Particles, Particles, Particles — Question 9

Question 9

A short resonance lifetime corresponds to a broad energy width

Two unstable resonances have mean lifetimes $10^{-22}$ s and $10^{-20}$ s. Use natural width $\Gamma=\hbar/\tau$ and $\hbar=6.582\times10^{-22}$ MeV s. A detector has characteristic energy resolution 1.00 MeV.

1. Calculate the natural widths.

2. Compare each width with instrumental resolution and identify which measured peak is more likely dominated by the apparatus.

Question 9 - Solution

A short resonance lifetime corresponds to a broad energy width

(a) Method and calculation

$$\boxed{\Gamma_1=6.582\,\mathrm{MeV},\quad\Gamma_2=.06582\,\mathrm{MeV}}.$$$$\boxed{\Gamma_1/\Gamma_2=\tau_2/\tau_1=100}.$$

(b) Interpretation and further calculation

$$\Gamma_1/(1\,\mathrm{MeV})=6.582,\quad\Gamma_2/(1\,\mathrm{MeV})=.06582.$$$$\boxed{\text{second peak: instrument dominated; first: natural width can be resolved}}.$$

Physical check

Natural and instrumental line shapes generally require convolution. Their widths should not automatically be added in quadrature without a specified shape model.