A Monte Carlo approach to diffusion applied to noble gas/helium thermochronology

Abstract

Knowledge of the diffusion domain is of primary importance for age interpretation in noble gas thermochronometers. We have developed a Monte Carlo method to solve the diffusion equation in three-dimensional space and have used it to examine the effect of realistic crystal geometries and anisotropy on noble gas diffusion. The method is based on the simulation of Brownian motion with a modified distribution of collision distances and with a variable mean free path. This approach drastically reduces calculation time while remaining accurate. This original approach is able to treat isotropic and anisotropic diffusion, any 3D shape, ejection and zonation. A code simulating production, ejection and diffusion from the grain to the external medium has been implemented to compute helium ages of minerals subjected to temperature histories. In parallel, another module has been developed to simulate diffusion experiments and diffusion coefficient determination for all types of He profiles in a grain (homogenous, depleted edge due to ejection, heterogeneous profile due to previous diffusion, etc). Both types of simulations are suitable for isotopic and anisotropic diffusion; we develop examples for apatite and zircon (U–Th)/He thermochronology but the method can be applied to any other noble gas thermochronometer. The Monte Carlo simulation reproduces the He age variation obtained by other calculation methods for simple geometries and for well-known thermal histories, demonstrating the viability of the tool. In the case of isotropic diffusion, we show that generally even for realistic shapes with many ridges the He age resulting from the diffusion can be well calculated by assuming a spherical shape of the same surface/volume (S/V) ratio. The only requirement for adequate representation of grains by spheres is thus accurate knowledge of their true shapes and dimensions. For anisotropic diffusion, we introduce a new concept termed “active radius”, which describes the complex anisotropic diffusion process by isotropic diffusion in a sphere. In this sense, the active radius can be seen as an extension of the sphere-equivalent radius to the anisotropic case. The active radius can be computed for any geometrical shape without Monte Carlo sampling, and a separate simple code is made available for its computation.

Introduction

For noble gas thermochronology, such as (U–Th)/He, (U–Th)/Ne and Ar–Ar (Zeitler et al., 1987, McDougall and Harrisson, 1999, Reiners and Farley, 1999, Reiners et al., 2004, Harrison and Zeitler, 2005, Farley and Clark, 2006, Gautheron et al., 2006, Farley, 2007), measured ages reflect the radioactive production of atoms from parent nuclei, their ejection and diffusion into the grain volume and leakage into the embedding medium. Proper estimation of the diffusion domain is important for the interpretation of the ages from these thermochronometers. In the case of (U–Th)/He thermochronology, for example, several calculations have been developed in the past ten years to describe the production–diffusion kinetics of helium, aiming at determining the closure temperature or the He age for a given temperature history. The first widely used model was developed by Wolf et al. (1998) and assumed simple sphere, infinite cylinder, and infinite plate geometries. Reiners and Farley (2001) have applied this method and used the sphere radius to illustrate the effect of grain size on He age. Meesters and Dunai (2002) extended the range of applicability by developing a numerical method limited to finite cylinder and rectangular block geometries. However the two aforementioned methods can hardly be extended beyond the simple geometries quoted above and do not allow simulation of complex shapes like hexagonal prisms with pyramidal terminations, which is a common feature of apatite grains. Herman et al. (2007) solved the diffusion equation for grains represented as clusters of cubes. More recently, Watson et al. (2010) studied anisotropic diffusion, but limited themselves to cylindrical geometries. Thus, the impact of the real geometrical shape on the diffusion rate out of the crystal remains an open question, which is even more crucial when anisotropic diffusion takes place as for example in zircon (Reich et al., 2007, Cherniak et al., 2009, Saadoune and De Leeuw, 2009, Saadoune et al., 2009).

To obtain better insight into these issues and to have at our disposal a numerical tool to calculate the diffusion process in any complex geometry, we have developed a simulation based on a Monte Carlo method. One part of this simulation has already been applied to a geological example (Gautheron et al., 2009). In the present article we focus on the description of the method, geometrical effects on the diffusion rate, and ultimately the age determined from the gas remaining in the crystal. In addition, a method to calculate a sphere equivalent surface/volume (S/V) ratio in case of isotropic and anisotropic diffusion is proposed. The main advantage and specific features of the Monte Carlo simulation lies in that it allows the treatment of diffusion in any complex geometry. More precisely, the description of diffusion for a grain of realistic shape, either for helium dating or for experimental determination of diffusion coefficients, can be easily handled. The Monte Carlo simulation in this paper is illustrated for the case of apatite and zircon (U–Th)/He and for isotopic and anisotropic diffusion examples, but can be extended to more general noble gas thermochronology.