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). This process is illustrated by Equation $43.26 .$ The emitting nucleus must recoil to conserve both energy and momentum. (a) Show that the recoil energy of the nucleus is $$E_{r}=\frac{(\Delta E)^{2}}{2 M c^{2}}$$ 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 ${ }^{57} \mathrm{Fe}$ nucleus when it decays by gamma emission from the 14.4-keV excited state. For this calculation, take the mass to be 57 u. Suggestion: Assume $h f<<M c^{2}$.

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). This process is illustrated by Equation $43.26 .$ The emitting nucleus must recoil to conserve both energy and momentum. (a) Show that the recoil energy of the nucleus is
$$E_{r}=\frac{(\Delta E)^{2}}{2 M c^{2}}$$
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 ${ }^{57} \mathrm{Fe}$ nucleus when it decays by gamma emission from the 14.4-keV excited state. For this calculation, take the mass to be 57 u. Suggestion: Assume $h f<<M c^{2}$.

Key Concepts

Non-relativistic Approximation

In cases where the energy of the emitted gamma photon is much smaller than the rest energy (Mc²) of the nucleus, a non-relativistic approximation can be used. This allows us to simplify the calculation of the recoil energy without invoking the full framework of relativistic mechanics, leading to a straightforward expression for the recoil energy.

Recoil Energy Calculation

The recoil energy of the nucleus is derived by applying both energy and momentum conservation principles. By relating the momentum carried by the photon to the kinetic energy of the recoiling nucleus and using the relation between energy, momentum, and mass, one can derive that the recoil energy is proportional to the square of the energy difference between the nuclear states divided by twice the rest energy of the nucleus.

Gamma Decay in Nuclei

Gamma decay is a process in which an excited nucleus releases excess energy by emitting a gamma-ray photon. This is a fundamental mechanism for nuclei to transition from a higher energy state to a lower one (typically the ground state), thereby stabilizing the nucleus after a reaction or disturbance.

Conservation of Momentum

In photon emission processes, conservation of momentum is crucial. When a nucleus emits a gamma-ray photon, the photon carries away momentum, and to conserve the total momentum of the system, the nucleus acquires an equal and opposite momentum, leading to a recoil effect.

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