There are no barriers or repulsive forces between a free proton and neutron, allowing them to fuse at low kinetic energies to form a deuterium nucleus. This nucleus has a mass slightly smaller than the sum of the free proton and neutron masses. This mass difference can be converted into energy using Einstein’s famous equation:
\( E = mc^2 \)
This energy release is 2.2 MeV, which is also the energy emitted as a γ photon in the reaction. To separate the two nucleons in the deuterium nucleus, at least 2.2 MeV must be added back into the system.
Q-value and Reaction Energy
The energy gained or lost in a nuclear reaction is referred to as the Q-value. In a nuclear reaction like \( ^{14}N(p, \alpha)^{11}C \), the Q-value is calculated as the difference between the sum of the mass of the particles before the reaction (proton and nitrogen nucleus) and the mass of the particles after the reaction (alpha particle and carbon-11 nucleus).
\( Q = \left[ (m_{\text{reactants}}) - (m_{\text{products}}) \right]c^2 \)
For the reaction \( ^{14}N(p, \alpha)^{11}C \), the Q-value is calculated to be \( -2923.056 \, \text{keV} \), meaning the proton must have at least 2.93 MeV of kinetic energy to induce the reaction.
Threshold Energy and Coulomb Barrier
Before the proton hits the \( ^{14}N \) nucleus, it has to overcome the Coulomb barrier, which is the repulsive force between the positively charged proton and the \( ^{14}N \) nucleus. This means the proton loses some energy during its journey, and the initial energy must be higher than the threshold value.
The threshold value for this reaction, as calculated by a Q-value calculator, is 3.14 MeV. This means the proton needs at least 3.14 MeV of kinetic energy to overcome the Coulomb barrier and initiate the reaction, ensuring that the proton has enough energy to overcome the Q-value and trigger the reaction.
Exothermic and Endothermic Reactions
The reaction energy (or Q-value) is positive for exothermic reactions (spontaneous reactions) and negative for endothermic reactions. Exothermic reactions release energy, while endothermic reactions require an input of energy to proceed.
For instance, radioactive decays are spontaneous, and as such, they always have positive Q-values. Some reactions used to produce radionuclides, such as those based on thermal neutrons, also have positive Q-values. On the other hand, reactions involving charged particles (like protons) often have negative Q-values, which means energy must be added for the reaction to occur.
Example Reaction: The reaction \( ^{14}N(p, \alpha)^{11}C \) is an endothermic reaction, with a Q-value of \( -2923.056 \, \text{keV} \), meaning additional energy must be supplied for the reaction to take place.
Conclusion
The Q-value, reaction threshold, and binding energy are crucial concepts in nuclear physics. The Q-value determines whether a reaction releases or absorbs energy, while the threshold energy accounts for the energy needed to overcome Coulomb barriers. Understanding these concepts helps explain the behavior of nuclear reactions, including fusion and radioactive decay.
Figure 1: Example of a nuclear reaction