Conclusions

This paper compared three ways to calculate an age from a single set of U, Th and He measurements and four ways to calculate the ``average'' of several aliquots of the same sample. U, Th and He form a ternary system, and the ternary diagram was introduced as an elegant way to make such a comparison. This reveals that the accuracy gained by the exact solution of the (U-Th)/He equation is easily lost if the average age of replicate measurements is calculated by the arithmetic mean. As a better alternative, the central age is calculated from the geometric mean composition of a dataset. In addition to the central age, the paper also introduced the pooled age and the isochron age as valuable alternatives to the arithmetic mean age in certain applications.

The pooled age is calculated by adding the U, Th, (Sm) and He contents of several single- and/or multi grain aliquots of the same sample. Pooled ages are biased to high U-Th-grains which may be affected by radiation damage, but are the only sensible way to average multi-grain aliquots. The isochron age is given by the slope of a linear fit of a diagram that plots helium content against present-day helium production rate. In order to reduce the bias towards large grains, it is a good idea to transform the input data to units of concentration, which can be done by dividing the atomic abundances by the estimated volume or mass of the component grains. Doing so will translate the datapoints along a straight line through the origin of the isochron plot and improves its power for detecting ``parentless helium''. If there is no parentless helium, data must plot on a single line going through the origin. But if, instead, the data do not define a line, or this line does not go through the origin of the isochron diagram, parentless helium or a similar problem may be present. The isochron age is less well suited for ages older than 100Ma because it uses the linearized age-equation (Equation 3).

Although most (U-Th)/He geochronologists are probably already aware that some accuracy is lost by calculating the $\alpha$-ejection correction by simply dividing the uncorrected (U-Th)/He age by the $\alpha$-retention factor F$_t$, it bears repeating that instead, the measured helium concentration should be divided by F$_t$ before the age calculation. The effects discussed in this paper are relatively minor, affecting the calculated ages by at most a few percent. Nevertheless, the added computational cost of following the above recommendations pales in comparison with the cost of collecting, separating and analyzing samples. Therefore, there is no reason why not to gain the extra percent of accuracy. To facilitate the calculation of the central age, a web-based calculator is provided at http://pvermees.andropov.org/central. It implements the calculations of central ages with or without Sm, and offers several options for propagating internal or external uncertainties. The web-calculator also allows the calculation of error-weighted central ages, and includes two dataset for testing purposes: the inclusion bearing (U-Th)/He data from Naxos which is also summarized in Table 1, and (U-Th-Sm)/He data from a Fish Lake Valley apatite lab standard.