Nuclei have energy levels just as atoms do. An excited...

Nuclei have energy levels just as atoms do. An excited nucleus can make a transition to a lower energy level by emitting a gamma-ray photon. The lifetime of a typical nuclear excited state is about 1 ps. What is the uncertainty in the energy of the gamma-rays emitted by a typical nuclear excited state? [Hint: Use the energytime uncertainty principle, Eq. $(28-5) .]$

Key Concepts

Gamma-Ray Emission in Nuclear Transitions

Gamma-ray emission is a common process in nuclear physics where an excited nucleus transitions to a lower energy state by releasing a photon in the gamma energy range. The characteristics of the emitted gamma-rays, such as their energy distribution, are influenced by the properties of the nuclear states involved. The intrinsic energy uncertainty (or broadening) in the gamma-ray spectrum is a direct consequence of the finite lifetime of the excited nuclear state as dictated by the energy-time uncertainty relation.

Lifetime and Energy Broadening

In quantum systems, particularly in nuclear physics, the lifetime of an excited state is directly linked to the precision with which its energy is defined. A very short lifetime results in a significant energy 'broadening' or width due to the energy-time uncertainty principle. This broadening is observable in the spectral lines of emitted radiation and signifies that the energy of the emitted gamma-rays is uncertain within a range approximately on the order of ? divided by the lifetime of the state.

Energy-Time Uncertainty Principle

This principle is a fundamental concept in quantum mechanics that states there is a limit to how precisely energy and time can be simultaneously determined. Specifically, a short-lived quantum state, such as an excited state with a very brief lifetime, inherently has a large uncertainty in its energy. The mathematical relationship is often expressed as ?E?t ? ?/2, which means that if the time duration (lifetime) of the state is very small, the range (or uncertainty) of the possible energy values becomes correspondingly large.

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