br>  ====
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The conservation of momentum demands that<br>
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&nbsp;&nbsp;&nbsp;&nbsp; <math>\overrightarrow{P}_n + \overrightarrow{P}_T = \overrightarrow{P}_{T+n} = \overrightarrow{P}_R</math>
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where P denotes the momentum, index ''n'' denots the neutron, index ''T'' the target nucleus, and index ''R'' the recoil.&nbsp;
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The general relationship between kinetic energy, ''E<sub>K</sub>'', and momentum ''p'' is given by:
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&nbsp;&nbsp;&nbsp; <math>E_K = \frac{\overrightarrow{p}^2}{2m}</math>
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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
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&nbsp;&nbsp;&nbsp; <math>E_{K,R} = \frac{\overrightarrow{P}^2_R}{2(A+1)}
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= \frac{\overrightarrow{P}^2_n m_n}{2 m_n (A+1)}
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=\frac{E_{K,n} m_n}{A+1}
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=\frac{E_{K,n}}{A+1}</math>
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(remember that the momemtun of the target nucleus initially is 0.)
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==== Recoil energy from γ emission<br>  ====
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For emission of the mass-less quantas we have the following relationship:
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&nbsp;&nbsp;&nbsp; <math>\overrightarrow{P}_R = \overrightarrow{P}_\gamma</math>
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and
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&nbsp;&nbsp;&nbsp; <math>P_\gamma = \frac{E_\gamma}{c}</math>
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In this case the nucleus has mass ''A+1'', then
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&nbsp;&nbsp;&nbsp; <math>E_{K,R}=\frac{\overrightarrow{P}^2_R}{2(A+1)}
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= \frac{\overrightarrow{P}^2_\gamma c^2}{2(A+1) c^2}
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= \frac{E^2_\gamma (MeV)}{2(A+1) 931.5 MeV}</math>
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==== The iodine case<br>  ====
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For iodine, A = 127. Thermal neutrons have E<sub>K,n</sub>&nbsp;= 0.025 eV. E<sub>γ</sub> will be around 3 MeV. We then get that for neutron capture the recoil energy is<br>
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&nbsp;&nbsp;&nbsp; <math>E_{K,R}= \frac{0.025 eV}{128} = 0.2 \; meV</math>&nbsp;
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and for the prompt γ-emission it is<br>
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&nbsp;&nbsp;&nbsp; <math>E_{K,R} = \frac{9 \cdot 10^6 eV}{2 \cdot 128 \cdot 931.5 \; MeV} = 38 \; eV</math>
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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 13:53, 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

     [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. 

The general relationship between kinetic energy, EK, 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

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

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

    [math]E_{K,R}= \frac{0.025 eV}{128} = 0.2 \; meV[/math] 

and for the prompt γ-emission it is

    [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.