Uncertainty in $^{10}$Be concentration

The uncertainty in the $^{10}$Be concentration in the quartz sample must take into account three sources of uncertainty: the uncertainty in the isotope ratio measurement (which we will call $\sigma R_{10/9}$), the uncertainty in the number of atoms in the process blanks ( $\sigma n_{10,B}$), and the uncertainty in the mass of Be added as carrier ( $\sigma M_{C}$). The isotope ratio uncertainty $\sigma R_{10/9}$ is supplied by the AMS lab. The blank uncertainty $\sigma n_{10,B}$ is discussed below. The uncertainty in the Be carrier mass comes from the fact that the Be carrier is added in solution, and the Be concentration in the solution has some uncertainty. The uncertainty in the Be concentration in typical commercial Be solutions is about 1%, so the uncertainty in the Be carrier mass is typically $\sigma M_{C} = 0.01M_{C}$.

Using standard error-propagation methods, the formula for the uncertainty in the $^{10}$Be concentration in quartz, that is, $\sigma N_{10}$, is:

$$\sigma N_{10} = \sqrt{ \left( \frac{\partial N_{10}}{\partial R_{... ...{2} + \left( \frac{\partial N_{10}}{\partial M_{C}} \sigma M_{C} \right) ^{2} }$$ (7)

where:

$$\frac{\partial N_{10}}{\partial R_{10/9}} = \frac{M_{C}N_{A}}{M_{q}A_{Be}}$$ (8)

$$\frac{\partial N_{10}}{\partial n_{10,B}} = \frac{{-1}}{M_{q}}$$ (9)

$$\frac{\partial N_{10}}{\partial M_{C}} = \frac{R_{10/9}N_{A}}{M_{q}A_{Be}}$$ (10)