A new phenomenological model for annealing of fission tracks in apatite: laboratory data fitting and geological benchmarking

Abstract

Rana track annealing model 2007, based on an Arrhenius approach, is significantly modified, including new assumptions, fitting parameters, and analysis, to yield a new phenomenological model for annealing of fission tracks in apatite. Construction and optimization of the new six parameter model have been described in detail. The model shows a hybrid behavior. In Arrhenius plot, for higher temperatures (laboratory data included), iso-annealing lines are curved while, for low temperatures (geological time scale), iso-annealing lines become parallel straight lines. This is a new feature for any such model and has produced promising results. C-axis projected lengths from laboratory experiments on Durango apatite are employed to find model parameters. KTB borehole fission-track data are added to build the temperature function that modifies the original equation. Partial annealing zone (PAZ), closure temperature ($T_{\mathrm{C}}$) and total annealing temperature ($T_{\mathrm{A}}$) were calculated and compared to geological benchmarks. The predictions of the present model agreed well with low and high temperature benchmarks. PAZ, $T_{\mathrm{C}}$ and $T_{\mathrm{A}}$ predictions were also compared to the Fanning Curvilinear Arrhenius model predictions, resulting in good agreement. The present model is flexible enough to be applied to other fission-track systems, like zircon, muscovite and titanite.

Appendix: Changing Rana (2007) model format

Here we show how to change the model Eq. (2) from its original format to the format $g(r)=f(t,T)$ that allow us to identify its behavior on the Arrhenius plot.

The model proposed by Rana (2007) is given by:

$$\begin{aligned} \ln r(t,T)=\beta \ln \left[ 1-\frac{t}{t_0} \exp \left( \frac{-E_{\mathrm{a}}}{kT}\right) \right] , \end{aligned}$$

(13)

Passing $\beta$ to the left side:

$$\begin{aligned} \frac{1}{\beta } \ln r = \ln r^\frac{1}{\beta } = ln \left[ 1-\frac{t}{t_0} \exp \left( \frac{-E_{\mathrm{a}}}{kT}\right) \right] \end{aligned}$$

(14)

Exponentiating both sides and rearaging terms:

$$\begin{aligned} 1-r^\frac{1}{\beta } = \frac{t}{t_0} \exp \left( \frac{-E_{\mathrm{a}}}{kT}\right) \end{aligned}$$

(15)

Applying the $\ln$ to both sides:

$$\begin{aligned} \ln \left( 1-r^\frac{1}{\beta }\right) = \ln t - \ln t_0 - \frac{E_{\mathrm{a}}}{kT} \end{aligned}$$

(16)

Changing parameters to $C_0 = -ln t_0$ ; $C_{1p}=1$; $C_{2p}= -E_{\mathrm{a}}/k$; $C_4=1/\beta$, the model equation becomes:

$$\begin{aligned} \ln \left( 1-r^{C_4}\right) = C_0 + C_{1p} \ln t + C_{2p} \frac{1}{T}, \end{aligned}$$

(17)

Equation (17) is a Parallel Linear model in Arrhenius pseudo-space.