The kinetic energy of a positron emitted from the β^+ decay of a ^13 N nucleus into a ^13 C nuc

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The kinetic energy of a positron emitted from the β^+ decay...

The kinetic energy of a positron emitted from the $\beta^{+}$ decay of $\mathrm{a}{ }^{13} \mathrm{~N}$ nucleus into $\mathrm{a}{ }^{13} \mathrm{C}$ nucleus is measured to be $1.190 \mathrm{MeV}$. What is the energy of the accompanying neutrino of the decay process? Neglect the recoil energy of the daughter nucleus. (See Exercise 23.)

Key Concepts

Beta-Plus Decay

Beta-plus decay is a type of radioactive process in which a proton in a nucleus is converted into a neutron with the simultaneous emission of a positron and a neutrino. This process helps maintain nuclear stability and satisfies conservation laws, such as charge, energy, lepton number, and momentum conservation.

Energy Conservation in Nuclear Reactions

In nuclear reactions, the total energy is conserved. This means that the Q-value, which represents the net energy change of the reaction, is distributed among all the products as kinetic energy or carried away by mass differences. In multi-particle decays, the sum of the kinetic energies of the emitted particles equals the Q-value, assuming negligible recoil of the residual nucleus.

Q-value Calculation

The Q-value of a nuclear decay process is the difference between the initial and final mass-energy of the system. It defines the total energy available to be shared among the decay products as kinetic energy. Understanding the Q-value is crucial when analyzing how the energy is partitioned in a nuclear decay, such as between a positron and a neutrino in beta-plus decay.

Neutrino Emission and Kinematics

In processes like beta-plus decay, a neutrino is emitted to satisfy conservation laws. Since neutrinos have extremely small or negligible mass, almost all of their energy is kinetic. The energy carried by the neutrino is determined by the balance between the total available energy (Q-value) and the kinetic energy imparted to the other emitted particles, typically calculated using energy conservation principles.

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