AN INCREMENTAL-ISOTHERMAL MODEL FOR RADIOGENIC ISOTOPE ACCUMULATION WITH CHANGING TEMPERATURE

Accumulation of radiogenic daughter isotope reflects decay and loss processes that are functions of time (t) and temperature (T).  The theory of cooling ages described by the classic work of Dodson provides a framework to consider apparent age, kinetic parameters for loss, and closure temperature.  Its application, however, is restricted by several assumptions, including a monotonic cooling process and an initial zero daughter concentration that preclude many situations, for example, where heating and partially resetting are involved.  A forward-modeling, incremental-isothermal approach is proposed to overcome these restrictions and reveal the accumulation of radiogenic isotope (and apparent age) with time in any given T-t path.  It assumes that daughter isotope loss in nature can be described by an Arrhenius relationship for a first-order reaction or diffusion.  Using the incremental-isothermal model, a known or prescribed T-t path is discretized into many small intervals in which the temperature (T_i) is constant as is the reaction coefficient, k_i.  The solution to the rate equation applied to an individual interval i is, in simplified and general form: x_i=(lC_p/k_i)(1-e^-kiDt)+x_i-1e^-kiDt , where: x_i is the concentration of daughter at the end of time interval i; l is the decay constant; C_p is the concentration of the parent (here neglecting change with time); and Dt is the time step.  Among the advanges of such modeling are: 1) it allows simulation of any complex T-t path, including multiple stages of cooling and heating; 2) it does not require an assumption of zero initial daughter concentration and, thus, is suitable for middle-low temperature processes; and, 3) it can explicitly yield concentration or apparent age evolution for different minerals along a complex T-t path.  The application of the method is illustrated using the K-Ar system in various minerals and a variety of T-t paths.  The results of the modeling provide a new perspective for understanding the meaning of apparent isotopic ages.