Some Nuclear Properties — Question 8

Question 8

Mass uncertainty propagates directly into binding-energy uncertainty

A nucleus has binding energy $B=[Zm_H+Nm_n-M]c^2$. Treat $m_H$ and $m_n$ as exact for this exercise. Its measured atomic mass is $M=(55.93494\pm .00002)\,\mathrm u$, with one-standard-deviation uncertainty. Use $1\,\mathrm u\,c^2=931.5\,\mathrm{MeV}$ and A=56.

1. Find the uncertainty in total binding energy and in binding energy per nucleon.

2. Two independent mass measurements each have uncertainty .00002 u. Find the uncertainty in a separation energy obtained from their difference.

Question 8 - Solution

Mass uncertainty propagates directly into binding-energy uncertainty

(a) Method and calculation

$$\boxed{\sigma_B=c^2\sigma_M=931.5(.00002)=.01863\,\mathrm{MeV}}.$$$$\boxed{\sigma_{B/A}=.01863/56=.0003327\,\mathrm{MeV}=.3327\,\mathrm{keV}}.$$

(b) Interpretation and further calculation

$$\sigma_{\Delta M}=\sqrt{\sigma_{M_1}^2+\sigma_{M_2}^2}=\sqrt2(.00002)\,\mathrm u.$$$$\boxed{\sigma_S=\sqrt2(.01863)=.02635\,\mathrm{MeV}}.$$

Physical check

Independent errors add in quadrature. Correlated mass calibrations can change the uncertainty of a difference and require a covariance term.