Parentless helium and the importance of being inclusion-free

``Erroneous'' apatite (U-Th)/He ages have often been attributed to U-Th rich mineral inclusions (e.g., Lippolt et al., 1994; House et al., 1997; Fitzgerald et al., 2006). A very substantial part of many (U-Th)/He studies is spent on selecting inclusion-free apatites under the binocular microscope. Under reflected and transmitted light, with or without polarizers, grains are scrutinized for imperfections and mineral inclusions, in order to avoid the parentless helium problem. But even when no inclusions can be detected with this method, it has been suggested that sub-micron sized inclusions, only visible by electron microscopy or fission-track mapping for uranium inhomogeneity, might produce significant amounts of parentless He (Farley and Stockli, 2002; Ehlers and Farley, 2003).

The validity of these concerns can be assessed by some simple order-of-magnitude calculations. Consider a spherical apatite of radius R

containing a spherical mineral inclusion with radius R

. If the inclusion is 10 times smaller than the apatite (R

= R

/10), then its cross-sectional area is 100 times smaller (A

/100) and the volume of the inclusion is 1000 times smaller than that of the host apatite (V

/1000). In other words, an apatite containing (an exceptionally low) 1 ppm U requires such an inclusion to be 1000 times more concentrated in U (i.e. 1000 ppm) for it to produce an equal amount of He (Figure 1). Identical arguments hold for non-spherical geometries. For example, consider a prismatic apatite with 10 ppm of U, containing an inclusion that is 1% of its length, 1% of its width and 1% of its height. Such an inclusion has one millionth the volume of the host grain (Figure 1). It would need to consist of pure uranium to increase the helium by just 10%. Typical apatites used in thermochronology have dimensions on the order of 100 $\mu$ m, and U-Th concentrations $\sim$ 10 ppm (Farley, 2002). Zircon inclusions have U and Th concentrations of typically 100-1000 ppm and sometimes up to 5000 ppm, whereas monazite can contain up to 30% of Th (Deer et al., 1992). Therefore, sub-micron sized inclusions may be a less significant source of parentless helium than previously thought, unless they are extremely numerous and their composite volume is more than a ten-thousandth or so of the host apatite. We will now shift our attention away from micro-inclusions and focus on somewhat larger inclusions which do contribute substantial amounts of parentless helium.

**Figure 1:** Illustration of the ``blessings of dimensionality''. The gray line shows, for example, how a tenfold decrease of the linear dimensions of a mineral inclusion (R = length, width or height of the inclusion, R = length, width or height of the apatite) corresponds to a thousandfold decrease of its volume (V for the inclusion, V for the apatite) and He-production. The black lines show the U or Th concentrations C which are required for the inclusion to produce x% of the helium produced by a host apatite with concentration C (for x = 2, 10, 50 and 100, respectively). Please note the different horizontal scale for the V/V and the C/C curves.