What is the relationship between a nuclear energy level’s width and its lifetime?

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Figure 1: Spectrometer utilizing Mössbauer effect concept Effect is usually observed by measuring transmission of gamma rays from radioactive source through absorber containing resonant isotope. — Mössbauer effect Figure 1: *Spectrometer utilizing Mössbauer effect concept* Effect is usually observed by measuring transmission of gamma rays from radioactive source through absorber containing resonant isotope.

The relationship between a nuclear energy level's width ( $Γ$ ) and its mean lifetime ( $τ$ ) is governed by the Heisenberg uncertainty principle. This relationship is expressed by the equation:

$Γ τ = \frac{h}{2 π}$

In this formula, $h$ represents Planck’s constant.1

Because of this inverse relationship, a state with a shorter lifetime will have a broader energy width, while a longer-lived state will have a narrower width. In the context of the Mössbauer effect, this natural width is fundamental; if the lifetime of an excited state is too short (less than $10^{- 11}$ seconds), the resulting resonance level becomes so broad that the distinction between ordinary and recoil-free emission is lost.1