This experiment was carried out on a single crystal of the Durango apatite from Mexico (Foshag, 1929; Young et al., 1969; McConnell, 1973). Durango apatite contains 7–22 ppm U and 150–380 ppm Th (Boyce and Hodges, 2005; Enkelmann et al., 2005a) and has a fission-track age of ∼ 31.5 Ma (Jonckheere et al., 1993; Jonckheere, 2003; Enkelmann et al., 2005b; Jonckheere et al., 2015, 2024a). We mounted a prism section of Durango apatite in resin, ground, and polished it with 6, 3, and 1 µm diamond suspensions and a 0.04 µm silica suspension to expose a flat internal surface. We covered the mount with a 180 µm Al degrader foil and 12 µm PET film, and irradiated it perpendicular to the c axis and at 1/4π dip angle with 25.0 MeV amu^{−1 86}Kr-ions at the HIRFL in Lanzhou, operated by the Institute of Modern Physics of the Chinese Academy of Sciences (Fig. 1). We etched the mount in 5.5 M HNO₃ at 21 °C for 20+10+10 s to reveal the ion tracks (Fig. 2). The sample had a sealed natural crack extending from the surface to tens of microns below it. Although no etchant could penetrate along it, the crack nevertheless constituted a break in the apatite lattice. The intersections of the Kr tracks with the crack are therefore convex, like their intersections with the surface, causing etch figures bounded by the fastest-etching faces to develop (Aslanian et al., 2021; Jonckheere et al., 2022). This produces uniform etch pits at the apatite surface, one side of which, with length Dpar, is parallel to the c axis (Burtner et al., 1994). The same process takes place on either side of the crack, giving rise to diamond-shaped etch figures, whose size decreases with increasing distance along the track from its surface intersection (Fig. 2a).

From the diameters (dS, dD) of circles tangent to their outside edges (Fig. 2b, c), and the immersion time (tI) we calculate the effective etch time tE=(dD/dS)tI, and access time, tA=tI-tE, of each diamond shape, i.e., the time that the etchant needs to advance from the surface to the crack. After correcting for dip, the etched distance to the crack is l=lD/cos⁡(1/4π) and the total etched length is lM=lT/cos⁡(1/4π).

Table 1 reports the measurements for the 20, 30, and 40 s cumulative immersion times. Figure 3a plots dD against l; Fig. 3b shows histograms of the calculated lengths lM. The solid lines in Fig. 3a are geometric mean regression lines anchored at the mean values of dS. These means have negligible error compared with the individual dD values, because dS (Dpar) exhibits almost no variation and was measured for each track and immersion time. Table 1 lists the slopes (s) of the regression lines, along with their d axis and l axis intercepts, and the calculated track etch rates: vT=-dS/(stI).

The vT value for the initial 20 s immersion (vT ≈60 µmmin-1) is the actual track etch rate. The value is comparable with that for induced fission tracks at high angles to the c axis in a prism section of Durango apatite (Fig. 5c in Jonckheere et al., 2024b). The values for 30 and 40 s immersion measure the fast but finite rate of etchant advance along pre-etched tracks. The sum of the immersion times of a step-etch experiment therefore overestimates the effective cumulative etch time. For the Kr tracks, this amounts to ≳4 s for the 10 s step following the initial 20 s immersion; i.e., the time required to travel the length (≈ 20 µm) of the pre-etched track section at vT≈275 µmmin-1 (20µm/4.6µms-1≈4.3 s). Step-etching thus broadens the effective-etch-time distributions of ion and fission tracks (Fig. 8 of Trilsch et al., 2023; Fig. 6a of Jonckheere et al., 2024b).

The difference between the measured mean length and the l axis intercept of the regression line for the first 20 s step (lM-lI) is negligible but increases at each subsequent step (Table 1; Fig. 3). This is because the high etch rates during later steps relate to the track sections etched during the preceding step but not to the unetched sections beyond them, which etch at the normal rate vT. We corrected the expected lengths (lE), based on the measured lengths and etch rates in Table 1 (caption of Fig. 3). The accord between the revised and measured lengths (lM-lE) and between the revised and actual immersion times (tM-tI) is reasonable, considering the uncertainties. In particular the high etch rates of the pre-etched sections affect the calculations because they are obtained from the small differences between the slopes of the regression lines fitted to the 20, 30, and 40 s data. The calculation assumes a constant vT over the entire etched track length up to 40 s (constant core; Tamer and Ketcham, 2020a). The results in Fig. 3a and Table 1 support this assumption.

A constant vT is consistent with the fact that fission tracks in apatite have straight sides, lacking curvature (Fig. 4a and b; “motorboat effect”; Fleischer et al., 1969, 1975; Paretzke et al., 1973; Somogyi and Szalay, 1973; Durrani and Bull, 1987; Nikezić, 2000; Nikezić and Yu, 2003; 2004). This means that vT is unrelated to the local stopping power (Tamer and Ketcham, 2020a; Ketcham and Tamer, 2021), or to the radius of the latent track (Li et al., 2012, 2014). The most compelling indication is that etched confined tracks show no constriction, interruption or other modification in their midsections, even though the effective fission fragment charges, and thus stopping power and track radius are assumed to be near zero at the locus of the fissioned nucleus (Li et al., 2023). We have earlier reported that the average fission track etch rate varies with Dpar (Fu et al., 2024) and with the track orientation, with minima parallel and perpendicular to the c axis, analogous to the apatite etch rate vR (Jonckheere et al., 2024b). This suggests that vT is more related to the chemical characteristics of the track detector than to the properties of the fission fragments.

The fact that the mean etch pit size increases less in relation to the immersion time during the initial 20 s immersion, than during the subsequent 10 s immersions (Table 1; ΔdS/ΔtI) is due to an etch induction time related to polishing damage (Jonckheere and Van den haute, 1996, their Fig. 19).

To investigate the effects of partial annealing on vT and vR, we carried out track measurements on four prism sections of Durango apatite that had been pre-annealed, neutron-irradiated and, all but one, re-annealed to create induced-track populations with mean lengths of ∼16 µm (sample 21-2), ∼14 µm (21-4), ∼12 µm (21-1), and ∼10 µm (21-3). The irradiation and annealing conditions are given in Ketcham et al. (2015). The track densities measured in transmitted and reflected light are reported in Aslanian et al. (2022). The samples were mounted in resin, polished with 6, 3, and 1 µm diamond and 0.04 µm silica suspensions, and etched for 20 s in 5.5 M HNO₃ at 21 °C (Carlson et al., 1999). The confined tracks were imaged and measured as in Jonckheere (2023). The measurements included the track lengths, orientations, widths, cone-angles, the positions of the host track intersections (iP), and Dpar. We calculated the effective etch times and the apatite and track etch rates using the equations of Aslanian et al. (2021) and Fu et al. (2024).

Figures 5 and 6 show the cumulative distributions of the measured track properties for different annealing stages, and Table 2 lists the main statistics. It is worth noting that the angular (azimuth) distribution remains unchanged after partial annealing (Fig. 5d). This is similar to the results of Ketcham (2003), except that our data cover mean track lengths from 15.9 to 10.1 µm, before c axis projection. The intersection-point distribution (distance from host track intersection to the nearest confined track tip) is also unaffected by annealing (Fig. 6a). In each sample, the number of intersections decreases from the centre towards the track tips (Fig. S4 in the Supplement) but there is no evidence that, in relative terms, short tracks are differently affected than long tracks. The decrease from the centre towards the tips could be attributed to the fact that a confined tracks intersected near its tip cannot be measured if it is obscured by the host track. In that case, one would expect that, in relative terms, the effect is greater for short tracks than for long tracks, which is not corroborated. One might conjecture that there is a countervailing bias of some sort, perhaps related to a change of the track etch rate due to annealing. We rule this out because our evidence shows that there is no vT-change, and because it is improbable that two unquantifiable biases exactly cancel (Fig. 6a). We therefore prefer to consider a factor that is constant across the annealing stages and mean lengths. There is e.g., the fact that it takes twice as long to etch a track intersected at one tip than a track intersected in its centre to the point that both tips are etched so that the track is accepted for measurement. Tracks intersected between their tips and centre will require an intermediate etching time. This could account for the gradual decrease of the intersection point frequencies starting from the track centre towards the track tips. This supports a constant-core vT-model, as a linear model entails that the time required to reach the furthest tip depends sensitively on the intersection point. These observations are counter to expectation based on length bias and anisotropic annealing (Galbraith et al., 1990; Donelick et al., 1999; Galbraith, 2002; Ketcham, 2003). It has been established that track selection is not just a matter of geometrical biases, but that etching and observer biases must also be taken into consideration. According to Aslanian et al. (2021) and Jonckheere (2023), the selection of a confined track depends on whether it exceeds a threshold width, which is not affected by annealing.

The cumulative vT-distributions are the same after 10 h annealing at 240 °C (21-4), 288 °C (21-1), and 310 °C (21-3; Fig. 6b) as that of unannealed induced tracks (21-2), despite experiencing mean length reductions of ∼2, ∼4, and ∼ 6 µm. This contradicts the finding that partial annealing of induced tracks at 235 to 280 °C causes up to a doubling of vT (Ketcham and Tamer, 2021). Taken together with the fact that vT exhibits no variation along the tracks (Sect. 2) and is unaffected by lab-annealing, the observations that vT correlates with Dpar (Fu et al., 2024) and varies with the angle to the apatite c axis (Jonckheere et al., 2024b) underscore the fact that vT is more a function of the detector properties than of the damage densities along the fission fragment trajectories.

The median effective etch time of the unannealed tracks (21-2) is 2 s longer than that of the most annealed (21-3; Fig. 6c). Both intermediate samples lie between the extremes, albeit in reverse order. Given that their track and apatite etch rates are almost the same, one might expect that tE correlates with the reciprocal track densities. That of 21-3 is ∼23 of 21-2 (Aslanian et al., 2022); the average separation between a host track and a potential confined track in 21-3 is thus greater than in 21-2. However, this must not be the case for the measured tracks, selected on the basis of their widths. For an average vT of ∼ 2 µms-1 (Table 2: ∼ 120 µmmin-1), a ∼2 s tE-difference corre­sponds to a ∼4 µm length difference in the etchant trajectories from the host track intersections to the confined-track tips. If the host track intersects a confined track in the middle, we have 2×4 µm = 8 µm; if it intersects it at the tip and etches in one direction, we have 4 µm. The average is 1/2 (4µm+8 µm) = 6 µm. This is correlates well with the ∼ 6 µm mean-length difference between 21-2 and 21-3.

All samples have narrow Dpar-distributions with medians (D50) decreasing from 1.89 µm for the unannealed apatite (21-2) to 1.80 µm for the most annealed (21-3). The difference is small but reflects a systematic shift of the Dpar-distribution to lower values with increasing annealing (Fig. 6d). Because Dpar is proportional to the apatite etch rate, it suggests a thermal relaxation of irradiation-induced strain from recoils following (n, γ)-reactions or uranium fission and track formation (Rivera et al., 2011; Dufour and Toulemonde, 2016; Schauries, 2018; Rymzhanov et al., 2019). As­suming Arrhenius kinetics, the D50-ratio (1.80(2)/1.89(2)=0.953(15)) of the most annealed and unannealed samples represents an activation energy increase for apatite etching of 118(27) J mol⁻¹, or 1.22(28) meV per atom. If an individual point defect stores ∼1 eV, the lowering of the apatite etch-rate results from annealing one in a thousand (FD=1.22×10-3) defects created by irradiation.

Assuming Arrhenius kinetics, the rate factor (kD) and activation energy (ED) for defect annealing are given by: 1kD=ln⁡1-FDtA=3.39(78)×10-8s-1 wherein tA=36000 s is the annealing time (10 h). For a pre-exponential factor k0=1013 s⁻¹, this gives: 2ED=RTAln⁡k0kD=228(1)kJmol-1 wherein R=8.314 J °C⁻¹ mol⁻¹ is the gas constant, and TA=310 °C the annealing temperature of 21-3. For comparison, we calculated corresponding nominal rate factors (kF) and activation energies (EF) for fission-track annealing using the fractional length reduction, FF=1-l/l0, to measure progress. The calculation shows that EF increases from 181(1) kJmol-1 at 240 °C to 202(1) kJmol-1 at 310 °C (Table 3). These are indicative average values, conditional on the assumed Arrhenius kinetics and the pre-exponential constant (k0). Without, therefore, attaching undue physical meaning to the numerical results, the calculation bears out that fission-track annealing and the lowering of the apatite etch rate due to partial elimination of point defects both occur under the annealing conditions to which the samples were exposed. On the whole, point defects begin to anneal at time-temperature conditions at which fission-track annealing is already well advanced.

For a constant-core model and high average track etch rate (vT≈120 µmmin-1; Table 2), the exact value of vT and the precise point from where a confined track begins to etch are of little consequence. For the same reason, the effect of multiple host track intersections, even occurring at the same time and far apart, is minimal. This is not so for a linear vT-profile with a central maximum (∼1.7 µms-1; unannealed induced tracks) decreasing to vR towards the track tips (Ketcham and Tamer, 2021). In that case, short tracks intersected towards one end are more likely than longer tracks to be etched at the opposite end to a point that satisfies an operator's selection criteria. This would produce a correlation between the track length and the position of the host track intersection (Fig. 8 of Jonckheere, 2023). The correlation would decrease depending on the extent of length reduction and the increase of the track etch rate following partial annealing (Ketcham and Tamer, 2021).

It is reasonable to suppose that the lengths of individual confined tracks depend to some extent on their host track intersections (iP), track etch rates (vT), and effective etch times (tE). To exclude spurious correlations due to length anisotropy, we plotted the c axis-projected lengths (lP) against iP, vT and tE. Regression lines to the lP-vs.-iP plots of 21-2 and 21-4 have small positive slopes whereas those to the lP-vs.-iP plots of 21-1 and 21-3 have small negative slopes, indicating the absence of a significant correlation (Fig. S7). Regression lines to the lP-vs.-vT plots of all the samples have small positive slopes (Fig. S8). Because of the large vT-range (50-250 µmmin-1; Fig. S5), this can add on average 0.1–0.3 µm to the lengths of fast-etching tracks compared to those of slow-etching tracks. Regression lines to three out of four lP-vs.-tE plots (except 21-4; Fig. S9) have weak positive slopes, which, nevertheless, add on average 0.1–0.3 µm to the lengths of well-etched tracks (tE≈20 s) compared to those of underetched tracks (tE≈5 s). Because the independent variables are not free from measurement error, least squares regression underestimates the true slopes and overestimates the intercepts of the regression lines due to attenuation bias. If half of the measured variation of vT and tE is true variation and the remainder due to measurement error, the true slopes would be twice as high, and the lP-ranges, should be doubled. We nevertheless conclude that, for practical purposes, within a sample of confined tracks selected for measurement, the intersection points, the track etch rates and the effective etch times have little effect on the length distribution (20 s; 5.5 M HNO₃; 21 °C). This does not mean, however, that they have no effect on the sampling itself, i.e., on the distinction between tracks selected for measurement and those that are not. E.g., the effective etch time influences the widths of confined tracks, and so their qualification for measurement.

We studied the FC1 and AS3 apatites from the Duluth intrusive complex in Minnesota to investi­gate the effects of geological ageing. FC1 and AS3 are reference materials and proposed age stand­ards with fission-track ages of 820(27)–882(44) Ma (FC1), and 924(36)–989(31) Ma (AS3; Iwano et al., 2019; Cogné and Gallagher, 2021; Härtel et al., 2022, 2023; Jonckheere et al., 2024a). Their interest lies in the fact that they are old and have long low-temperature thermal histories, and so accumulated extensive radiation damage, the most important factor responsible for ageing. Based on the data of Härtel et al. (2023), the total accumulated alpha dose for FC1 is ∼7×1016 α g⁻¹ (0.8–1.0 dpa), although it is not known what fraction of the damage survived till the present. We are not aware of such data for AS3.

We annealed an aliquot of FC1 and AS3 for 24 h at 450 °C, and irradiated it with thermal neutrons in channel Y4 of the BR1 reactor in Mol, Belgium. We mounted irradiated and untreated fractions of each sample in resin, and ground, polished and etched them using the techniques described in Sect. 3.1. As there also, we measured the track lengths, orientations, widths, cone-angles, and Dpar, but not the host track intersections, and calculated the effective etch times and the apatite and track etch rates.

Consistent with their similar Dpar-values, the induced-track length distributions of FC1 and AS3 are almost identical. The fossil tracks in both samples underwent >20 % shortening; both fossil-track length distributions are also similar, although those in FC1 are somewhat shorter. We in­ter­pret this difference as significant; c axis projection highlights the greater fraction of short tracks (Table 4; Fig. 7a, b). The semi-axes of ellipses fitted to the induced-track data are close to the trend for unannealed apatites (Donelick et al., 1999); those fitted to the fossil-track data reveal them to be somewhat less anisotropic than induced tracks of the same mean length (Fig. 7c). This could be due to different length bias affecting high- and low-angle tracks, different angular distributions, which control the relative weights of the constituent populations, or under-etching (Jonckheere, 2023; Jonckheere et al., 2024b). However, the angular distributions of the fossil, as well as induced, tracks are almost identical (Fig. 7d) and the effective etch times of the fossil tracks are 1–2 s longer than those of the induced (Table 4; Fig. 8b). This argues against experimental factors, leaving length-bias as the likeliest cause, barring unknown differences between fossil and induced tracks.

As in the case of the annealed Durango samples, the vT-distributions of all the Duluth samples are almost identical (Fig. 8a). The ∼5 % difference between the means for the fossil and the induced tracks (Table 4) is too small to be considered significant at this stage. The overall agreement is striking given that the track etch rates vT are calculated from the apatite etch rates vR (Fu et al., 2024): 3vT=s12d1-d2vRS, wherein s12 is the distance between two inscribed circles, with diameters d1 and d2, touching opposite sides of the confined track, and vRS is the apatite etch rate perpendicular to the track axis. Dpar-measurements indicate that the apatite etch rate differs 7 %–8 % between FC1 and AS3 and 16 %–17 % between the fossil and induced tracks (Table 4). We applied the empirical correlation between Dpar and vR (Trilsch et al., 2023; Fu et al., 2024) for calculating the apatite etch rate of each sample: 4vRS=DparSDparDvRD. [Dpar]_S and [Dpar]_D refer to the sample and to Durango apatite, and vRD is the etch rate of Durango (Aslanian et al., 2021). The agreement between the vT-distributions of FC1 and AS3 is perhaps unsurprising. Both are from the Duluth complex anorthosite series, albeit from locations >100 km apart, and, according to their ages and track length distributions, both had similar thermal histories. The less obvious finding is that, in contrast to the substantial length reduction, ageing, or geological low-temperature annealing, appears to have had a minimal effect on the etch rates of the fossil tracks.

As in the case of the annealed Durango samples, the average effective etch-times (tE) of FC1 and AS3 are just over half the immersion time (tI=20 s; Table 4). The means, medians and standard deviations for both induced-track samples are similar, those for both fossil-track samples somewhat less so, reflecting their geological histories. There is, on the other hand, a definite offset between the tE-distributions of the fossil and induced tracks (Fig. 8b, d), which is not due to vT. The longer effective etch times of the fossil tracks are instead due to the greater vR of the un­annealed grains, which allow faster etchant access to the confined tracks. There are contributing factors: the much higher fossil than induced track densities (Jonckheere et al., 2024a) lowers the average separation between the host tracks and confined tracks; on the other hand, the shorter fossil- than induced-track lengths have an opposite effect. It is difficult to separate these factors because the stock of confined tracks is sampled using the experimenter's track selection criteria.

The major difference between samples FC1 and AS3, and between the fossil and induced tracks in each, is between their Dpar-values (Table 4; Fig. 8c). The fossil-track values are ∼16 % (AS3) and ∼ 17 % (FC1) greater than those of induced tracks. We propose that this is due to distributed radiation damage from α-disintegration of ²³⁸U, ²³⁵U, ²³²Th, and ¹⁴⁷Sm, and from collateral recoils from ²³⁸U-fission, accumulated over geological time. The resulting lattice damage increases the overall apatite etch rate, producing larger Dpar. Figure 9 shows the openings of fossil and induced tracks in FC1. The ageing of the fossil-track samples appears to have had a greater effect on the widths (Dper) than on the lengths (Dpar) of the track-surface intersections. This support the radiation damage idea: Dpar depends on the etch rates of fast etching faces while Dper depends on that of slow-etching prism faces flanking each track to either side (Jonckheere et al., 2020, 2022). This can account for the doubling of Dper while Dpar increases <20 % (Fig. 9b), as slow-etching faces are more susceptible to radiation damage than fast etching faces (Jonckheere et al., 2019; their Fig. 4). Tamer and Ketcham (2020b), in contrast, found no difference between the Dpar of fossil and induced tracks in the Durango, Fish Canyon Tuff, and Tioga apatites. Depending on uncertain estimates of the U, Th-concentrations and damage accumulation ages, it is just possible that the FC1 apatite is, or once was, past the first damage percolation point, whereas the younger samples are, or were, not.

Figure S15 plots the c axis-projected lengths (lP) of individual confined tracks against their effective etch times (tE), and Fig. S16 against their track etch rates (vT). The results for the induced tracks in FC1-I and AS3-I are in part similar to those for the unannealed Durango sample (21-2; Table 2). As in sample 21-2, the variation of vT adds 0.1–0.2 µm to the lengths of the fastest etching tracks, compared to those of the slowest etching tracks. In contrast, the variation of tE causes a 0.6 µm length difference over the full tE-range, compared to 0.1 µm for 21-2. The results for the fossil tracks in AS3-S are similar to those for the induced tracks in AS3-I (0.2 µm length difference due to vT; 0.6 µm due to tE), but those for the fossil tracks in FC1-S are not (0.7 µm length difference due to vT; 0.2 µm due to tE). On the whole, the vT- and tE-variation between tracks has little effect on the length distribution of a confined track population etched in a single 20 s step. However, as mentioned, the ordinary least squares regression underestimates the true slopes of the regression lines.