The effect of -emitting mineral inclusions on closure temperatures

Besides complicating the $\alpha$ -ejection correction, the inhomogeneous U-Th-distribution inherent to inclusion-bearing apatites also affects the diffusive behavior of the helium. This problem is well studied in the case of concentrically zoned spherical crystals (Meesters and Dunai, 2002b). The ``closure temperature'' (defined below) of the (U-Th)/He system depends on the spatial distribution of the helium. This does not constitute a significant problem for relatively rapidly cooled rocks (e.g.

C/Ma for rocks older than $\sim$ 10 Ma). However, for slowly cooled rocks, the effect of mineral inclusions on helium diffusion might induce significant scatter in the apparent ages, depending on the spatial distribution of the inclusions and their relative $\alpha$ -emitting activity (G-factor, equation 2), which are nearly impossible to measure. To get a handle on the significance of this effect, this section will consider the ``worst case scenario'' of a spherical U-Th-free apatite containing an $\alpha$ -emitting micro-inclusion at its core or rim. Such a grain should have the most different diffusive behavior compared to the default case of an inclusion-free apatite containing a homogeneous U-Th-distribution.

Before proceeding with the analysis, it is useful to recall the definition of ``closure temperature'' by Dodson (1973). First consider the case of isothermal diffusion for a simple geochronometer defined by one radioactive parent P (e.g., $^{40}$ K) decaying to a radiogenic daughter D (e.g., $^{40}$ Ar). At high temperatures, the radiogenic daughter escapes from mineral grains as soon as it is formed whereas at low temperatures, thermally activated diffusion is negligible and the daughter products are quantitatively retained. Now consider the case of monotonic cooling. In the high-temperature part of the cooling-curve, the D/P-ratio stays zero. Under transitional temperatures, the D/P-ratio increases at a rate that increases with time. Finally, at low temperatures, D/P increases linearly with time. The apparent age is the time intercept of the linear section of the D/P vs time curve. This age corresponds to a particular temperature in the cooling history, the so-called ``closure temperature'' T

. If the cooling curve is linear in the temperature T, then the closure temperature (T

) is the temperature intercept of the linear section of the T vs D/P curve (Figure 6). If the cooling curve is linear in 1/T, then T

can be calculated analytically (Dodson, 1973). Things are a little more complicated for (U-Th)/He because there are not one but three radioactive parents ( $^{238}$ U, $^{235}$ U, and $^{232}$ Th) with different half-lives, causing their relative contributions to change with time. However, the helium content of young rocks does increase linearly with time to a very good approximation (t $\approx$ [He]/ $\pi$ with $\pi$ as defined in equation 3). Therefore, the closure temperature concept can also be used for (U-Th)/He. Because we are interested in the relationship between mineral inclusions and cooling rate (dT/dt), we will assume linear cooling (dT/dt = constant with t). We will not calculate T

using Dodson's approximate analytical solution (an exact solution is only possible if T $\propto$ 1/t). Instead, we will calculate T

numerically using the DECOMP program of Bikker, Meesters and Dunai (Dunai, 2005).

As said before, in the case of linear cooling T

is the temperature-intercept of a curve plotting temperature versus apparent (U-Th)/He age (Figure 6). Thanks to a combination of the ``superposition principle'' (Meesters and Dunai, 2002a & b) and the spherical symmetry, a concentrated $\alpha$ -emitting inclusion is mathematically equivalent to a thin spherical shell at the same distance of the rim (A.G.C.A. Meesters, written communication, July 2006). A spherical shell of a certain U-Th concentration produces as much He as a half-shell with twice, or a quarter-shell with four times this U-Th concentration. By induction it follows that a U-Th rich spherical shell is equivalent to a point source of equal $\alpha$ -emitting activity located at the same distance from the rim. Therefore, individual inclusions can be adequately modeled as spherical shells in DECOMP. The ``worst case scenario'' of a single $\alpha$ -emitting micro-inclusion located in the center of a U-Th-free apatite was modeled in DECOMP by a spherical zone of 1 $\mu$ m radius containing all the U and Th, contained in a much larger spherical host apatite (e.g., R = 60 $\mu$ m in Figure 6), whereas an inclusion located on the rim of such an apatite was modeled by a spherical shell of 1 $\mu$ m thickness. Given a user-defined cooling curve (10

C/Ma for Figure 6), DECOMP forward-models the apparent age through time using an eigenmode-decomposition method (Meesters and Dunai, 2002a & b). In this paper, we used 100 eigenmodes. The apparent age increases linearly with time in the lower temperature part of the linear cooling model (Figure 6). As expected, the linear section of the T-t curve begins at higher temperatures for the apatite containing an inclusion at its core than for inclusion-free apatite. Intuitively, this makes sense because the helium produced by the inclusion must ``travel further'' before reaching the edge of the apatite. The opposite is true for apatites with an inclusion on their rim.

The closure temperature calculation was repeated for different grain-sizes (40 $\mu$ m $\leq$ R $\leq$ 130 $\mu$ m) and cooling rates (10

C/Ma and 1

C/Ma) (Figure 7). In the ``worst case scenario'' of an inclusion-bearing apatite, T

is 5-10

C higher or lower than in the absence of inclusions. This is a small effect in several respects. First of all, the ``worst case scenario'' used for the calculations of Figure 7 is very unlikely to ever occur in nature. As explained in Section 2, mineral inclusions never contain 100% of $\alpha$ -emitting activity, because their relative volume is small and the U-Th content of apatite is never zero. Consider the more realistic scenario of a mineral inclusion contributing 50% of the total helium (G = 0.5). Because of the superposition principle, such an inclusion will change the closure temperature by only 2.5-5

C. The effects of grain size (15-20

C difference in T

between 40 and 130 $\mu$ m, Figure 7) and cooling rate ( $\sim$ 15

C difference in T

between 1 and 10 K/Ma, Figure 7) are greater than the inclusion-effect. Therefore, it should also be possible to apply the HF-dissolution method to slowly cooled rocks. However, several studies have reported irreproducible apatite (U-Th)/He ages for slowly cooled terranes (Fitzgerald et al., 2006) and (U-Th)/He ages older than apatite fission track ages (Soderlund et al., 2005; Green and Duddy, 2006). Therefore, the next section will test the HF-dissolution method on rapidly cooled rocks.

**Figure 6:** Closure temperature (T) calculation using the DECOMP program (Dunai, 2005). The inset shows a linear cooling curve with a cooling rate of 10C/Ma. In this case, the closure temperature can be defined as the y-intercept of the linear section of the temperature vs. age curve (Dodson, 1973). (a) ``worst case'' scenario of a U-Th-free apatite with a U-Th-bearing inclusion at its center (closure temperature T = 86.85 C); (b) inclusionless apatite of uniform U, Th concentration (T = 77.8 C). (c) ``worst case'' scenario of a U-Th-free apatite with a U-Th-bearing inclusion at its edge (T = 73.06). Radius of the host apatite = 60 $\mu$ m, $\alpha$ -stopping distance S = 20 $\mu$ m.

**Figure 7:** Repeating the numerical experiment of Figure 6 for different grain sizes and cooling rates, this plot shows the evolution of closure temperature T with grain size, (a & c) with and (b) without the presence of a mineral inclusion in a spherical apatite with radius R, for relatively rapidly cooled (10C/Ma, thick lines) and very slowly cooled (1C/Ma, thin lines) rocks. (a), (b) and (c) are as defined in Figure 6.