The effect of $\alpha$-emitting mineral inclusions on $\alpha$-ejection corrections

In the previous section, we discussed the magnitude of the parentless helium problem for small mineral inclusions. As will be demonstrated later, it is possible to avoid this problem altogether (even for relatively large inclusions) by dissolving the apatites and their mineral inclusions in aggressive acids such as HF. However, this does not solve a second problem, caused by the inhomogeneous U-Th concentrations associated with mineral inclusions. $\alpha$-decay of U, Th and their radioactive daughters is associated with energies of 4-8 MeV (Farley et al., 1996). $\alpha$-particles with such high energies travel on average 20 $\mu$m in apatite before coming to rest. Consider a spherical apatite with radius R and an $\alpha$-emitting nuclide located at a radial distance X from its center. Let S be the $\alpha$-stopping distance (e.g., 20 $\mu$m). $\alpha$-emitting nuclides located at a distance R-S$\leq$X$\leq$R have a non-zero probability of ejecting an $\alpha$-particle outside the boundaries of the apatite grain (Figure 2). For any given spatial distribution of U and Th, it is possible to predict the fraction (1-F$_t$) of radiogenic He lost by $\alpha$-ejection (Farley et al., 1996; Meesters and Dunai, 2002; Hourigan et al. 2005). In most cases, the U-Th distribution is not known and assumed to be uniform. This assumption often constitutes the bulk of the analytical (U-Th)/He age uncertainty. If significant He is produced by small mineral inclusions, the assumption of uniform composition is violated. We will address this problem mathematically for spherical grain geometries. The physical dimension of the mineral inclusions will be neglected, i.e. they will be considered point sources of $\alpha$-particles, making the He-retentivity of the inclusion itself irrelevant.

If F$_t^a$ is the $\alpha$-retention fraction of the apatite, and F$_t^i$ is the fraction of $\alpha$-particles that are ejected from the inclusion but remain inside the apatite, then the total $\alpha$-retention factor F$_t$ can be defined as:

$$F_t = G F_t^i + (1-G) F_t^a$$ (1)

where G is the fraction of $\alpha$-decay activity ($\pi$) associated with the mineral inclusion:

$$G = \frac{\pi(inclusion)}{\pi(inclusion)+\pi(apatite)}$$ (2)

$$\pi = 8^{238}\lambda[^{238}U] + 7^{235}\lambda[^{235}U] + 6^{232}\lambda[^{232}Th] + ^{147}\lambda[^{147}Sm]$$ (3)

with $^n\lambda$ the decay constant and [n] the number of atoms or moles of nuclide n (for n = 238, 235, 232 or 147). Note that equation 3 considers He-production to be a linear function of time, which is a good approximation for relatively young samples (t $\ll$ 1/ $^{n}\lambda$ $\forall$ n). Our goal is to derive the probability distribution of F$_t$. To achieve this goal, we first compute the cumulative distribution function (cdf) of the $\alpha$-retention factor F$_t^i$:

$$cdf_{F_t^i}(f_t^i) \equiv Pr(F_t^i \leq f_t^i) = \begin{cases} \... ...)/4 \leq f_t^i < 1 \ \hfill 1 &\text{if} \quad f_t^i \geq 1 \ \end{cases}$$ (4)

$$cdf_{X^*}(x^*) \equiv P(X^* \leq x^*) = \left(x^*\right)^3$$ (5)

Where X$^*$ is the nondimensional radial distance X$^*$=X/R corresponding to the $\alpha$-retention factor F$^i_t$. cdf$_{F_t^i}$ can be computed because there exists a unique mapping between F$_t^i$ and X$^*$ (Figure 2), derived from equation 1 of Farley et al. (1996):

$$F_t^i\left(X^*\right) = \frac{1-\left(X^*-\frac{S}{R}\right)^2}{4\frac{S}{R}X^*}$$ (6)

$$X^*\left(F_t^i\right) = \left(\frac{S}{R}\right)\left(1-2F_t^i\right) + \sqrt{4\left(\frac{S}{R}\right)^2\left(F_t^i-1\right)F_t^i + 1}$$ (7)

Figure 2: Bottom: cross-section through the lower hemisphere of a spherical apatite with radius R. Top: $\alpha$-retention factor F$_t^i$ for U-Th-bearing mineral inclusions as a function of the non-dimensional radial distance X$^*$. All $^4$He produced by inclusions located at X$^*<$1-S/R will remain inside the host apatite ( $\Rightarrow$ F$_t^i$=1), with S the $\alpha$-stopping distance. However, inclusions located at X$^*>$1-S/R will eject a fraction (1-F$_t^i$) of helium outside the boundaries of the apatite.

The probability density functions (pdfs) are then easily obtained by taking derivatives of the cdfs:

$$pdf_{F_t^i}(f_t^i) = \frac{d(cdf_{F_t^i}(f_t^i))}{d(f_t^i)}$$ (8)

$$pdf_{X^*}(x^*) = \frac{d(cdf_{X^*}(x^*))}{d(x^*)} = 3(x^*)^2$$ (9)

Using equations 8 and 9, we can calculate $\bar{F_t^i}$, the expected value of F$_t^i$ assuming that the inclusions have a spatially uniform distribution. Here we use ``expected value'' in the statistical sense of the word, meaning the average F$_t^i$ of many apatites containing a few inclusions, or the average F$_t^i$ of a few apatites containing many inclusions. Thanks to the mapping between F$_t^i$ and X$^*$ (equations 6 and 7 and Figure 2), $\bar{F_t^i}$ can be calculated either by integrating over F$_t^i$ or over X$^*$. Not surprisingly, both approaches yield the same result, which turns out to be the analytical solution for F$_t^a$ under spherical geometry calculated by Farley et al. (1996) for compositionally homogeneous apatite:

$$\bar{F_t^i} = \int_{0}^{1} f_t^i  pdf_{F_t}(f_t^i)  df_t^i = ... ... = 1 - \frac{3}{4}\frac{S}{R} + \frac{1}{16}\left(\frac{S}{R}\right)^3 = F_t^a$$ (10)

The probability distribution of F$_t$ (equation 1) can be calculated for any G from the probability distribution for F$_t^i$ (equation 8) and the expression for F$_t^a$ given by Farley et al. (1996) and equation 10 (Figure 3). Although G is not known in most cases, our ignorance about G can be quantified by assigning a probability density function pdf$_G$ to it. Again, to derive the probability distribution of F$_t$, we must first define its cumulative density function cdf$_{F_t}$:

$$cdf_{F_t}(f_t) = \int_{0}^{1} cdf_{F_t^i}\left(\frac{f_t+(g-1)F_t^a}{g}\right)  pdf_G(g)  dg$$ (11)

Figure 3: The expected spread of $\alpha$-retention factors F$_t$ for grains of a single size (S/R = 0.5) if the inclusions are located in different radial positions within a spherical apatite grain. The six curves correspond to inclusions of different $\alpha$-emitting activity (G-value, equation 2). More helium will be retained from apatites containing an $\alpha$-emitting mineral inclusion at their core than from inclusion-free apatites or apatites containing an inclusion near their rim.

with cdf $_{F_t^i}(\cdot)$ as defined in equation 4, after which pdf$_{F_t}$ is obtained by taking the derivative:

$$pdf_{F_t}(f_t) = \frac{d(cdf_{F_t}(f_t))}{d(f_t)}$$ (12)

Figure 4 shows pdf$_{F_t}$ for a uniform pdf$_G$ distribution and various S/R-values. The mode of the distribution is always at F$_t^a$, with heavy tails, especially toward high $\alpha$-retentivities. If a ``normal'' $\alpha$-ejection correction is made (F $_t \equiv$ F$_t^a$) the most frequently measured age will be accurate, but some other measurements will not. ``Undercorrected'' ages will generally be further removed from the true age than ``overcorrected'' ages. It would be relatively easy to compute F$_t$- and corresponding age-distributions for different, and possibly more realistic pdf$_G$s such as the logistic normal distribution. However, such an exercise is of limited interest because in reality, pdf$_G$ is not known. Nevertheless, the main features of Figures 3 and 4 are robust: the mean value of F$_t^i$ equals F$_t^a$ and therefore the mean value of F$_t$ is independent of pdf$_G$. The distribution of F$_t$ has a sharp mode at F$_t^a$, with tails towards lower and higher values (Figure 4).

Figure 4: Probability density of the true $\alpha$-retention factor F$_t$ for uniform G-distribution ($pdf_G$(g) dg = 1/dg for 0$\leq$g$\leq$1) and various grain-sizes (S/R-values). Each of the curves is the result of ``stacking'' the curves of Figure 3.

Given the probability distribution of F$_t$ (equation 12), the standard deviation of F$_t$ can be calculated as:

$$\sigma(F_t) = \int_0^1 \left(f_t - \bar{F_t}\right)^2 pdf_{F_t}(f_t)  d(f_t)$$ (13)

Where $\bar{F_t}$ = $F_t^a$ (equations 1 and 10). The relative spread ($\sigma$($F_t$)/F$_t$) of the single grain $\alpha$-retention factor F$_t$ depends on the grain size (Figure 5). For very small grains (S/R $>$ 2), the spread is zero because all $^4$He is ejected (F$_t$ = 0), irrespective of the presence or absence of mineral inclusions. For very large grains (S/R $\approx$ 0), the spread of F$_t$ is also zero, because all $^4$He stays within the apatite (F$_t$ $\approx$ 1) and the chance that a mineral inclusion is located within the outermost fraction S/R of such an apatite is negligible. However, between these two extremes, the spread of the $\alpha$-retention factors is non-zero, reaching a maximum at S/R $\approx$ 0.6, where $\sigma(F_t)$/F$_t$ $\approx$ 0.2. Please note that because the F$_t$-distribution is not normally distributed (Figure 5), the 2$\sigma$-value of 40% must not be interpreted as the usual 95% confidence interval. However, by virtue of the Central Limit Theorem, the average of n single-grain measurements converges to a normal distribution with standard deviation $\sigma$(F$_t$)/$\sqrt{n}$. For example, the expected 2$\sigma$-spread (corresponding to a 95% confidence interval) of F$_t$-values for multi-grain packages containing n=10 grains each with one inclusion, or single grains with n = 10 inclusions is less than 12%. If S/R = 1/3 (e.g., S = 20 $\mu$m and R = 60 $\mu$m), then the 2$\sigma$ confidence interval for F$_t$ is $\sim$ 10% (Figure 5). These estimates are conservative because they assume that all the U and Th is contained in the mineral inclusions and that the host apatite itself contains no U or Th. If this is not the case, then the spread of the multi-grain ages will be smaller.

Figure 5: Relative spread of the $\alpha$-retention factors F$_t$ as a function of grain-size (S/R) and for different multi-grain sample sizes (n = 1, 2, 4, 10 and 50 grains, with $\sigma$(F$_t\vert$n) = $\sigma$(F$_t\vert$1)/$\sqrt{n}$).

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