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The role of crystallographic angle in characterizing and modeling apatite fission-track length data
Radiation Measurements 39 (2005) 595 – 601 www.elsevier.com/locate/radmeas
The role of crystallographic angle in characterizing and modeling apatite fission-track length data Richard A. Ketcham∗ Jackson School of Geosciences, The University of Texas at Austin, Austin, Texas 78712, USA Received 13 February 2004; accepted 13 July 2004
Abstract Recent experimental work on apatite fission-track length measurement has documented a number of factors that can exert a considerable influence on determinations of mean length. The principal source of length variation is anisotropy of annealing and etching behavior with respect to crystallographic angle. Procedural and environmental factors that influence which angular populations are preferentially sampled during measurement lead to variations in mean length that exceed standard statistical predictions. This paper evaluates the possibility of using c-axis projection to remove angular effects, and thus make length data more reproducible and informative. © 2005 Elsevier Ltd. All rights reserved. Keywords: Fission-track; Apatite; Annealing; Biasing; Thermal history inversion
1. Introduction As part of a systematic study of apatite fission-track thermochronology, Barbarand et al. (2003b) documented several sources of variation in measurement of confined horizontal track lengths. Among the topics they addressed were reproducibility of measurements by a single observer, compatibility of measurements among different observers, and the effects of Cf irradiation. Based on their observations they deemed it “essential” to take the effect of track angle to the apatite crystallographic c-axis into account when interpreting or utilizing track length data quantitatively. It has long been recognized that fission-track annealing in apatite is anisotropic (Green and Durrani, 1977). Unannealed or lightly annealed fission tracks exhibit only minor anisotropy, which may be attributable to etching effects. As annealing progresses, however, tracks at high angles to the c-axis shorten more quickly than tracks at low angles (Green and Durrani, 1977; Green et al., 1986; Galbraith and ∗ Tel.: +1 512 471 6942; fax: +1 512 471 9425.
E-mail address: [email protected] (R.A. Ketcham). 1350-4487/$ - see front matter © 2005 Elsevier Ltd. All rights reserved. doi:10.1016/j.radmeas.2004.07.008
Laslett, 1988; Galbraith et al., 1990; Crowley et al., 1991; Donelick, 1991; Donelick et al., 1999; Ketcham, 2003). Apatite etching is also anisotropic, as reflected by faster etch figure widening rates in the c-axis direction. Ravenhurst et al. (2003) searched for and found no strong evidence of a link between degree of track length anisotropy and etching strength, but etching does have subtle biasing effects that influence which horizontal confined tracks are intersected and measured. Etch figures, where semi-tracks intersect the polished grain surface, tend to be oblong in the direction of the c-axis, and confined horizontal tracks at high angles tend to be wider, easier to see, and appear more completely etched than tracks at low angles. Tracks very near to perpendicular to the c-axis are an exception to this trend, as they also frequently appear incompletely etched. The physical process behind length anisotropy in apatite is unknown, but in view of the above evidence it is probably linked most strongly to annealing rather than etching or registration effects such as channeling. Annealing and etching anisotropy manifest themselves in fission-track length measurements in three primary ways: length bias, under-etching bias, and track loss. Length bias
R.A. Ketcham / Radiation Measurements 39 (2005) 595 – 601
is the increased probability of intersecting longer tracks relative to shorter ones with etchant pathways, leading to the preferential detection of long tracks (Laslett et al., 1982, 1984). For TINT (track-in-track) measurements, the length bias is modified by etching anisotropy, as the inclined semi-tracks used as etchant pathways present a larger crosssection for intersection for high-angle tracks than low-angle ones (Galbraith et al., 1990; Ketcham, 2003). Under-etching bias is caused by thinness and a tendency toward apparent incomplete etching of low- and very high-angle tracks, particularly when strong etchants (5 or 5.5 M HNO3 ) are used. Such tracks tend to be either overlooked or bypassed in favor of better-defined tracks. For apatites featuring high etching anisotropy, including all F-apatites, under-etching bias exerts a far greater influence on probability of track measurement than length bias. This is evidenced by low relative frequency of observation of these angular populations on apatites at low levels of annealing (Donelick et al., 1999, Fig. 10); the same trends are apparent in Fig. 2 of Barbarand et al. (2003b). Track loss occurs at advanced stages of annealing, when the rate of shortening of high-angle tracks accelerates greatly relative to low-angle tracks. Although some measurable high-angle tracks may persist and be measured, some proportion of them will anneal below the detection limit, which is defined by the requirement to reliably identify a track and distinguish its endpoints. The evidence for this phenomenon is a steadily increasing relative probability of detection for low-angle tracks that is far in excess of what is predicted by length biasing (Ketcham, 2003). Although the importance of track angle for interpretation of apatite fission-track data has long been recognized in theoretical studies (e.g., Galbraith and Laslett, 1988; Galbraith et al., 1990), to date the only fully developed means of utilizing angular data is c-axis projection (Donelick et al., 1999; Ketcham, 2003). The goal of c-axis projection is to remove crystallographic orientation effects from length data while retaining the thermal history information. The technique consists of constructing a transform function that, for any individual track measurement of length and angle (li , i ), estimates the length distribution of tracks with = 0, or parallel to the c-axis (lc , c ), that have experienced the same annealing conditions (Donelick et al., 1999). The model was subsequently refined to include an estimate of the uncertainty in the projection (lc ) (Ketcham, 2003). The purpose of this paper is to briefly examine some of the results presented by Barbarand et al. (2003b) and its companion paper (Barbarand et al., 2003a) in the context of c-axis projection. In addition, the prospects for constructing an empirical annealing calibration based on the annealing data of Barbarand et al. (2003a) will be discussed.
2. Reproducibility and observer compatibility Barbarand et al. (2003b) documented that the reproducibility of mean track length measurements of experi-
mentally annealed apatites is frequently outside the bounds estimated by the standard error of the mean (SEM). A larger than predicted variation was observed both in repeated measurements by a single observer and between different observers. In the case of highly annealed populations, this divergence is in part traceable to the fact that the SEM is not an appropriate estimator for uncertainty when the underlying distribution being sampled is heavily skewed or bimodal. Also, a substantial component of variation is likely to be based on observer tendencies and thus is systematic, rather than random. In all cases where larger than expected variation was observed, Barbarand et al. (2003b) found that the divergence could be explained by inspecting plots of track length versus angle. These plots indicate that there exists a regular length–angle relationship that characterizes the level of annealing produced in an experimental mount (i.e. in which all natural tracks were annealed, and fresh tracks induced in a nuclear reactor were annealed under controlled conditions in a laboratory furnace), and that poor reproducibility was brought about by differential sampling of this distribution. Because track length varies markedly with angle, particularly in highly annealed populations, when different angular ranges are preferentially sampled the measured mean length is biased accordingly. Because c-axis projection seeks to characterize the relation between length and angle at a given level of annealing, it provides a reasonable approach for solving the problem of reproducibility for single and multiple observers. By projecting all track lengths to a common orientation, the biasing effects of sampling different angular populations are eliminated. As an example, Fig. 1 shows a selection of contours of 20
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φ (angle to c axis, degrees) Fig. 1. c-axis projection contours from Ketcham (2003) superimposed in Fig. 4b from Barbarand et al. (2003b). Contours from top to bottom are for lc values of 12.7, 11.9, 11.1, and 10.3 m. Track lengths are converted to their c-axis-projected equivalents by tracing along the contours to the = 0 axis.
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the Ketcham (2003) c-axis projection model superimposed on the data shown in Barbarand et al. (2003b, Fig. 4b). The data are fairly well bounded by contours for lc of 12.7 and 10.3 m, which matches well with the expectation that the distribution of c-axis projected tracks should have a standard deviation (c ) of about 0.8 m. It is likely that the two data sets have very similar mean c-axis projected lengths, but this cannot be tested because the individual length data are unavailable. It should also be noted that the Ketcham (2003) c-axis projection model is based on the Carlson et al. (1999) annealing data set, which was compiled by a different observer using slightly different etching conditions; it is unknown whether these differences will have a substantial effect on c-axis projection.
3. Cf irradiation Cf irradiation is a means of increasing the number of measurable tracks by bombarding the polished mount with fission particles from a 232 Cf source, creating more etchant conduits (Donelick and Miller, 1991). In order to assess the possible biasing effects of Cf irradiation, it is necessary to consider how the population of latent horizontal tracks within several microns of the polished surface of a grain mount is sampled. First we envision a “normal” case in which there are relatively few etchant conduits penetrating the crystal surface. One version of the standard Poissonian model is to assume that detection of a particular latent horizontal track is very unlikely. In this case, the line segment model predicts that the probability that a latent track will be intersected is proportional to its length (Parker and Cowan, 1976; Laslett et al., 1982), modified by geometric factors such as etchant path shape (Galbraith et al., 1990; Ketcham, 2003). In fact, the under-etching bias is a more influential effect than length, but the underlying principle of sampling being very unlikely still applies. If very few of the available tracks are detected and measured, the resulting set of lengths will be a biased representation of the true underlying distribution. The opposite case is a grain surface so perforated with Cf-induced tracks that every available latent track is detected: all short tracks are intersected, and all tracks susceptible to under-etching are impinged multiple times, facilitating more complete etching. In such a case sampling would no be longer Poissonian, but complete. This is only a theoretical end-member case, as a completely perforated grain surface would be obliterated thus making tracks unmeasurable, but it demonstrates that there exists a continuum of possible biasing, from Poissonian to no bias. Grains that have a high density of either Cf-induced or naturally occurring tracks traverse some of the distance between Poissonian and complete sampling, with the effect on each bias being the same: tracks that are under-represented under Poissonian sampling become more frequently seen and measured.
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It should be mentioned that the more traditional Poissonian model for length biasing is based on the rarity of intersections between etchant pathways and latent confined horizontal tracks, rather than detections, which can involve multiple intersections. This model led Laslett et al. (1982) to suggest that tracks that are intersected twice should have their measurements counted twice, to ensure consistent length biasing. Although this practice would also apply for Cf-irradiated mounts, it is not widely used, in part because it is often not straightforward to discern if nearby tracks are impinging or not. Furthermore, if the concept of under-etching bias is valid, it is arguable that this protocol actually has the opposite of the desired effect of maintaining consistent biasing. If a low-angle track needs to be intersected twice to be fully etched and thus measured, it should be counted zero times rather than twice to match the biasing inherent in low-track-density mounts. Neither the Carlson et al. (1999) nor the Barbarand et al. (2003a) annealing data sets were acquired using this protocol. Barbarand et al. (2003b) ran a series of experiments in which mounts with and without Cf irradiation were measured. They documented among populations with mean track lengths below 14 m an increase in mean track length of roughly 5% when Cf irradiation was used. In interpreting these results, they postulate that “the introduction of additional etching conduits may increase the probability of intersecting longer tracks at the expense of shorter tracks, in effect a variation of the normal TINT length bias”; but this explanation is incorrect. As outlined above, the introduction of numerous additional etching conduits decreases biasing, and thus should lead to proportionally more short tracks being measured if the only biasing factor was track length. A more probable explanation for their results is that, along with the length bias, the under-etching bias was also decreased, leading to increased revelation of longer, low-angle tracks. The Carlson et al. (1999) data set also includes several experiments in which mounts were measured using Cf irradiation, then re-polished, re-etched and re-measured to obtain length measurements without the aid of Cf-induced tracks. These data are somewhat divergent from Barbarand et al. (2003b), but share the same underlying features (Fig. 2). Mean track length measurements (closed symbols) in lessannealed populations were relatively unaffected by Cf irradiation, but at advanced stages of annealing populations featured higher mean track lengths with Cf irradiation, with divergences significantly beyond statistical expectation. Also plotted in Fig. 2 are the corresponding mean c-axis projected lengths for these experiments (open symbols), which show much less effect from Cf irradiation. The reason is analogous to that cited in the previous section: Cf irradiation does not change the underlying length-angle distribution, but it does change which tracks in this distribution are sampled. By projecting all tracks to a common orientation, this source of variation is removed. Plots of individual experiments from Carlson et al. (1999) again illustrate this trend (Fig. 3). In the highly annealed DR
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and RN populations, Cf irradiation preferentially revealed long, high-angle tracks, showing that its main effect was to diminish the under-etching bias. In the near-end-member Cl apatite B3, Cf-irradiated mounts gave slightly shorter mean lengths, though not outside the bounds of expected variation (Cf: 11.82 ± 0.09 m versus no Cf: 11.93 ± 0.15 m for run 63 (shown) and Cf: 11.38 ± 0.13 m versus no Cf: 11.53 ± 0.13 m for run 66). However, these results are consistent with the expectation of under-etching bias reduction in this unusual apatite. B3 shows very little under-etching bias for low-angle tracks, but near-perpendicular tracks are underrepresented (Donelick et al., 1999, Fig. 10). Because these tracks are shorter than those at lower angles, increasing their representation would tend to slightly lower mean length. The experiment with the worst match between mean c-axis projected lengths with and without Cf irradiation is also shown (F-apatite PQ, run 63). In this case, the non-Cf tracks fall within the length-angle distribution defined by the Cf-irradiated population, but do not sample it evenly. This experiment is in the very final stages of annealing, in which virtually all angular populations show evidence of
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Fig. 3. Polar diagrams of four experiments from Carlson et al. (1999) showing difference between length measurements with and without Cf irradiation. Note that analyses of mounts with plentiful confined tracks were halted after 110 measurements.
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accelerated length reduction. This case is not well described by current c-axis projection models. However, another feature of such fission-track populations is that they have extremely low density, so they will rarely be sampled in practice, and when they are will only have a small influence on thermal history inversions.
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Ketcham et al. (1999) developed a means of combining apatites with disparate annealing behaviors into a single multi-kinetic model by relating pairs of apatites to each other with the equation: rmr − rmr0 rlr = , (1) 1 − rmr0 where rlr and rmr are the reduced (normalized) lengths of apatites that are less resistant and more resistant to annealing, respectively, and rmr0 and are fitted parameters. The rmr0 parameter is the reduced length of the more resistant apatite at the point where the reduced length of the less resistant apatite falls to zero (i.e. it is fully annealed). The equation is fitted by comparing length measurements of apatites that were annealed in the same furnace at the same time, and thus have identical thermal histories. It was found that Eq. (1) adequately describes the variation observed between apatites at laboratory time scales, and that extrapolation to million-year time scales does not significantly change predictions of thermal behavior in geological settings. Creation of a multi-kinetic annealing model based on the Barbarand et al. (2003a) data, or perhaps a combined model utilizing the Carlson et al. (1999) data as well, could be possible using this approach. Fig. 4 shows an example rlr − rmr plot for mean length data from Carlson et al. (1999), and the effect that differential biasing due to Cf irradiation can have on the fit. The two measurement pairs that utilized Cf irradiation (on both apatites) plot off of the main trend defined by the other experiments. Interestingly, removal of both the positive displacement of the DR mean length and the negative displacement of the B3 mean length caused by Cf irradiation brings these experiments into line with the main trend. Conversely, B3 and DR rc,mod values are not significantly different between experiments with and without Cf irradiation (mean change of 0.10 m). The Barbarand et al. (2003a) data are mixed with respect to the use of Eq. (1) (Fig. 5), but promising overall. Their BAM-DUR data (Fig. 5a) line up well, and are well modeled by the fitted line. Some of the Cf-irradiated measurements of GIL apatite plot off of the trend defined by other experiments (Fig. 5b), suggesting that a Cf-induced shift in bias has resulted in an incompatibility in their respective mean length data. Fig. 5c shows a problematic case, in which the experiments with annealing times of 10 h plot to either side of a line that fits the longer duration experiments. It is un-
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known whether this scatter is caused by problems in the annealing schedule or from other sources of variability, including the use of non-projected lengths.
5. Discussion The examples shown above corroborate the claim by Barbarand et al. (2003b) that angular data are necessary for proper interpretation of annealing experiments and inversion of thermal histories. c-Axis projection appears to be a promising avenue for removing many of the sources of variability that beset track length measurements, including interand intra-observer reproducibility and Cf irradiation. It also may provide a remedy for other concerns listed by Barbarand et al. (2003b). In particular, they note that the measurement of short tracks is very influential on mean track length, and thus rigorous measurement criteria for short tracks need to be defined and standardized. Although standardization is never a bad idea if it is practical, the use of c-axis projection may obviate much of the need for it. Likewise, Barbarand et al. (2003b) propose that etching procedures be standardized, but it is likely that c-axis projection can provide a fully effective conversion between measurements made using different etching methods. Both of these assertions are based on the same principle discussed above: the population of latent tracks available for measurement exists independent of the observer, etching technique, number of etchant
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Fig. 5. Apatite-apatite plots for various experiments by Barbarand et al. (2003a). Error bars are 2SEM when available, and experiments that retained some tracks but at a density too low to measure are plotted as zero with error bars extending to a reduced length of 0.35.
pathways, and all other procedural and environmental variables. The principle source of length variation in this population is linked to angle to the apatite c-axis, and the tendency of various factors to influence which angular populations are sampled more frequently than others. Projection to a common angle may correct for virtually all such factors in a single step. It should be noted that projection may not be the only approach—an alternative is to develop a schema based on E(l|) (expected confined length conditional on c-axis angle ) whereby angular data may enable derivation of additional thermal history information (Galbraith, 2002). However, no such alternative has yet been sufficiently developed. A possible impediment is that it may re-emphasize user-specific effects that may be difficult to calibrate for, such as whether certain short tracks are measured or not, which will have a large effect on E(l|) at some values of . A different and perhaps more promising approach may be to construct a method based on E(lc , lc |l, ), as lc provides an index of thermal equivalence while lc estimates the precision of that index.
Once angular effects have been removed, the only remaining issue for thermal history inversion is the relative probability of sampling the different populations present in geological samples that reflect varying amounts of annealing. This is usually estimated (e.g., Willett, 1997; Ketcham et al., 2000) using the length versus density relationship reported by Green (1988), although it is only an approximate solution, and is never re-calibrated. It is likely that any adjustments would be small and subtle, but their effect on inversion has not been studied. It should also be noted that the issue of proper consideration and calibration of the length versus density relationship is not fully resolved and exists independent of c-axis projection (Jonckheere, 2003). A valid critique of c-axis projection is to question whether this fairly complex transform, with six free parameters, is warranted, or if it diminishes the thermal history information contained in track lengths. In view of the available experimental data, however, it may be more reasonable to ask how continued use of length data without angular information can be justified. The recent work has demon-
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strated that “standard” mean track length data are prone to shifting biases that are subtle to quantify and predict but exert a considerable influence on results, to the extent of rendering currently used estimates of statistical uncertainty demonstrably inadequate. With respect to calibrating empirical or physical models of fission track annealing based on length reduction (e.g., Laslett et al., 1987; Carlson, 1990; Crowley et al., 1991; Laslett and Galbraith, 1996; Ketcham et al., 1999), it is important to recognize that measured mean lengths are not a direct depiction of length reduction, but rather a strongly skewed observation. Because zero-length tracks are not included in calculations of the mean, the link between measured and actual physical length reduction is very indirect. Conversely, models based on c-axis projected data are self-consistent, as the c-axis parallel population they characterize is not subject to accelerated length reduction and track loss until complete annealing is imminent. Furthermore, the onset of accelerated length reduction once tracks fall below ∼ 10 m is likely caused by a change in the physical process of annealing (e.g., Carlson, 1990); depicting only one annealing process, rather than two, is less demanding for a physical model and less of a stretch for an empirical one. Ultimately, the concept underlying time-temperature history inversion from fission-track length data is to treat each track as an indicator of thermal input. c-Axis projection can be viewed not so much as a length conversion as an improved means of measuring this thermal signature, which uses more of the available information to arrive at a more reliable solution.
Acknowledgements This work benefited from continuing conversations with R. Donelick, and answering insightful questions posed by S. Guedes. It was also improved thanks to comments from two anonymous reviewers. Support for sharing this research was provided by UNESP and UNICAMP.
References Barbarand, J., Carter, A., Wood, I., Hurford, A.J., 2003a. Compositional and structural control of fission-track annealing in apatite. Chem. Geol. 198, 107–137. Barbarand, J., Hurford, A.J., Carter, A., 2003b. Variation in apatite fission-track length measurement: implications for thermal history modeling. Chem. Geol. 198, 77–106. Carlson, W.D., 1990. Mechanisms and kinetics of apatite fissiontrack annealing. Am. Mineral. 75, 1120–1139. Carlson, W.D., Donelick, R.A., Ketcham, R.A., 1999. Variability of apatite fission-track annealing kinetics I: experimental results. Am. Mineral. 84, 1213–1223. Crowley, K.D., Cameron, M., Schaefer, R.L., 1991. Experimental studies of annealing etched fission tracks in fluorapatite. Geochim. Cosmochim. Acta 55, 1449–1465.
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Donelick, R.A., 1991. Crystallographic orientation dependence of mean etchable fission track length in apatite: an empirical model and experimental observations. Am. Mineral. 76, 83–91. Donelick, R.A., Miller, D.S., 1991. Enhanced TINT fission track densities in low spontaneous track density apatites using 252Cfderived fission fragment tracks: a model and experimental observations. Nucl. Tracks 18, 301–307. Donelick, R.A., Ketcham, R.A., Carlson, W.D., 1999. Variability of apatite fission-track annealing kinetics II: crystallographic orientation effects. Am. Mineral. 84, 1224–1234. Galbraith, R.F., 2002. Some remarks on fission-track observational biases and crystallographic orientation effects. Am. Mineral. 87, 991–995. Galbraith, R.F., Laslett, G.M., 1988. Some calculations relevant to thermal annealing of fission tracks in apatite. Proc. R. Soc. London A 419, 305–321. Galbraith, R.F., Laslett, G.M., Green, P.F., Duddy, I.R., 1990. Apatite fission track analysis: geological thermal history analysis based on a three-dimensional random process of linear radiation damage. Philos. Trans. R. Soc. London A 332, 419–438. Green, P.F., 1988. The relationship between track shortening and fission track age reduction in apatite: combined influences of inherent instability, annealing anisotropy, length bias and system calibration. Earth Planet. Sci. Lett. 89, 335–352. Green, P.F., Durrani, S.A., 1977. Annealing studies of tracks in crystals. Nucl. Tracks 1, 33–39. Green, P.F., Duddy, I.R., Gleadow, A.J.W., Tingate, P.R., Laslett, G.M., 1986. Thermal annealing of fission tracks in apatite 1. A qualitative description. Chem. Geol. 59, 237–253. Jonckheere, R., 2003. On methodical problems in estimating geological temperature and time from measurements of fission tracks in apatite. Rad. Meas. 36, 43–55. Ketcham, R.A., 2003. Observations on the relationship between crystallographic orientation and biasing in apatite fission-track measurements. Am. Mineral. 88, 817–829. Ketcham, R.A., Donelick, R.A., Carlson, W.D., 1999. Variability of apatite fission-track annealing kinetics III: extrapolation to geological time scales. Am. Mineral. 84, 1235–1255. Ketcham, R.A., Donelick, R.A., Donelick, M.B., 2000. AFTSolve: a program for multi-kinetic modeling of apatite fission-track data. Geol. Mater. Res. 2 (electronic). Laslett, G.M., Galbraith, R.F., 1996. Statistical modeling of thermal annealing of fission tracks in apatite. Geochim. Cosmochim. Acta 60, 5117–5131. Laslett, G.M., Kendall, W.S., Gleadow, A.J.W., Duddy, I.R., 1982. Bias in measurement of fission-track length distributions. Nucl. Tracks 6, 79–85. Laslett, G.M., Gleadow, A.J.W., Duddy, I.R., 1984. The relationship between fission track length and track density in apatite. Nucl. Tracks 9, 29–38. Laslett, G.M., Green, P.F., Duddy, I.R., Gleadow, A.J.W., 1987. Thermal annealing of fission tracks in apatite 2. A quantitative analysis. Chem. Geol. 65, 1–13. Parker, P., Cowan, R., 1976. Some properties of line segment processes. J. Appl. Probab. 13, 255–266. Ravenhurst, C.E., Roden-Tice, M.K., Miller, D.S., 2003. Thermal annealing of fission tracks in fluorapatite, chlorapatite, manganoapatite, and Durango apatite: experimental results. Can. J. Earth Sci. 40, 995–1007. Willett, S.D., 1997. Inverse modeling of annealing of fission tracks in apatite 1: a controlled random search method. Am. J. Sci. 297, 939–969.
The role of crystallographic angle in characterizing and modeling apatite fission-track length data Richard A. Ketcham∗ Jackson School of Geosciences, The University of Texas at Austin, Austin, Texas 78712, USA Received 13 February 2004; accepted 13 July 2004
Abstract Recent experimental work on apatite fission-track length measurement has documented a number of factors that can exert a considerable influence on determinations of mean length. The principal source of length variation is anisotropy of annealing and etching behavior with respect to crystallographic angle. Procedural and environmental factors that influence which angular populations are preferentially sampled during measurement lead to variations in mean length that exceed standard statistical predictions. This paper evaluates the possibility of using c-axis projection to remove angular effects, and thus make length data more reproducible and informative. © 2005 Elsevier Ltd. All rights reserved. Keywords: Fission-track; Apatite; Annealing; Biasing; Thermal history inversion
1. Introduction As part of a systematic study of apatite fission-track thermochronology, Barbarand et al. (2003b) documented several sources of variation in measurement of confined horizontal track lengths. Among the topics they addressed were reproducibility of measurements by a single observer, compatibility of measurements among different observers, and the effects of Cf irradiation. Based on their observations they deemed it “essential” to take the effect of track angle to the apatite crystallographic c-axis into account when interpreting or utilizing track length data quantitatively. It has long been recognized that fission-track annealing in apatite is anisotropic (Green and Durrani, 1977). Unannealed or lightly annealed fission tracks exhibit only minor anisotropy, which may be attributable to etching effects. As annealing progresses, however, tracks at high angles to the c-axis shorten more quickly than tracks at low angles (Green and Durrani, 1977; Green et al., 1986; Galbraith and ∗ Tel.: +1 512 471 6942; fax: +1 512 471 9425.
E-mail address: [email protected] (R.A. Ketcham). 1350-4487/$ - see front matter © 2005 Elsevier Ltd. All rights reserved. doi:10.1016/j.radmeas.2004.07.008
Laslett, 1988; Galbraith et al., 1990; Crowley et al., 1991; Donelick, 1991; Donelick et al., 1999; Ketcham, 2003). Apatite etching is also anisotropic, as reflected by faster etch figure widening rates in the c-axis direction. Ravenhurst et al. (2003) searched for and found no strong evidence of a link between degree of track length anisotropy and etching strength, but etching does have subtle biasing effects that influence which horizontal confined tracks are intersected and measured. Etch figures, where semi-tracks intersect the polished grain surface, tend to be oblong in the direction of the c-axis, and confined horizontal tracks at high angles tend to be wider, easier to see, and appear more completely etched than tracks at low angles. Tracks very near to perpendicular to the c-axis are an exception to this trend, as they also frequently appear incompletely etched. The physical process behind length anisotropy in apatite is unknown, but in view of the above evidence it is probably linked most strongly to annealing rather than etching or registration effects such as channeling. Annealing and etching anisotropy manifest themselves in fission-track length measurements in three primary ways: length bias, under-etching bias, and track loss. Length bias
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is the increased probability of intersecting longer tracks relative to shorter ones with etchant pathways, leading to the preferential detection of long tracks (Laslett et al., 1982, 1984). For TINT (track-in-track) measurements, the length bias is modified by etching anisotropy, as the inclined semi-tracks used as etchant pathways present a larger crosssection for intersection for high-angle tracks than low-angle ones (Galbraith et al., 1990; Ketcham, 2003). Under-etching bias is caused by thinness and a tendency toward apparent incomplete etching of low- and very high-angle tracks, particularly when strong etchants (5 or 5.5 M HNO3 ) are used. Such tracks tend to be either overlooked or bypassed in favor of better-defined tracks. For apatites featuring high etching anisotropy, including all F-apatites, under-etching bias exerts a far greater influence on probability of track measurement than length bias. This is evidenced by low relative frequency of observation of these angular populations on apatites at low levels of annealing (Donelick et al., 1999, Fig. 10); the same trends are apparent in Fig. 2 of Barbarand et al. (2003b). Track loss occurs at advanced stages of annealing, when the rate of shortening of high-angle tracks accelerates greatly relative to low-angle tracks. Although some measurable high-angle tracks may persist and be measured, some proportion of them will anneal below the detection limit, which is defined by the requirement to reliably identify a track and distinguish its endpoints. The evidence for this phenomenon is a steadily increasing relative probability of detection for low-angle tracks that is far in excess of what is predicted by length biasing (Ketcham, 2003). Although the importance of track angle for interpretation of apatite fission-track data has long been recognized in theoretical studies (e.g., Galbraith and Laslett, 1988; Galbraith et al., 1990), to date the only fully developed means of utilizing angular data is c-axis projection (Donelick et al., 1999; Ketcham, 2003). The goal of c-axis projection is to remove crystallographic orientation effects from length data while retaining the thermal history information. The technique consists of constructing a transform function that, for any individual track measurement of length and angle (li , i ), estimates the length distribution of tracks with = 0, or parallel to the c-axis (lc , c ), that have experienced the same annealing conditions (Donelick et al., 1999). The model was subsequently refined to include an estimate of the uncertainty in the projection (lc ) (Ketcham, 2003). The purpose of this paper is to briefly examine some of the results presented by Barbarand et al. (2003b) and its companion paper (Barbarand et al., 2003a) in the context of c-axis projection. In addition, the prospects for constructing an empirical annealing calibration based on the annealing data of Barbarand et al. (2003a) will be discussed.
2. Reproducibility and observer compatibility Barbarand et al. (2003b) documented that the reproducibility of mean track length measurements of experi-
mentally annealed apatites is frequently outside the bounds estimated by the standard error of the mean (SEM). A larger than predicted variation was observed both in repeated measurements by a single observer and between different observers. In the case of highly annealed populations, this divergence is in part traceable to the fact that the SEM is not an appropriate estimator for uncertainty when the underlying distribution being sampled is heavily skewed or bimodal. Also, a substantial component of variation is likely to be based on observer tendencies and thus is systematic, rather than random. In all cases where larger than expected variation was observed, Barbarand et al. (2003b) found that the divergence could be explained by inspecting plots of track length versus angle. These plots indicate that there exists a regular length–angle relationship that characterizes the level of annealing produced in an experimental mount (i.e. in which all natural tracks were annealed, and fresh tracks induced in a nuclear reactor were annealed under controlled conditions in a laboratory furnace), and that poor reproducibility was brought about by differential sampling of this distribution. Because track length varies markedly with angle, particularly in highly annealed populations, when different angular ranges are preferentially sampled the measured mean length is biased accordingly. Because c-axis projection seeks to characterize the relation between length and angle at a given level of annealing, it provides a reasonable approach for solving the problem of reproducibility for single and multiple observers. By projecting all track lengths to a common orientation, the biasing effects of sampling different angular populations are eliminated. As an example, Fig. 1 shows a selection of contours of 20
15 l (track length, µm)
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φ (angle to c axis, degrees) Fig. 1. c-axis projection contours from Ketcham (2003) superimposed in Fig. 4b from Barbarand et al. (2003b). Contours from top to bottom are for lc values of 12.7, 11.9, 11.1, and 10.3 m. Track lengths are converted to their c-axis-projected equivalents by tracing along the contours to the = 0 axis.
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the Ketcham (2003) c-axis projection model superimposed on the data shown in Barbarand et al. (2003b, Fig. 4b). The data are fairly well bounded by contours for lc of 12.7 and 10.3 m, which matches well with the expectation that the distribution of c-axis projected tracks should have a standard deviation (c ) of about 0.8 m. It is likely that the two data sets have very similar mean c-axis projected lengths, but this cannot be tested because the individual length data are unavailable. It should also be noted that the Ketcham (2003) c-axis projection model is based on the Carlson et al. (1999) annealing data set, which was compiled by a different observer using slightly different etching conditions; it is unknown whether these differences will have a substantial effect on c-axis projection.
3. Cf irradiation Cf irradiation is a means of increasing the number of measurable tracks by bombarding the polished mount with fission particles from a 232 Cf source, creating more etchant conduits (Donelick and Miller, 1991). In order to assess the possible biasing effects of Cf irradiation, it is necessary to consider how the population of latent horizontal tracks within several microns of the polished surface of a grain mount is sampled. First we envision a “normal” case in which there are relatively few etchant conduits penetrating the crystal surface. One version of the standard Poissonian model is to assume that detection of a particular latent horizontal track is very unlikely. In this case, the line segment model predicts that the probability that a latent track will be intersected is proportional to its length (Parker and Cowan, 1976; Laslett et al., 1982), modified by geometric factors such as etchant path shape (Galbraith et al., 1990; Ketcham, 2003). In fact, the under-etching bias is a more influential effect than length, but the underlying principle of sampling being very unlikely still applies. If very few of the available tracks are detected and measured, the resulting set of lengths will be a biased representation of the true underlying distribution. The opposite case is a grain surface so perforated with Cf-induced tracks that every available latent track is detected: all short tracks are intersected, and all tracks susceptible to under-etching are impinged multiple times, facilitating more complete etching. In such a case sampling would no be longer Poissonian, but complete. This is only a theoretical end-member case, as a completely perforated grain surface would be obliterated thus making tracks unmeasurable, but it demonstrates that there exists a continuum of possible biasing, from Poissonian to no bias. Grains that have a high density of either Cf-induced or naturally occurring tracks traverse some of the distance between Poissonian and complete sampling, with the effect on each bias being the same: tracks that are under-represented under Poissonian sampling become more frequently seen and measured.
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It should be mentioned that the more traditional Poissonian model for length biasing is based on the rarity of intersections between etchant pathways and latent confined horizontal tracks, rather than detections, which can involve multiple intersections. This model led Laslett et al. (1982) to suggest that tracks that are intersected twice should have their measurements counted twice, to ensure consistent length biasing. Although this practice would also apply for Cf-irradiated mounts, it is not widely used, in part because it is often not straightforward to discern if nearby tracks are impinging or not. Furthermore, if the concept of under-etching bias is valid, it is arguable that this protocol actually has the opposite of the desired effect of maintaining consistent biasing. If a low-angle track needs to be intersected twice to be fully etched and thus measured, it should be counted zero times rather than twice to match the biasing inherent in low-track-density mounts. Neither the Carlson et al. (1999) nor the Barbarand et al. (2003a) annealing data sets were acquired using this protocol. Barbarand et al. (2003b) ran a series of experiments in which mounts with and without Cf irradiation were measured. They documented among populations with mean track lengths below 14 m an increase in mean track length of roughly 5% when Cf irradiation was used. In interpreting these results, they postulate that “the introduction of additional etching conduits may increase the probability of intersecting longer tracks at the expense of shorter tracks, in effect a variation of the normal TINT length bias”; but this explanation is incorrect. As outlined above, the introduction of numerous additional etching conduits decreases biasing, and thus should lead to proportionally more short tracks being measured if the only biasing factor was track length. A more probable explanation for their results is that, along with the length bias, the under-etching bias was also decreased, leading to increased revelation of longer, low-angle tracks. The Carlson et al. (1999) data set also includes several experiments in which mounts were measured using Cf irradiation, then re-polished, re-etched and re-measured to obtain length measurements without the aid of Cf-induced tracks. These data are somewhat divergent from Barbarand et al. (2003b), but share the same underlying features (Fig. 2). Mean track length measurements (closed symbols) in lessannealed populations were relatively unaffected by Cf irradiation, but at advanced stages of annealing populations featured higher mean track lengths with Cf irradiation, with divergences significantly beyond statistical expectation. Also plotted in Fig. 2 are the corresponding mean c-axis projected lengths for these experiments (open symbols), which show much less effect from Cf irradiation. The reason is analogous to that cited in the previous section: Cf irradiation does not change the underlying length-angle distribution, but it does change which tracks in this distribution are sampled. By projecting all tracks to a common orientation, this source of variation is removed. Plots of individual experiments from Carlson et al. (1999) again illustrate this trend (Fig. 3). In the highly annealed DR
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lc,mod Fig. 2. Difference between mean length determinations on mounts with and without Cf irradiation (Cf and nCf , respectively), from Carlson et al. (1999). Error bars are 2SEM. Mean value of unprojected lengths (closed symbols) increases greatly with Cf-irradiation at high levels of annealing (i.e. low lc,mod ), while mean of c-axis projected lengths is less affected.
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and RN populations, Cf irradiation preferentially revealed long, high-angle tracks, showing that its main effect was to diminish the under-etching bias. In the near-end-member Cl apatite B3, Cf-irradiated mounts gave slightly shorter mean lengths, though not outside the bounds of expected variation (Cf: 11.82 ± 0.09 m versus no Cf: 11.93 ± 0.15 m for run 63 (shown) and Cf: 11.38 ± 0.13 m versus no Cf: 11.53 ± 0.13 m for run 66). However, these results are consistent with the expectation of under-etching bias reduction in this unusual apatite. B3 shows very little under-etching bias for low-angle tracks, but near-perpendicular tracks are underrepresented (Donelick et al., 1999, Fig. 10). Because these tracks are shorter than those at lower angles, increasing their representation would tend to slightly lower mean length. The experiment with the worst match between mean c-axis projected lengths with and without Cf irradiation is also shown (F-apatite PQ, run 63). In this case, the non-Cf tracks fall within the length-angle distribution defined by the Cf-irradiated population, but do not sample it evenly. This experiment is in the very final stages of annealing, in which virtually all angular populations show evidence of
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Fig. 3. Polar diagrams of four experiments from Carlson et al. (1999) showing difference between length measurements with and without Cf irradiation. Note that analyses of mounts with plentiful confined tracks were halted after 110 measurements.
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accelerated length reduction. This case is not well described by current c-axis projection models. However, another feature of such fission-track populations is that they have extremely low density, so they will rarely be sampled in practice, and when they are will only have a small influence on thermal history inversions.
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Ketcham et al. (1999) developed a means of combining apatites with disparate annealing behaviors into a single multi-kinetic model by relating pairs of apatites to each other with the equation: rmr − rmr0 rlr = , (1) 1 − rmr0 where rlr and rmr are the reduced (normalized) lengths of apatites that are less resistant and more resistant to annealing, respectively, and rmr0 and are fitted parameters. The rmr0 parameter is the reduced length of the more resistant apatite at the point where the reduced length of the less resistant apatite falls to zero (i.e. it is fully annealed). The equation is fitted by comparing length measurements of apatites that were annealed in the same furnace at the same time, and thus have identical thermal histories. It was found that Eq. (1) adequately describes the variation observed between apatites at laboratory time scales, and that extrapolation to million-year time scales does not significantly change predictions of thermal behavior in geological settings. Creation of a multi-kinetic annealing model based on the Barbarand et al. (2003a) data, or perhaps a combined model utilizing the Carlson et al. (1999) data as well, could be possible using this approach. Fig. 4 shows an example rlr − rmr plot for mean length data from Carlson et al. (1999), and the effect that differential biasing due to Cf irradiation can have on the fit. The two measurement pairs that utilized Cf irradiation (on both apatites) plot off of the main trend defined by the other experiments. Interestingly, removal of both the positive displacement of the DR mean length and the negative displacement of the B3 mean length caused by Cf irradiation brings these experiments into line with the main trend. Conversely, B3 and DR rc,mod values are not significantly different between experiments with and without Cf irradiation (mean change of 0.10 m). The Barbarand et al. (2003a) data are mixed with respect to the use of Eq. (1) (Fig. 5), but promising overall. Their BAM-DUR data (Fig. 5a) line up well, and are well modeled by the fitted line. Some of the Cf-irradiated measurements of GIL apatite plot off of the trend defined by other experiments (Fig. 5b), suggesting that a Cf-induced shift in bias has resulted in an incompatibility in their respective mean length data. Fig. 5c shows a problematic case, in which the experiments with annealing times of 10 h plot to either side of a line that fits the longer duration experiments. It is un-
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known whether this scatter is caused by problems in the annealing schedule or from other sources of variability, including the use of non-projected lengths.
5. Discussion The examples shown above corroborate the claim by Barbarand et al. (2003b) that angular data are necessary for proper interpretation of annealing experiments and inversion of thermal histories. c-Axis projection appears to be a promising avenue for removing many of the sources of variability that beset track length measurements, including interand intra-observer reproducibility and Cf irradiation. It also may provide a remedy for other concerns listed by Barbarand et al. (2003b). In particular, they note that the measurement of short tracks is very influential on mean track length, and thus rigorous measurement criteria for short tracks need to be defined and standardized. Although standardization is never a bad idea if it is practical, the use of c-axis projection may obviate much of the need for it. Likewise, Barbarand et al. (2003b) propose that etching procedures be standardized, but it is likely that c-axis projection can provide a fully effective conversion between measurements made using different etching methods. Both of these assertions are based on the same principle discussed above: the population of latent tracks available for measurement exists independent of the observer, etching technique, number of etchant
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Fig. 5. Apatite-apatite plots for various experiments by Barbarand et al. (2003a). Error bars are 2SEM when available, and experiments that retained some tracks but at a density too low to measure are plotted as zero with error bars extending to a reduced length of 0.35.
pathways, and all other procedural and environmental variables. The principle source of length variation in this population is linked to angle to the apatite c-axis, and the tendency of various factors to influence which angular populations are sampled more frequently than others. Projection to a common angle may correct for virtually all such factors in a single step. It should be noted that projection may not be the only approach—an alternative is to develop a schema based on E(l|) (expected confined length conditional on c-axis angle ) whereby angular data may enable derivation of additional thermal history information (Galbraith, 2002). However, no such alternative has yet been sufficiently developed. A possible impediment is that it may re-emphasize user-specific effects that may be difficult to calibrate for, such as whether certain short tracks are measured or not, which will have a large effect on E(l|) at some values of . A different and perhaps more promising approach may be to construct a method based on E(lc , lc |l, ), as lc provides an index of thermal equivalence while lc estimates the precision of that index.
Once angular effects have been removed, the only remaining issue for thermal history inversion is the relative probability of sampling the different populations present in geological samples that reflect varying amounts of annealing. This is usually estimated (e.g., Willett, 1997; Ketcham et al., 2000) using the length versus density relationship reported by Green (1988), although it is only an approximate solution, and is never re-calibrated. It is likely that any adjustments would be small and subtle, but their effect on inversion has not been studied. It should also be noted that the issue of proper consideration and calibration of the length versus density relationship is not fully resolved and exists independent of c-axis projection (Jonckheere, 2003). A valid critique of c-axis projection is to question whether this fairly complex transform, with six free parameters, is warranted, or if it diminishes the thermal history information contained in track lengths. In view of the available experimental data, however, it may be more reasonable to ask how continued use of length data without angular information can be justified. The recent work has demon-
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strated that “standard” mean track length data are prone to shifting biases that are subtle to quantify and predict but exert a considerable influence on results, to the extent of rendering currently used estimates of statistical uncertainty demonstrably inadequate. With respect to calibrating empirical or physical models of fission track annealing based on length reduction (e.g., Laslett et al., 1987; Carlson, 1990; Crowley et al., 1991; Laslett and Galbraith, 1996; Ketcham et al., 1999), it is important to recognize that measured mean lengths are not a direct depiction of length reduction, but rather a strongly skewed observation. Because zero-length tracks are not included in calculations of the mean, the link between measured and actual physical length reduction is very indirect. Conversely, models based on c-axis projected data are self-consistent, as the c-axis parallel population they characterize is not subject to accelerated length reduction and track loss until complete annealing is imminent. Furthermore, the onset of accelerated length reduction once tracks fall below ∼ 10 m is likely caused by a change in the physical process of annealing (e.g., Carlson, 1990); depicting only one annealing process, rather than two, is less demanding for a physical model and less of a stretch for an empirical one. Ultimately, the concept underlying time-temperature history inversion from fission-track length data is to treat each track as an indicator of thermal input. c-Axis projection can be viewed not so much as a length conversion as an improved means of measuring this thermal signature, which uses more of the available information to arrive at a more reliable solution.
Acknowledgements This work benefited from continuing conversations with R. Donelick, and answering insightful questions posed by S. Guedes. It was also improved thanks to comments from two anonymous reviewers. Support for sharing this research was provided by UNESP and UNICAMP.
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