G.M. Laslett

Thermal annealing of fission tracks in apatite 2. A quantitative analysis

Abstract An extensive set of laboratory annealing data, relating the reduction in mean confined track length of induced fission tracks in Durango apatite (Mexico) to annealing temperature and time, has been used to construct an empirical mathematical description of the annealing process. Firstly, Laplacian smoothing splines are used to reveal the gross nature of the dependence of the length reduction, r , on logarithm of time (ln( t )) and inverse absolute temperature ( T −1 ). This suggests that contours of equal length reduction in an Arrhenius plot can be described by parallel or only slightly fanning straight lines. In seeking a more rigorous description, we construct a series of models relating some function g ( r ) and another function f {ln( t ), T −1 }, which contain most previously published relationships, as well as a number of novel forms. Within this composite model, both parallel and fanning Arrhenius plots are considered. The models are fitted and compared using formal statistical methods. None of the previously suggested relationships satisfactorily describe the data, and a novel form is proposed with g ( r )=ln(1− r ) for the parallel Arrhenius plot. The best fitting model accounts for 96.7% of the variation of g ( r ), but residual plots show some structure; suggesting that some improvement in the model is possible. The best fanning model accounts for 98.0% of the variation in g ( r ), and gives a significantly better fit than the parallel model, with residual plots showing no obvious structure. The degree of fanning is much less than in most previously published Arrhenius plots for apatite, which may be due to the presence of apatites of various compositions in those previous studies, whereas the present study relates to only a single composition. The slight amount of fanning observed in this study may be an artefact introduced by several intermediate steps between the physical processes taking place during annealing and their manifestation as the reduction in mean track length.

Statistical models for mixed fission track ages

Abstract In fission track analysis it is common to find that the true ages of different crystal grains vary within a sample, and this may be important for geological interpretation. There are at least two well-recognized geological processes that lead to mixed ages: grains from multiple sources, and differential annealing between grains of differing composition. Data from multiple sources may be represented statistically by a finite mixture model, usually with two or three components, but data arising from the multicompositional annealing process may be better modelled as an infinite mixture. We discuss finite mixtures and two new infinite mixture models: a random effects model whose parameters describe the location and spread of the population grain ages, and a more general model encompassing both two-component mixtures and random effects. We illustrate with case studies how to use these models to estimate various features of interest such as the minimum age, the other component ages and the age dispersion.

Thermal annealing of fission tracks in apatite 4. Quantitative modelling techniques and extension to geological timescales

Abstract A methodology is presented for the prediction of fission-track parameters in geological situations from a laboratory-based description of annealing kinetics. To test the validity of extrapolation from laboratory to geological timescales, the approach is applied to a number of geological situations for which apatite fission-track analysis (AFTA) data are available and where thermal history is known with some confidence. Predicted fission-track parameters agree well with observation in all cases, giving confidence in the validity of the extrapolation, and suggesting that fission-track annealing takes place by a single pathway in both laboratory and geological conditions. The precision of predicted track lengths is considered in some detail. Typical levels of precision are ∼ ±0.5 μm for mean lengths ⪷ 10 μm, and ∼ ±0.3 μm for length ⪆ 10 μm. Precision is largely independent of thermal history for any reasonable geological thermal history. Accuracy of prediction is limited principally by the effect of apatite composition on annealing kinetics. The development of fission-track parameters is illustrated through a series of notional thermal histories to emphasise various key aspects of the response of the system. Temperature dominates over time in determining final fission-track parameters, with an order of magnitude increase in time being equivalent to a ∼ 10°C increase in temperature. The final length of a track is determined predominantly by the maximum temperature to which it is subjected. Aspects of AFTA response are further highlighted by prediction of patterns of AFTA parameters as a function of depth and temperature from a series of notional burial histories embodying a variety of thermal history styles. The quantitative understanding of AFTA response not only affords the basis of rigorous paleotemperature estimation, but also allows a better understanding of the situations in which AFTA can be applied to yield useful information.

Distinguishing dose populations in sediment mixtures: a test of single-grain optical dating procedures using mixtures of laboratory-dosed quartz

Many natural deposits contain grains that have different burial histories, but reliable procedures to extract the component doses from mixed-dose samples have not been developed in optical dating. Here we present results for synthetic two- and three-component mixtures of data derived from laboratory-dosed sedimentary quartz using a single-aliquot regenerative-dose protocol and statistical models for finite mixtures. Composite sets of data were created from the doses measured for individual grains that had been bleached by sunlight and then given a beta dose of 5, 10 or 20 Gy. We found that the correct number of dose components, corresponding doses, and relative proportions of each component may be estimated if the within-component dispersion is small or is known. We also found, however, that recuperation of the OSL signal in sun-bleached quartz may produce artificially high dose estimates for some grains, an effect that should be taken into account when modelling natural mixtures.

Bias in measurement of fission-track length distributions

Abstract Problems in the measurement of fission-track length distributions include biases in various methods of sampling and the amount of information about the true lenght distribution that can be recovered. It is concluded that all length measurements are biased and that this bias must be corrected before meaningful geological interpretations can be made. It is recommended that track-length measurements in minerals be restricted to horizontal confined fission tracks, because the length bias is then simple and easy to correct. Projected track-length measurements are not recommended because of complicated bias and insensitivity to important features of the true length distribution. These points are illustrated by length measurements on two Australian apatite samples.

Fission-track annealing in apatite: Track length measurements and the form of the Arrhenius plot

Abstract Previous fission-track annealing studies have described the reduction in fission-track density in terms of a series of fanning lines on an Arrhenius plot. This has been interpreted in terms of a range of activation energies corresponding to different degrees of annealing, with activation energies varying by a factor of 2 or 3 from complete retention to total erasure. New High precision measurements of confined track lengths in annealed Durango apatite, however, seem to be described by a single activation energy or only a very narrow range of energies (about 30%), implying a near parallelism of lines for various degrees of length reduction in an Arrhenius plot. Borehole studies have shown that different apatite grains respond to the same annealing conditions to differing degrees. Electron microprobe analyses of these apatites indicate that the annealing properties of individual grains are strongly controlled by their Cl/F ratio. The interpretation of laboratory annealing studies, and to a lesser extent borehole studies, in terms of fanning Arrhenius plots may be understood as the result of the superposition of a series of near parallel Arrhenius plots corresponding to the range of compositions present, each characterised by different activation energies.

Thermal annealing of fission tracks in apatite 3. Variable temperature behaviour

Abstract Despite considerable attention to isothermal behaviour in the literature, no satisfactory description of the variable temperature annealing behaviour of fission tracks in apatite has been given to date. Here, we show that our recently published constant-temperature annealing description can be adapted to temperatures which vary with time using the “principle of equivalent time”. This assumes that at any moment, a track which has been annealed to a certain degree r (= l/l0) behaves during further annealing in a manner which is independent of the conditions which caused the prior annealing, but which depends only on the degree of annealing that has occurred, and the prevailing conditions of temperature and time. Comparison of predictions of mean track length based on this assumption with observed values in a large number of laboratory variable-temperature annealing experiments shows good agreement, suggesting that the assumption is valid. Detailed inspection of the behaviour of tracks during heating and cooling shows that annealing is much more rapid at higher temperatures, and that temperature is the dominant factor in fission-track annealing in apatite. Extension of this treatment to geological situations is not straightforward and is left to a future paper.

Extending the age range of optical dating using single ‘supergrains’ of quartz

Abstract Luminescence dating methods have been used to obtain reliable age estimates for quartz sediments deposited within the last 500 ka, but it has proven difficult to extend the age range much beyond this limit. Here we report the results of a study of individual quartz grains from Australian sedimentary deposits that range in age from ∼250 to ∼950 ka. A small number of the grains examined are strongly luminescent and saturate at unusually high doses. These ‘supergrains’ may permit reliable age determination to 1 Ma, and possibly beyond. Some other grains are in, or close to, dose-saturation, so that only minimum age estimates may be obtained. Most of the grains examined are very weakly luminescent and have palaeodoses much less than expected, while the palaeodoses of some grains cannot be estimated because of anomalous dose–response characteristics. We offer some possible explanations for the behaviour of aberrant grains.

The relationship between fission track length and track density in apatite

Abstract Fission track dating is based upon an age equation derived from a random line segment model for fission tracks. This equation contains the implicit assumption of a proportional relationship between the true mean length of fission tracks and their track density in an isotropic medium. Previous experimental investigation of this relationship for both spontaneous and induced tracks in apatite during progressive annealing has not confirmed the line segment model in an obvious fashion. Corrected equations relating track length and density for apatite, an anisotropic mineral, show that the proportionality in this case is between track density and a length factor which is a generalization of the mean track length combining the actual length and crystallographic orientation of the track. This relationship has been experimentally confirmed for induced tracks in Durango apatite, taking into account bias in sampling of the track lengths, and the effect of the bulk etching velocity.

Apatite fission track analysis: geological thermal history analysis based on a three-dimensional random process of linear radiation damage

Spontaneous fission of uranium atoms over geological time creates a random process of linearly shaped features (fission tracks) inside an apatite crystal. The theoretical distributions associated with this process are governed by the elapsed time and temperature history, but other factors are also reflected in empirical measurements as consequences of sampling by plane section and chemical etching. These include geometrical biases leading to over-representation of long tracks, the shape and orientation of host features when sampling totally confined tracks, and ‘gaps’ in heavily annealed tracks. We study the estimation of geological parameters in the presence of these factors using measurements on both confined tracks and projected semi-tracks. Of particular interest is a history of sedimentation, uplift and erosion giving rise to a twocomponent mixture of tracks in which the parameters reflect the current temperature, the maximum temperature and the timing of uplift. A full likelihood analysis based on all measured densities, lengths and orientations is feasible, but because some geometrical biases and measurement limitations are only partly understood it seems preferable to use conditional likelihoods given numbers and orientations of confined tracks