# Diffusion chronometry
## Introduction

Diffusion chronometry is a geological technique that examines the chemical zoning within phenocrysts to estimate the duration of events in a magmatic system (e.g. magma mixing, magma decompression, magma degassing, etc.) and their associated eruptions. In simple terms, the phenocryst acts as a “crystal clock” to determine how long a magmatic process took. Diffusion chronometry takes advantage of the common chemical disequilibrium found in volcanic phenocrysts and examines their kinetic window to obtain timescale information. Diffusion chronometry should not be confused with geospeedometry. Both utilize the theory of chemical diffusion, but the latter is more commonly used for analyzing rates of metamorphic events whereas the former is used to analyze features of eruptive events.
Eruptive events can occur on a time scale of days or months which, geologically speaking, is very hard to analyze using traditional radiometric dating techniques. The use of diffusion chronometry on past eruptions can help in forecasting eruptions by gaining a better understanding of the timescales of magmatic processes and can aid in determining timescales for volcano alert levels during an eruptive event.

## Kinetics

If there is a chemical gradient present in the crystal, then Fick’s second law can be applied to calculate the timescale of the magmatic process that produced the gradient(Chakraborty 2008, Lasaga 2014, Costa et al 2020). Fick’s second law is:
$\partial C/\partial t=D*\partial ^{2}C/\partial x^{2}$
Where C is the concentration of the element in question, t is time, D is the diffusion coefficient, and x is distance. A common analytical solution to Fick’s second law that is used in diffusion chronometry is:
$C=C_{2}+((C_{1}-C_{2})/2)*Erfc(x/(2{\sqrt {Dt}}))$
Where C is concentration at the current point, C$_1$ is the concentration of the depleted region, C$_2$ is the concentration of the enriched area, Erfc is the complementary error function, D is the diffusion coefficient, and t is the time. A common finite difference numerical solution to Fick’s second law is:
$C_{2}new=C_{2}old+(D\Delta t/\Delta x^{2})*(C_{3}old-2C_{2}old+C_{1}old)$
Where C$_{2new}$ is the concentration of the second point at the current timestep, C$_{2old}$ is the concentration of the second point at the previous timestep, C$_{3old}$ is the concentration of the next point at the previous timestep, C$_{1old}$ is the concentration of the previous point at the previous timestep, Δt is the timestep size in seconds, and Δx is the step distance, and D is the diffusion coefficient. If the (DΔt/Δx$^2$) term is larger than 0.5, this finite difference model is unstable and will not produce a usable result.

## Experimental Methods

Geochemical experiments are performed in order to determine the diffusivity of an element in a mineral. The three most common types of experiments are diffusion couple experiments, thin-film experiments, and powder source (constant surface) experiments. In the diffusion couple experiment, 2 of the same minerals, with different concentrations of the intended diffusant element, are fused together and put in a high temperature furnace to induce diffusion across the mineral boundaries. Thin film experiments are set up with a thin film (nm to μm thickness) doped with the diffusant element is fused to the surface of the mineral and is put in a high temperature furnace to induce diffusion from the thin film to the mineral. The powder source experiments are set up with the mineral packed into and surrounded by a powder source doped in the diffusant element within an alumina or quartz glass crucible and placed in a high temperature furnace to induce diffusion from the powder source to the mineral. Experiments are typically run at varying temperatures (and to a lesser extent pressure and oxygen fugacity), with the goal of producing an Arrhenius plot. From the Arrhenius plot the frequency factor (D$_0$) and the activation energy (E$_a$) can be determined for the specific mineral-element pair, these terms are then used in applying the diffusion equation to determine the diffusivity of the element in the mineral at the magmatic conditions of the system.
With the growth of new technology and therefore new experimental methods and analytical methods, the number of available element-mineral diffusivities are growing rapidly.

## Measuring Diffusion Profiles

Samples are typically prepared as thin sections or crystal separates. Thin sections are thin slices (typically 30 um) of rock that allow the mineral to be analyzed in-situ. Crystal separates involve either crushing a rock or sampling a tephra layer and then picking through the material to find the required mineral and then mounting it in epoxy for analysis.
Depending on the relative abundance of the element of interest, this will dictate what analytical methods you use. For major elements, on the order of a few weight percent, the common analytical tool electron microprobe analyses. For trace elements, on the scale of ppm, the common analytical tool is LA-ICP-MS. If the study is interested in analyzing the diffusion of isotopes, then Secondary Ion Mass Spectrometry (SIMS) or Multi-Collector LA-ICP-MS are the most common analytical tools.
When selecting analytical points to measure a diffusion profile, a certain spacing should be used based on the ratio of the diffusion length to spot size (ratio greater than 2) to avoid convolution effects.

## Applying Diffusion Chronometry

To apply diffusion chronometry an appropriate mineral-element pair which represents the process that is to be timed must be chosen. As seen in ure 7 different mineral-element pairs capture different timescales and therefore different magmatic processes. From there, a temperature estimate for the system in question must be determined. For example, temperature can be calculated through geothermobarometry. Though the uncertainty on temperature can propagate through calculations and greatly influence the timescale extracted from the crystal. Using experimentally derived D$_0$ and E$_a$, temperature is plugged into the diffusion equation to get the diffusivity of the mineral-element pair in the system of question.
$D=D_{0}\exp(-E_{a}/RT)$
Where D is the diffusion coefficient, D$_0$ is the frequency factor specific to your mineral element pair, E$_a$ is the activation energy in Joules, R is the gas constant, and T is temperature in Kelvin. This diffusivity (D) is then used in the appropriate analytical or numerical solutions to Fick’s second law to calculate a timescale. Diffusion chronometry timescales typically have large uncertainties or ranges of timescales, but this information is critical when forecasting an eruption.

## Recent Applications of Diffusion Chronometry

The diffusion profiles of crystals within a single eruption have been used to constrain the time of magmatic processes at a variety of systems. This section will briefly highlight a number of studies where diffusion chronometry was successfully applied. Note that this list is not comprehensive.
Diffusion chronometry was applied to the 1999 eruption of Shishaldin volcano and displayed the connection between volcano monitoring signals and diffusion timescales from Fe-Mg interdiffusion in olivines from the eruptive product. Diffusion timescales showed that eruption run-up started 9 months before the eruption, with more immediate run-up occurring 50 and 10 days prior to the eruption. The 9-month diffusion timescales correlate well with the start of deep earthquakes, a typical precursor to eruption, the ~50 day timescales correlate well with a temperature anomaly (i.e. intruding magma), steam, and a 5.2 magnitude earthquake, and the ~10 day timescales correspond to rapid remobilization of the magma during its final ascent to the surface. This study by Rassmusen and colleagues in 2018, is a prime example of how diffusion chronometry can aid in forecasting volcanic eruptions.
Shamloo and Till (2019) applied Ba and Sr in sanidine and Ti in quartz diffusion chronometry to the Lava Creek Tuff in Yellowstone Caldera, to understand the timing of eruption triggers for super eruptions. Here they determined that the super eruption occurred years to at most a decade after magma rejuvenation occurred, triggering the Lava Creek Tuff super eruption.
Diffusion chronometry profiles helped reconstruct the dynamics of the magma plumbing system underneath Mount Etna during 1991-1993. This study employed Fe-Mg interdiffusion in olivines to determine the timescales of magmatic processes that a number of crystal populations experienced and determined 3 significant timescales. The first timescale tracked magma transfer between differing magmas and occurred 3–6 months before eruption. The second timescale is a short residence time of less than a month which is thought to be the final mixing process. The third and longest timescale, up to 1 year, represents crystals that were recycled through different magmatic environments prior to eruption.