Page 34 - Caldera Volcanism Analysis, Modelling and Response

Residence Times of Silicic Magmas Associated with Calderas 9

record any prolonged magmatic history. From this perspective, subtracting the ages
39
obtained from U- and Th–Pb methods form those of K–Ar or 40 Ar/ Ar should
be a reliable proxy for magma residence time. One can also start with Fick’s second
law of diffusion and include an additional term for the production of the daughter
isotope:
2
@C DA @ C DA
¼ DðtÞ DA þ lN o expð ltÞ (2)
@t @x 2
where C DA is the concentration of the daughter, N o the number of parent atoms
present initially, D(t) DA the diffusion coefficient which depends on time through
the thermal history, D(t)=D o exp [ E/RT(t)], but not on composition or position,
and x is distance (in one dimension). Solving this equation with finite difference
numerical techniques can also address the closure temperature and enables the
implementation of almost any initial and boundary condition and cooling history.
The attainable information using the closure temperature concept has not been
much exploited for magmatic processes but has a long history in metamorphic
rocks (e.g., Ganguly, 2002) and in low-temperature chronology (e.g., thermo-
chronology; Reiners and Ehlers, 2005). Recent detailed geochronological results
40 39
of the Fish Canyon system (Bachmann et al., 2007b) show that Ar/ Ar total
fusion ages of sanidine are slightly younger than those of the other minerals
(plagioclase, hornblende and biotite) which could be explained by the effects the
closure temperatures discussed here.

2.2. Chemical diffusion, relaxation of mineral zoning

The discussion above has already introduced the notion of using diffusion
modelling to retrieve time scales of magmatic processes. The technique exploits the
presence of gradients of concentration (or chemical potential) which dissipate or
tend to equilibrium at a rate that has been experimentally calibrated (e.g., diffusion
coefficient). The idea is that the zoning in minerals records the conditions and
composition from the environment in which they reside. Depending on the rate of
chemical diffusion of the elements in the crystal and surrounding matrix, different
elements will reequilibrate to different extents. Because diffusion is exponentially
dependent on temperature, and diffusion rates of typical elements in most
geological materials are quite slow at room temperature, the mineral compositions
that we measure reflect conditions of much higher temperatures; i.e., closure
temperature above. To solve Fick’s second law (e.g., Equation (2) without the
production term on the RHS) we need initial and boundary conditions. It is crucial
to have criteria to establish these as close as possible to the natural system we are
studying. In this sense, the uncertainties in initial and boundary conditions are
similar to those discussed above for the radioactive isotopes techniques.

2.2.1. Initial and boundary conditions
The initial conditions are the concentration distribution within the crystal prior
to the main episode of diffusion. They vary depending on the problem at hand.