When, after a reaction or disturbance of any kind, a nucleus...

When, after a reaction or disturbance of any kind, a nucleus is left in an excited state, it can return to its normal (ground) state by emission of a gamma-ray photon (or several photons). Equation 13.25 describes this process. The emitting nucleus must recoil in order to conserve both energy and momentum. (a) Show that the recoil energy of the nucleus is $$E_{\mathrm{r}}=\frac{(\Delta E)^{2}}{2 M c^{2}}$$ where $\Delta E$ is the difference in energy between the excited and ground states of a nucleus of mass M. (b) Calculate the recoil energy of the $^{57}$ Fe nucleus when it decays by gamma emission from the 14.4 -keV excited state. For this calculation, take the mass to be 57 u. (Hint: When writing the equation for conservation of energy, use $(M v)^{2} / 2 M$ for the kinetic energy of the recoiling nucleus. Also, assume that $\left.h f<<M c^{2} .\right)$

Key Concepts

Conservation of Energy

This principle states that the total energy in an isolated system remains constant over time. In the context of nuclear decay, the energy released as a gamma photon is equal to the difference between the excited and ground state energies of the nucleus, with any excess energy appearing as kinetic energy of the recoiling nucleus.

Conservation of Momentum

Momentum conservation requires that the total momentum of a system remains constant if no external forces act on it. When a nucleus emits a photon, the momentum carried away by the photon must be balanced by the recoil momentum of the nucleus, ensuring that the overall momentum before and after the emission remains zero.

Photon Energy-Momentum Relationship

Photons, despite having no rest mass, carry momentum related to their energy by the equation p = E/c. This relationship is fundamental in connecting the energy release from nuclear transitions to the recoil momentum (and therefore kinetic energy) acquired by the nucleus during gamma emission.

Non-Relativistic Approximation

In many nuclear decay processes, the energy of the emitted gamma photon is small compared to the rest mass energy of the nucleus (hf << Mc²). Under this approximation, classical formulas for kinetic energy, such as K = p²/(2M), become valid, greatly simplifying the derivation of recoil energy.

Derivation of Recoil Kinetic Energy

This concept involves deriving the kinetic energy of the recoiling nucleus by using momentum conservation and the non-relativistic kinetic energy formula. By equating the momentum imparted to the nucleus with the photon’s momentum and substituting into K = p²/(2M), one arrives at the recoil energy expression E_r = (?E)²/(2Mc²), showing how the energy difference between nuclear states translates into nuclear recoil energy.

Transcript

Please give Ace some feedback