Multi-grain ages

Equations 2-6 can be used to calculate (U-Th)/He ages from individual U, Th and He measurements, but do not explain how to calculate the ``average'' value of multiple analyses. Traditionally, the average has been estimated by the arithmetic mean of the single grain ages. This section will introduce two alternative methods for calculating average ages, and the next section will add a third. Each of these new methods is more appropriate than the arithmetic mean age in specific applications.

Helium can be extracted from the host grain either in a resistance furnace or by laser-heating in a micro-oven (House et al., 2000). In the former, but sometimes also in the latter case, it may be necessary to analyze multiple mineral grains together (e.g., Persano et al., 2007). ``Pooling'' several grains boosts the signal strength and sometimes averages out $\alpha$ -ejection correction errors caused by zoning and mineral inclusions. Vermeesch et al. (2007) introduced the ``pooled age'' as the best way to compare multiple single-grain ages with one or more multi-grain ages, or to compare two sets of multi-grain ages with each other (Figure 1). The pooled age is calculated by adding the respective U, Th and He abundances (in moles) of several measurements together, thereby generating one ``synthetic'' multi-grain measurement. The age of the pooled measurement can then be calculated using any of the equations given in Section 2.

An obvious disadvantage of pooling compositional data is that the resulting age is biased to the high U, Th or He compositions. Because such bias can be associated with anomalous grains affected by radiation damage or implanted helium, the pooled age may be wrong by effectively giving extra weight to outliers. However, all these objections are also true for standard multi-grain analyses, which cannot be avoided when dating small, young, or U-Th-poor grains (e.g., Persano et al., 2007). In short, the pooled age must be used for and only for averaging multi-grain aliquots.

**Figure 1:** Linearized (U-Th)/He diagram with a subset of the HF-treated inclusion-bearing apatite data of Vermeesch et al. (2007). 30% uncertainty (2 $\sigma$ ) was added to the helium-abundances to account for $\alpha$ -ejection correction induced scatter (Vermeesch et al., 2007). (a) According to the linear age equation, (U-Th)/He ages are given by the slopes of lines connecting each point (P,[He]) with the origin; (b) the ``pooled'' age is a ``synthetic multi-grain age'' calculated from the summed production rates and helium-abundances of all the measurements. The box in 1.b marks the outline of 1.a. The pooled age of the sample is 11.28 $\pm$ 0.14 Ma.
(a) (b)

The previous section showed that (U-Th)/He data can be visualized on a two-dimensional plot of helium abundance or concentration versus -production (Figure 1). To calculate a pooled age, it is important that [U], [Th] and [He] are elemental abundances, expressed in moles. If the data are recast in units of concentration, some of the bias towards high U-Th-grains disappears and the He-P diagram can be used to define a (U-Th)/He isochron. This is an unconstrained linear fit through a series of single-grain (P,[He]) measurements.

For an application of the isochron method, consider the U-Th rich mineral inclusions in apatite which are often held responsible for erroneously old (U-Th)/He ages, because they produce ``parentless'' He. This problem can be detected with the (U-Th)/He isochron. In the absence of mineral inclusions, the isochron goes through the origin (P=[He]=0). However, in the presence of $\alpha$ -emitting inclusions, the isochron is either not defined or does not go through the origin. For example, consider the worst-case scenario of an $\alpha$ -emitting zircon inclusion contained in an apatite without U and Th. The inclusion ejects He into the surrounding apatite that is measured following degassing by heating with a laser or in a resistance furnace. However, the zircon inclusion will not dissolve in the concentrated HNO

that is commonly used to digest apatites prior to U-Th analysis. Therefore, the apparent (U-Th)-production of such a sample is zero, and its isochron does not go through the origin of the [He]-P diagram.

Vermeesch et al. (2007) solved the parentless helium problem by dissolution of the apatite and its inclusions in hot HF. The effectiveness of this technique is illustrated by comparing an inclusion-rich sample from Naxos using the traditional HNO

method with an HF-treated aliquot of the same sample. The latter defines a well-constrained (U-Th)/He isochron with zero intercept, whereas the former does not (Figure 2). Calculation of the isochron age, including error propagation, can easily be done using the Isoplot Excel add-in (Ludwig, 2003).

**Figure 2:** (U-Th)/He isochron plots for inclusion-bearing apatites from Naxos, Greece (Vermeesch et al., 2007). (a) HNO-treated apatites do not plot on a line, indicating ``parentless helium'' caused by undissolved mineral inclusions containing ``missing'' U and Th; (b) HF-treated apatites from the same sample do form a well-defined isochron intersecting the origin, indicating that all parent and daughter nuclides are accounted for. The isochron age is 12.0 $\pm$ 4.2 Ma.
(a) (b)