Apatite fission-track modeling reconstructs the low-temperature histories of geological samples based on measurements of the lengths of etched confined fission tracks and counted surface tracks. The duration for which each confined track was etched can be calculated from its width given the apatite etch-rate νR. We measured νR as a function of crystallographic orientation for fourteen samples from the igneous and metasedimentary basement of Tian Shan, Central Asia, to optimize the track-length distribution for modeling the thermal histories of apatites with varying chemical compositions. To first order, νR scales with the size of the track intersections with the mineral surface in the range of Dpar = 1.4–2.6 µm. We use νR for calculating the effective etch time tE of confined tracks measured after 20–60 seconds of immersion in 5.5 M HNO3 at 21°C. Considering only tracks within a predetermined etch-time window improves the reproducibility of the track-length distributions. Because an etch-time window allows excluding under- and over-etched tracks, sample immersion times can be optimized to increase the number of confined tracks suitable for modeling. Longer immersion times also allow the longest-etched tracks to develop a clear geometrical outline from which the orientation of the apatite c-axis can be inferred. We finish by comparing thermal histories obtained with a conventional 20-second immersion protocol, without tE selection, with those using the length of tracks within the range of tE = 15–30 seconds. Overall, the alternative models fit better to independent AHe data than the conventional ones.

The fission-track method is a thermochronological method for determining the ages and thermal histories of rocks. It is based on counting and measuring damage trails from uranium fission, that is, fission tracks, in minerals and is most applied to apatite. An important problem remains relating the tracks counted and measured after etching to the original damage trails. On the one hand, confined track length measurements are affected by geometrical biases that include length bias, orientation bias, intersection (fracture and host track thickness) bias, and edge and surface proximity biases [1-7]. These biases are mathematical consequence of the line-segment model [1] and their influence on track length measurements is understood. On the other hand, the etching conditions, like the nature, concentration, temperature of the etchant, and etching time, control the length measurements of confined tracks as well [8-13]. Although there is agreement on a few protocols (e.g., 20 seconds, 5.5 M HNO3), recent inter-analyst comparison revealed a general lack of reproducibility of length measurements [14]. It was argued that this results, to a large degree, from the inclusion, or otherwise, of weakly etched tracks and from the subjective nature of decisions which tracks are “well etched,” that, for lack of a quantitative track selection criterion, each analyst makes based on experience and the appearance of the etched tracks under the microscope [14-17]. The low reproducibility of track length measurements due to the subjective user-dependent selection of well-etched tracks and the consequence that the selection criteria will not always coincide with those used for establishing the annealing equations increases the uncertainties of thermal history modeling.

The physical–chemical process of fission-track etching is not well understood. The standard model describes etching as the combined action of two etch rates: the etch rate vT of the damaged material along the track axis and the bulk etch rate vB of the undamaged material in all other directions [17-21]. Following Gross (1918)[22], Jonckheere et al. (2019, 2022) [23, 24] applied the principles of crystal growth and dissolution to fission-track etching in apatite. Instead of each point on the plane etching at the bulk etch rate vB, crystallographic planes are displaced during etching at a radial etch rate νR perpendicular to the plane. This model accounts for the shapes of etched fission tracks depending on their crystallographic orientation and that of the etched surface (Figure 1). Confined tracks widen at a rate νR perpendicular to the track axis. Aslanian et al. [25] determined νR as a function of orientation for Durango apatite etched in 5.5 M HNO3 at 21°C by measuring the widths of horizontal confined tracks after 30 and 45 seconds etching. From the maximum width of confined tracks and the etch rates perpendicular to their track axes, we calculate the true duration for which each track has been etched. This effective etch time tE [24, 25] is shorter than the immersion time because of the access time, required for the etchant to travel down the host track and across to the confined track, differs from track to track [2, 26].

If the maximum track width is measured after two consecutive etch steps, vR can be calculated as:


Herein ΔW is the width increase and ΔtI the time increment between the first and second etch step [24, 25]. As the latent confined track is as narrow as a few nm [27], track widening can be considered as etching undamaged apatite. Consequently, it is unlikely that the apatite etch rate will change during the etching process.

The effective etch time of the track is then [25]:

tE(s)=60 W (μm)νR (μm/min)

Effective etch-time calculations offer a practical quantitative criterion for distinguishing between under-, well-, and over-etched tracks. We report step-etch experiments on apatites from geological samples with different compositions and etch rates. We examine how their etch-rate plots (vR vs. orientation) are related to each other and their Dpar values (the size of the track openings in apatite prism faces parallel to its c-axis after etching for 20 seconds [28]) and compare them to published values for Durango apatite. We use the results for calculating the effective etch times of the confined tracks and consider how track selection based on the effective etch time affects the length distributions and modeled thermal histories of the samples. We discuss how calculating the effective etch times of confined tracks supports more efficient etch protocols and reproducible length measurements.

We selected 27 samples from a study of the Mesozoic–Cenozoic exhumation of the Tian Shan in Tajikistan [29]; unpublished data. We used 14 samples for step-etch experiments and the remaining 13 samples for track-width measurements at 20 seconds of immersion. Table 1 provides their lithologies and characteristics related to their thermochronological analysis. The samples are from magmatic (granite, granodiorite, dolerite) and metasedimentary (sandstone, siltstone, phyllite, schist) rocks. The analyzed apatites span a broad range of fission and ion-track (2–4 × 106 cm−2) densities, apatite fission-track (AFT) ages (7.8–161.0 Ma), Dpar’s (1.4–2.6 µm), mean c-axis projected track lengths (12.3–14.5 µm), track-length distributions and (U-Th)/He (AHe) ages. Most samples were irradiated with ions from a linear accelerator. The mounts were covered with 60 µm Al degrader foil and irradiated at 15° from normal incidence with 11.1 MeV/amu 197Au ions from the UNILAC linear accelerator at the GSI Helmholtzzentum für Schwerionenforschung, Darmstadt, Germany. Ion track densities are ~4 × 106 cm−2 and ~2 × 106 cm−2 for sample no’s starting with “6” and for all other samples, respectively. These ion tracks conduct the etchant into the apatite interiors and so increase the number of etched confined tracks [30]. The wide Dpar range of the Tian Shan apatites allows for exploring whether the etch-rate model for Durango apatite [25] applies to apatites with different compositions. The different sample properties made it necessary to adapt the immersion times of our step-etch experiments for each sample (Table 1).

We first etched all samples (but one) for 20 seconds in a 5.5 M HNO3 solution at 21°C (hereafter called the “conventional etch protocol”) [31]. We arrested etching by consecutive immersion in two 250 ml beakers with deionized water, rinsed the mounts with ethanol, and dried them in a curing cabinet at 35°C. Following the first series of measurements, we etched 14 samples for a second and in some cases for a third time, using 10 seconds increments for samples with high fossil or ion track densities, for which a total etch time of more than 30 seconds would create overlap problems. We re-etched samples with low fossil and ion track densities for 20 seconds and applied a third 10- or 20-second step if required for collecting sufficient width measurements for calculating vR. The values for successive steps did not show systematic differences and were pooled.

After each step, we measured the length, the maximum width, and the c-axis angle of subhorizontal confined tracks (dip ≤15°). For the remaining 13 samples, we measured the confined track widths only once after 20 seconds immersion. In order to assess the increase in the number of suitable confined tracks with immersion time [30], including those with effective etch times between 15 and 30 seconds, we recorded and measured all new confined tracks not measured during the preceding step in 7 mounts.

We measured the lengths and orientations with a Zeiss AxioImager M2m microscope and Axiovision software. We scanned the samples in transmitted light at 250× optical magnification (100× dry objective and 2.5× post-magnification with a pixel size of 0.4 × 0.4 μm²). We measured all tracks for which both ends could be distinguished with confidence. We determined the widths of tracks at 30–80° to the c-axis at the microscope (Figure 1(b–c)) and recorded transmitted-light image stacks of all confined tracks. We used the CorelDraw graphic software for measuring the width of the thinnest tracks at 0–30° and 80–90° to the c-axis (Figure 1(a) and (d-f)). The images allowed to check the microscope measurements and to re-measure occasional outliers. A comparison of the results for one sample confirmed that the width measurements at the microscope and on the images showed no significant difference (online supplementary Figure S1), although it is more difficult to make consecutive measurements at the same location with the microscope. It is often assumed that the precision of microscope measurements is limited by its resolution, which depends on the wavelength of the light source and the numerical aperture of the objective lens. Interference produces a dark border around the track, which is somewhat wider than the resolution. The optical resolution of our microscope and a wavelength of 500 nm is 0.33 µm. By measuring at the same position, for example, in the center of that dark band, one can reproduce width measurements with about five times better precision than the resolution limit. We estimate the uncertainty of our width measurements to be ~0.1 μm. Thus, for the thinnest measurable tracks with a thickness of ~0.4 μm, the relative error is as high as 25% but it decreases significantly for thicker tracks. The error for measuring the orientation to the c-axis is ±1° [32].

Tracks at ≥80° to the c-axis develop a diamond-shaped etch figure, at their intersections with the host tracks, bounded by the fastest etching faces (Figure 1(d–f)) [23]. Tracks at 80–85° to the c-axis are wider, as their orientations approach those of the fastest etching faces and their shapes are for the most part bounded by those faces (Figure 1(d)). The diamond shape is more distinct for tracks at 85–90° to the c-axis, which have narrower channels bounded by slower etching planes (Figure 1(e)). For such tracks, vR is obtained as before from the width of the track channel, while tE is calculated from the size of the diamond-shaped etch figure and the maximum etch rate (Figure 1(f)) [24]. The precision of measuring the widths in the diamond-shaped etch figures is as high or even higher as measuring the narrow track channels due to their large widths and the fact that they are confined by two faces.

We modeled the samples' thermal histories with HeFTy v.1.9.3 (temperature–time (T-t) modeling; [7]); first using all length data for a conventional immersion time of 20 seconds, and, for comparison, using the data for tracks with effective etch times between 15 and 30 seconds. In the latter case, we excluded tracks at <20° to the c-axis, because we considered that their tE estimates are not accurate enough (section 3.2). If two consecutive measurements of the same track gave tE values between 15 and 30 seconds, we used the mean of the two measured lengths. We modeled the data using c-axis projection [33] to correct for anisotropic annealing. The c-axis projected lengths were calculated using the “user l0” calibration [14] with l0 = 16.1 μm, the operator’s (FT) mean-induced track length for Durango apatite. Modeling with l0 calculated from the Dpar measurements did not produce significantly different T-t patterns. Supplementary Material Text S1 provides the modeling details.

In the following, we focus on three aspects. First, we discuss the results of the etch-rate measurements for different apatites and their relationship to Dpar (section 3.1). We then use them for calculating the effective etch times (i) for a conventional single 20 seconds immersion, (ii) for a subsequent immersion, and (iii) for tracks measured first after 30 or 40 seconds immersion (section 3.2). We then discuss the advantages and drawbacks of track-width measurements and effective etch-time calculations for confined track-length measurements and T-t modeling (section 3.3).

3.1. νR-calculation and Scaling

Different apatites etch at different rates, so that samples etched under the same conditions exhibit different confined track widths; Figure 2 illustrates this for samples with a Dpar range of 1.4–2.6 µm. The tracks in our sample with the highest Dpar (2.6 µm; Figure 2) that was etched for 30 seconds have similar widths (black lines) as in the sample with a Dpar of 1.8 µm etched for 40 seconds, indicating its higher etch rate. Given the correlation between Dpar and Dper [33] (r = 0.97; Figure 3(a)), it seems reasonable that the etch rates in different directions scale by the same factor. We infer that, in the first approximation, the etch rates of our geological samples can be related to those of Durango apatite [25] via a scaling factor dependent on Dpar.

We first fitted a two-part empirical function to the etch-rate data for Durango apatite of Aslanian et al. [25] (vR = F(ϕW); vR in µm/min; ϕW = angle of vR to the c-axis in degrees):

F(ϕW)=0.0071 ϕW2+0.2807 ϕW+0.2495 (ϕW25)
F(ϕW)=0.00025 ϕW20.0633 ϕW+4.2500 (ϕW25)

To account for the different etch rates of our samples, we used a simple linear scaling of their model.


Figure 4 shows plots of vR against c-axis angle for our samples (a) and the Durango apatite (b) and fits to the Durango data (black lines, equation 3) and their scaled counterparts (scaling factor aRequation 4); aR ranges from 0.76 to 1.20 (online supplementary Table S1). For Dpar <1.87 μm νR values at >40° to c are sometimes lower than predicted by the fits. This suggests that the magnitude of νR and the precise shape of the etch-rate plot may both depend on the apatite composition. At this stage, the limited data for low-angle tracks do not warrant a more sophisticated fit.

Next, we determined a best-fit linear relationship between aR and Dpar for our 14 samples (Figure 3(b)):


Figure 3(b) shows that aR = 1 corresponds to Dpar ≈ 2.03 μm, somewhat higher than most measurements for Durango apatite (1.7–1.9 μm; [14], [31], [34]). The difference between our νR estimates and those of Aslanian et al. [25] (Figure 4(b)) likely results from a difference in the experimental procedures, here starting with an immersion of 20 instead of 30 seconds, resulting in a greater fraction of less-etched confined tracks with less-distinct boundaries.

3.2. Effective Etch-Time Calculations

Figure 5 and 6 show tE plotted against c-axis angle and tE distributions for characteristic samples measured after 20 seconds immersion and an additional 10–20 seconds immersion for selected samples characteristic for our Dpar range of 1.4–2.6 µm; online supplementary Figures S2 and S3 show the complete dataset. The calculated effective etch times for a single 20 seconds immersion range from 3 to 27 seconds (Table 2; 2850 tracks). Some outliers as high as tE = 45 seconds occur at low and high angles to c. At low angles to the c-axis, the tracks are thin and vR is lowest. tE calculations for those tracks are imprecise due to the limited precision of the measurements and the uncertainties on the fitted etch rates. Hence, we excluded these tracks from here on by setting a threshold at 20° to c. The effect on the length measurements and thermal models is negligible as there are few such tracks. Outliers at high angles to c correspond to tracks that did not develop a distinct etch pyramid and for which we calculated tE from the narrow channel widths, as at low angles to c. In most of the samples etched for 20 seconds, we measured all suitable confined tracks. Our tE-data thus reflect their effective etch-time distributions for this protocol. The mean effective etch times range from 7.1 to 12.6 seconds, which is about half the immersion time. There is a negative linear relationship between the mean tE and Dpar (Figure 3(c)). This is due to the fact that the effective etch time at which the tracks become measurable decreases from ~5 to ~3 seconds, as Dpar increases from 1.4 to 2.6 µm (Figures 5 and 6; online supplementary Figures S2; S3), corresponding to a common threshold width of ~0.4 μm [25, 35]. Hence, for an immersion time of 20 seconds, more tracks started etching and became measurable in samples with high Dpar, resulting in overall lower mean tE values. In principle, the maximum effective etch time corresponds to the immersion time, which is 20 seconds for confined tracks closest to the polished surface. Some tE estimates up to 27 seconds (Tables 2 and 3; Figures 5 and 6, online supplementary Figures S2; S3) are no doubt in part due to the limited precision of the width measurements and perhaps in part result from low-νR grains in an inhomogeneous apatite population.

Figure 6 also shows tE distributions of selected samples after 20 s immersion and re-measured after a second immersion for 10–20 s. The horizontal bands mark the expected tE ranges from 15–30 and 25–40 s. Figure 7 shows tE distributions of tracks first measured after 40 s; online supplementary Figures S3; S4 show all samples for which tracks were recorded after an immersion longer than 20 s, with some of them used for step-etching. Most tE estimates are in the expected range (horizontal bands), although some tE are below the assumed lower limit. For step-etched tracks, one would expect that tE2 = tE1 + tI, wherein tE1 and tE2 are its effective etch times after the first and second immersion, and tI the duration of the latter. Figure 8 plots tE2 against tE1 + tI for four samples with different Dpar; online supplementary Figure S5 shows the plots for all samples. For several tracks tE2 < tE1 + tI. More tracks plot below the 1:1 line at low tE1. In general, tracks with high tE1 are located closer to the polished surface than ones with low tE1. Their tips are thus harder to reach, which accounts for the greater delay during subsequent etching. From the scatter about the 1:1 line at higher tE2, we estimate a statistical uncertainty of ~2.5 s on tE.

3.3. Practical Application

This section considers applications of effective etch-time calculations to apatites with different chemical compositions. Ketcham et al. (2015) [14] showed in an inter-laboratory comparison that the reproducibility of confined track-length measurements continues to be a serious problem. A series of recent studies addressed the issue of track selection [15, 16, 36]; the last of these considered tracks well-etched if they are terminated by the slowest etching basal and prism faces [23-25, 37, 38]. Effective etch-time calculations provide a quantitative criterion for selecting suitable tracks for measurement [23-25, 37, 38].

Tamer and Ketcham (2023) [36] emphasized the need for a new etch protocol, suggesting a two-step (20 + 10 seconds) procedure. Confined tracks should be selected after 20 seconds etching, but their lengths measured after etching for another 10 seconds. Such a two-step procedure is time-consuming compared to width measurements, which can be done at the same time as the length measurements. It was previously proposed to select tracks based on their track width for zircon fission track dating (e.g., [32]). For apatite such a selection based directly on the track width has the problem that the apatite etch rate vR is anisotropic and tracks along the fastest-etching orientations etch 5–6 times faster than ones along the slowest-etching ones. Using the width as a selection criterion for confined tracks would, hence, require a variable threshold width, depending on track orientation, instead of a constant value. Applying equation (5) to unknown samples enables to convert track width to tE and to allow for anisotropic widths. We therefore propose to define an effective etch-time window that excludes tracks that are under- or over-etched and use the remainder as input for T-t modeling [25]. This approach allows to adjust the immersion time for individual samples or grains depending on their fossil track densities and surface etch rates (quantified by Dpar).

Long immersion times increase the number of measurable confined tracks [30, 39]. In principle, their lengths cannot be used for T-t modeling based on annealing equations for a 20-second protocol. Effective etch times can overcome this limitation. For low track densities, an immersion time of 40 seconds is appropriate; for high track densities or fast etching apatites, for which overlap or over-etching is a concern, a shorter immersion is more suitable. For example, in sample 0817I1, we measured four times more tracks after 40 seconds immersion and twice as many in the tE window 15–30 seconds than Käßner et al. (2016) [29] after 20 seconds (Tables 1 and 2; Figure 7 and online supplementary Figure S4). The gain is however variable and depends on the sample properties. For our samples first measured after 20 seconds immersion, from 0.6 to 1.5 times more additional tracks were measurable after 40 seconds (online supplementary Table S2); the latter increase is considered an ideal case. Lower values are common due to tracks being obscured by overlap in samples with high track densities or due to the widening of etched cracks in fractured grains. Our approach allows to combine track lengths measured during consecutive etch steps without re-polishing to increase the number of measured tracks, which is important for young and U-poor samples (Figures 6 and 7, online supplementary Figures S3; S4). Tamer and Ketcham (2020) [16] showed earlier that lengths measured after a first etch and remeasured after a second etch have longer mean track lengths than tracks measured only after the second immersion time, as—on average—tracks with shorter effective etch times will be selected using the second approach. That problem should not occur with our etch-time window approach, as measuring the widths and calculating the effective etch times gives the same results for one or for consecutive steps and results in roughly the same track selection. Thus, a combination of tracks using both approaches should be unproblematic. Not using tracks at <20° to the c-axis for effective etch-time calculations causes the loss of a few tracks, but the number of such tracks is anyway low (5,6,Figures 5-7; [6]). Better measurements can make our widths more precise so that these tracks can perhaps be included in future work.

For samples with high fission or ion-track densities, the maximum immersion time is limited. The limit is ~30 seconds for our samples irradiated with high fluences of 132Xe-ions (6813X1, 6814D1, and 6910F1). In our older samples, only one in four tracks measured after 20 seconds was also measurable after 40 seconds, due to overlap (18912C1 and 18912E1; online supplementary Table S3). A fixed 20 + 10 seconds protocol, as proposed by Tamer and Ketcham (2023) [36], is therefore not optimal for all samples. In difficult cases, etching with the conventional 20 seconds, 5.5 M protocol still gives reliable tE-estimates for all measured tracks and allows to use them for validating and interpreting T-t modeling results.

It is debatable how a practical etch-time window should be defined. Here, we use the interval of 15–30 seconds (Figures 6 and 7, online supplementary Figures S3; S4; right panel, green bands). Above tE = 10 seconds, most confined tracks are etched to their ends in Durango apatite, consistent with an average fossil track-etch rate νT of ~75 μm/min [25], and the observation that the slowest etching faces begin to develop at the track tips after tE = 10 seconds [23-25]. 35% to 85% of the tracks in our samples are etched for less than tE = 10 seconds after 20 seconds immersion (Figure 5, online supplementray Figure S2). A tE = 15 seconds fixed lower limit thus ensures that no tracks are under-etched. An upper limit of tE = 30 seconds excludes most over-etched tracks.

In some cases, c-axis projection results in projected track lengths increasing with angle to the c-axis. Figures 9 and 10 show the c-axis projected lengths plotted against the c-axis angles for some and online supplementary Figure S6 for all samples. This might indicate that in certain samples the fossil tracks are to various degrees under-etched compared to induced tracks after the 20 seconds immersion. Therefore, a regression line to a plot of c-axis projected lengths against c-axis angles is useful for identifying such samples. A flat trend indicates well-etched tracks and a valid c-axis projection, which eliminates the length anisotropy [33]. In the opposite case, the samples should not be modeled because they do not meet the requirements, for example, of the anisotropy model [33, 40, 41]. For a majority of our samples, the effective etch-time window 15–30 seconds improves the c-axis projection to the point of eliminating anisotropic length conditions, for which the c-axis projected lengths become isotropic. It should be investigated if the etching conditions for which the c-axis projected lengths become isotropic define the optimal etch-time window for each specific sample.

We modeled the T-t paths of 14 samples for which we have confined track lengths after 20 seconds immersion and in the 15–30 seconds tE-window. Figure 9 shows the results for Cenozoic thermal histories, Figure 10 for Mesozoic-Cenozoic histories; online supplementary Figure S6 compiles all results. We used an initial length (l0) of 16.1 µm but nearly the same results were obtained with l0 from Dpar. The mean c-axis projected lengths for the 15–30 seconds tE-windows exceed those after 20 seconds immersion by 0.4–1.1 µm (Table 3). The difference increases with Dpar, with length increases of 0.8–1.1 µm for samples with Dpar’s of ~2.2 μm.

We have independent AHe ages for several samples (Table 1). They are discussed in Text S1. In the following, we compare the modeling results obtained with the lengths measured after 20 seconds immersion with those obtained with the lengths of tracks in the 15–30 seconds tE-window and discuss their fit to the AHe data. A late cooling event, sometimes interpreted as an artifact (e.g., [42, 43]), is a common feature of the thermal histories of samples measured with the 20 seconds immersion protocol [29]. It is prominent in the T-t results for sample 0817I1, but absent when modeled with lengths in the 15–30 seconds tE interval; the T-t solution also fits better to the AHe date (Figure 9(a)). The same is true for GW30 (online supplementary Figure S6(b)), although the late cooling is less pronounced. Overall, the good-fit T-t paths of most Cenozoic samples are more consistent with their AHe ages when using a 15–30 seconds tE-window than when using the 20 seconds immersion lengths (Table 1; 6814D1, Figure 9(b); 6813X1 (online supplementary Figure S6(e)) and 18911C (online supplementary Figure S6(g). Sample 1899D1 is an exception: its continuous cooling T-t model based on the 20 seconds lengths agrees better with its ~4.8 Ma AHe age than the rapid cooling at 10–7 Ma based on the etch-time window lengths (Table 1; online supplementary Figure S6(f)). It is also the only sample for which the slope of the c-axis projected lengths plotted against the orientation to the c-axis is significantly negative using the etch-time window approach. The latter may indicate pronounced etching with the slowest radial etch rate vR, causing over-correction for tracks at low angles to the c-axis. For samples 0818C1 (Figure 10(c)) and 18912C1 (online supplementary Figure S6h), with Mesozoic cooling ages (146 and 122 Ma), the Cenozoic cooling segment is not well constrained and, hence, the fit to the Cenozoic AHe ages is the same for both modeling approaches.

We have no independent constraints for the Mesozoic evolution. The main difference is a 5–10°C offset during the >100 Ma cooling through the partial annealing zone (Figure 10(a) and online supplementary Figure S6(h)). For these samples, modeling with tE = 15–30 seconds gives the lower temperatures. For the conventional-protocol models suggesting long-term continuous cooling, the 15–30 seconds tE-models indicate faster cooling from 250 to 150 Ma, in better agreement with the regional Mesozoic evolution of the southwestern Tian Shan (Figure 10(b) and online supplementary Figure S4(k)) [44]. Both modeling approaches indicate pronounced Cenozoic reheating for sample 0818C1 but to a ~10°C lower maximum temperature in the case of the tE-window approach (Figure 10(c)).

Overall, modeling with an effective etch-time window of 15–30 seconds improves the agreement of our T-t solutions with independent geological evidence and with the AHe ages compared to modeling the confined track lengths obtained after a conventional 20 seconds immersion. A main reason for the difference between both modeling approaches may be the high number of tracks etched for <10 seconds (section 3.2.) in the 20 seconds dataset (Figure 5, online supplementary Figure S2). Such tracks may be underetched compared to induced tracks used for annealing experiments [8, 10, 31] which seem to etch faster than fossil tracks [29]. We argue that the deviation from the annealing equations caused by the slight over-etching of tracks in the 15–30 seconds tE-window has a comparatively minor impact on the modeling results. “Over-etching” proceeds by etching the slowest-etching faces at both track ends with rates of 0.5–0.7 µm/min for Durango Apatite [25], similar to bulk-etch rates of ~1.3 μm/min reported by Tamer et al. [16]. Over-etching for 10 seconds, thus, results in an additional track length of only ~0.2 μm.

The etch rates νR (5.5 M HNO3 at 21°C) of geological apatite samples depend on their chemical compositions and our results show a linear dependence of νR with Dpar in the range of 1.4–2.6 µm. We introduced a scaling factor aR relating the etch rates of unknown apatite samples to those of Durango apatites [25] in the same orientations. We fitted a linear function aR = 0.40 Dpar + 0.19 (r = 0.87) to data for fourteen apatite samples with Dpar in the range of 1.4–2.6 µm. A scaling factor aR = 1 corresponds to Dpar ≈ 2.05 µm, which is 0.1–0.2 µm higher than most reported values for Durango apatite. The difference, likely due to the limited precision of the fit and different experimental setups compared to the study of Aslanian et al. (2021) [25], highlights that protocols for how track-width measurements have to be performed need to be defined to be consistent with a scaling function.

This equation enables us to calculate the effective etch times of confined tracks in different apatites from their widths and Dpar. This gives reasonable estimates for tracks at >20° to the c-axis but less so at <20° because the tracks are thin and vR is low. The difference between the effective etch times of individual tracks at two consecutive steps is often less than the intervening immersion time, in particular for low initial tE. This suggests that even after a first etch it takes up to several seconds for the etchant to reach the tips of hard-to-reach tracks. This raises some concerns about vR-calculations based on step-etch experiments.

Effective etch-time calculations serve as a quantitative criterion for selecting suitable confined tracks for modeling thermal histories. A tE window of 15–30 seconds combined with a total immersion time of up to 40 seconds increases the number of measurable confined tracks up to 2.5 times. Pooling 15–30 seconds track lengths from consecutive steps increases the database for T-t modeling. This approach also allows to adjust the immersion time for geological samples depending on their characteristics, such as etch rates, spontaneous track densities, cracks, and inclusions.

We calculated the effective etch times of confined fission tracks in geological apatite samples with different lithologies, fission-track and (U-Th)/He ages, spontaneous- and ion-track densities, and confined-track-length distributions. We compared the T-t solutions using length measurements obtained after a conventional 20 seconds immersion in 5.5 M HNO3 at 21°C with those for a tE window of 15–30 seconds. Overall, the latter agrees better with the AHe ages and geological constraints. Plots of c-axis projected lengths against c-axis angle lead us to suspect that the fossil tracks in geological samples are to various extents under-etched. An etch-time window can in part compensate this artifact, for the most part by eliminating the tracks with the shortest effective etch times, which can be as short as 5 seconds for a 20 seconds immersion. We propose that a plot of c-axis projected length against c-axis angle can be useful for detecting cases of under-etching, estimating its magnitude, and adjusting the sample immersion time.

Hence, summarized, to us, as discussed above, the main advantage of using effective etch-time calculations and a tE window for T-t modeling is to provide an explicit track-selection criterion. This can improve the reproducibility of confined track-length measurements. It is nevertheless obvious that effective etch times >20 seconds differ from those used in the annealing experiments [8, 10, 31], which are the basis for the annealing equations [40, 41]. On the other hand, these experiments were carried out on induced fission tracks, which appear to etch faster than fossil tracks [29, 35]. One possible solution is to conduct new annealing experiments, another is to correct for differences in effective etch time, or better, in width.

Raw data for all measurements can be found in Table S3

The authors declare that they have no conflict of interest.

Alexandra Käßner provided apatite mounts and grain separates. We are grateful to B. Wauschkuhn and C. Aslanian for discussion. We are indebted to B. Van Houdt and J. Wagemans (SCK BR1 reactor, Mol, Belgium) for the neutron irradiations and fluence measurements. We are grateful to C. Trautmann and E. Toimil-Molares (GSI, Darmstadt) for the heavy ion irradiations.

Text S1. This is a word file describing details of our T-t modeling approach and chosen constraints.

Figure S1. Radial etch rates νR plotted against c-axis angle for an apatite sample etched in 5.5 M HNO3 at 21°C, calculated from microscope measurements using Zeiss Axiovision software and measured on images imported in the CorelDraw graphic software. The red and black lines are fitted polynomials.

Figure S2. (a) Effective etch times tE plotted against angle to the c-axis for 20 seconds immersion in 5.5 M HNO3 at 21°C (Table 1). Red arrows show outliers; red dashed lines mark the lower limit of the c-axis angles up to where we consider our tE calculations reliable. Pinkish bands highlight the 5–20 seconds tE-intervals. (b) Histograms and cumulative distributions (gray lines) of the effective etch times in (a). Vertical dashed black lines at 10 seconds indicate the lower tE limit at which all tracks are considered to have completed the vT stage in Durango apatite (Figure 8c of [25, 28]). Horizontal thin dashed black lines show the proportion of measurable tracks that fall below that threshold after single etch with the 20 seconds protocol [35].

Figure S3. Comparison of effective etch times tE after a single immersion in 5.5 M HNO3 at 21°C for 20 seconds and subsequent immersions to a total of 30 seconds (blue) and 40 seconds (gray). Colored bands indicate the expected tE-ranges. Histograms present the tE-distributions. Red, blue, and gray bins refer to immersion times of 20, 30, and 40 seconds, respectively. Dashed black lines and green bars indicate the effective etch-time window used for thermal history modeling.

Figure S4. (a) Effective etch times tE of tracks recorded after a first 30 or 40 seconds step and a subsequent immersion to a total of 40 or 60 seconds. Blue and gray bars indicate the expected tE-ranges. Histograms show the tE-distributions. Blue, gray, and dark gray bars refer to immersion times of 30, 40, and 60 seconds. Dashed black lines and green bars show the effective etch-time window used for thermal history modeling. (b) As in (a) but for tracks measured after 40 seconds immersion only.

Figure S5. Measured (tE2) versus predicted (tE1 + tI2) effective etch times for step-etch experiments (Table 1).

Figure S6. Comparison of thermal histories for a conventional etch protocol (5.5 M HNO3, 20 seconds, 21°C; [35]) and for a 15–30 seconds effective etch time window. Apatite fission-track and (U-Th)/He) data are from Trilsch et al. (in prep.) and Käßner et al. (2016) [32].

Table S1. Scaling factor aR to the νR-equation of Aslanian et al. [25, 28] fitted to each sample.

Table S2. Number and effective etch-time calculations for tracks detected and recorded after immersion times of 30 and 40 seconds.

Table S3. Raw data.

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Supplementary data