On etching, selection and measurement of confined fission tracks in apatite

Jonckheere, Raymond Charles

doi:https://doi.org/10.5194/gchron-2023-13

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https://doi.org/10.5194/gchron-2023-13

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16 Jun 2023

| 16 Jun 2023

Status: this discussion paper is a preprint. It has been under review for the journal Geochronology (GChron). The manuscript was not accepted for further review after discussion.

On etching, selection and measurement of confined fission tracks in apatite

Raymond Charles Jonckheere

Abstract. This work investigates the selection of horizontal confined tracks for fission-track modelling. It is carried out on prism sections of Durango apatite containing induced tracks with mean lengths of ~16, ~14, ~12, and ~10 μm. Suitable tracks are identified during systematic scans in transmitted light. The explicit selection criteria are that the tracks are horizontal and measurable. We measure the length, width, orientation, and cone angle of each selected track and in some cases other dimensions.

The confined track selection is in the first place dependent on a threshold width and in the second place on the requirement that the tracks are etched to their ends. In most cases the first condition implies the second, which decreases in importance as the tracks are shortened following annealing. The widest confined tracks, which must also be the shallowest, come to intersect the surface and are excluded. In general, the selection is dominated by the width of the etched tracks. This, in turn, depends on their orientation relative to the c-axis and the apatite etch rates, and their effective etch times. Despite the different geometrical configuration of the unetched host tracks and confined tracks, neither the angular distribution nor the etch time distribution of the confined track sample depends on the degree of annealing. This illustrates the general principle that those entities are selected that have the right properties for being selected. In this case etching-related factors determining the track width are the most important, while the known geometrical biases are second order. The track etch rate exhibits no demonstrable variation along the track, but signifi-cant differences from track to track. Moreover, although the track etch rate of induced tracks is not correlated with the extent of partial annealing, it is on average twice as high as the value for fossil tracks.

Our length measurements are in good agreement with the annealing models for this apatite and etch protocol. We submit that this is not fortuitous and that it is possible to select a representative confined track sample, and perform reproducible and meaningful confined track length measurements. Deliberate or inadvertent biasing, carelessness or inexperience will of course give different results, but these should be treated as statistical outliers, not as an indication that track lengths are fluid.

Received: 30 May 2023 – Discussion started: 16 Jun 2023

Publisher's note: Copernicus Publications remains neutral with regard to jurisdictional claims made in the text, published maps, institutional affiliations, or any other geographical representation in this preprint. The responsibility to include appropriate place names lies with the authors.

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Raymond Charles Jonckheere

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CC1: 'Comment on gchron-2023-13', Richard A. Ketcham, 13 Jul 2023

Comments on “On etching, selection and measurement of confined fission tracks in apatite”, by Raymond Jonckheere
By Richard Ketcham, July 12 2023
This is a very interesting study that builds upon previous innovative work by the author and his student. Utilizing track shape is an extremely promising way to wrest more information out of fission-track data. As the technology of data acquisition and image analysis continues to progress, some shape data will undoubtedly become routinely available with little effort, and can certainly be utilized. That time is not quite here yet, but the author’s group has invested considerable effort in try to build the foundations for what is to come, both observational (Aslanian et al. 2021) and theoretical (Jonckheere et al. 2022). The future of fission-track analysis will certainly include aspects of this work.
That said, there are some critical errors in the present manuscript, which in my opinion make it unsuitable for publication at this time. First, the author attempts to compare his results to the (possibly competing) predictions of the variable along-track etching rate model of Ketcham and Tamer (2021), but rather than using the equations therein attempts to rederive them anew. In doing so, he made a critical error by taking the etch rate at the track tip to be zero, rather than the bulk etch rate. Simply put, his equations do not correctly reflect my model, rendering a number of plots and assertions incorrect. Second, he fails to “remove the log in one’s own eye before removing the mote from his brother’s” in not investigating the limitations of his own data. The wavelength of visible light (0.4-0.7 µm) induces an unavoidable limit to the precision of optical microscopy data, which is not too influential for track length measurements but is a much larger issue for track width, and larger still for measuring angles from which to infer etch rates. Thus far I’m not aware of any attempt to quantitatively investigate the uncertainties of these data, either by mathematical analysis or brute force repeated measurements. This makes it very difficult to critically think through the implications of these data. In addition, it appears that each half-track is measured once, and so it’s unclear how the author would detect a change in along-track etch rate. Third, the author neglects to discuss data that appear plainly contradictory to his assertions. Tamer and Ketcham (2020) present clear indications of changes of track etch rate, such as annealed tracks being longer than non-annealed ones after 10s of etching, as well as the overall sequence of lengthening over a step etch experiment. As discussed in the detailed comments below, the model proposed here makes a number of further predictions about minimum or maximum track etch times that appear well tested and definitively excluded by the Tamer and Ketcham (2020) results. Given that the same individual (M.T. Tamer) produced both data sets, the scientific process suggests that this evident disparity in results at least be discussed, and credible ideas offered on where the problem might be, or what alternatives are available. For example, it may be plausible that the huge variation in track etch rate inferred from these measurements could be from measuring different sections of tracks – arguably a more parsimonious explanation than asserting that etch rates vary from track to track by a factor of 10. Or, if Tamer’s previous data are unreliable, where might that have stemmed from?
The Ketcham and Tamer (2021) variable along-track etching model is very simplified because it is based on mean confined length data from step-etch experiments, and with few data points to fit one can only test a simple model. That said, the constant-core etch rate model preferred by the author was the first one I derived and tested because I thought going into it that it would be the correct answer. It was the exercise of actually trying to fit the Tamer and Ketcham (2020) data (replicated in Tamer and Ketcham 2023, Chem. Geol.) that pointed to the linear model as adequate for most (but not necessarily all) levels of annealing. Shape data promises to be an excellent independent or complementary data source for deciphering track structure amidst the uncertainty stemming from not knowing when any given track starts etching, and it may well allow development of a more detailed and comprehensive model closer to physical reality. The author may be repeating my mistake in believing he knows the answer ahead of time, and thus limiting his field of consideration.
Finally, I note that the acknowledgements state that M.T. Tamer “made a substantial contribution to the measurements but desires not to be listed as a co-author.” I gather that this was because he was being asked to be on a manuscript that both contradicted and ignored his own measurements, putting him in an impossible place. This is a shame, and should be remedied if possible.
Detailed comments:
[line 10] The statement that the widest tracks “must also be the shallowest” is incorrect in two ways. First, width depends heavily on crystallographic orientation, and so one needs to consider this within angular bins of some sort. Second, within a given angular bin, width depends on time of intersection by the etchant channel (i.e. effective etching time). A shallower track is more likely to be intersected early, but given a limited number tracks within a bin, one or more deeper tracks easily could be intersected earlier, and thus be wider.
[line 18] It’s unclear whether the measurements described here actually tested for variation along the track length; only one rate was measured per half-track in most cases.
[line 102-103] Indeed, this is a significant bias. It might be clearer to just compare this to a 16-micron track (4.4 degree dip). It probably only has a small effect on this study, but is not something one would want to do when measuring unknowns.
[Figure 1] Needs scale bar or statement of image width.
[line 120] How do you know if there was a gap if you did not pierce it? It might eb better to state this as an interpretation, rather than an observation.
[line 130] Again, if only one pair of measurements is made for a half-track, it’s not clear whether or how that permits a change in etch rate along the track to be detected. It’s also not clear how reproducible these measurements are with respect to (a) where one places the two circles along the track, and (b) how precise the circle margins are given the limited resolution of these measurements imposed by the wavelength of light. For example, some circles in Figure 1 go out to the edges of the blurry (resolution-limited) track boundaries, while others are set noticeably within those boundaries. How does this affect the rate measurement? Has a multiple-measurement, come-back-to-it-later study been done?
[line 154] It’s unclear what is meant by one “participant” versus the other, and what they both did. Did two people make all of these measurements (thus providing a repeatability analysis it would eb good to report), or did one make all of the measurements and the other just check to make sure they looked OK?
[line 204] Unclear what is meant by “both projections”, and which is the former versus latter.
[line 242] Not really; one must impose additional assumptions concerning monotonicity, or let the uncertainty going back in time propagate to very large values.
[line 245] The reference is an abstract; I guess one can use it to claim that someone once said uncertainties can be taken into account, but it’s not a source of information for how to go about it. In any event, how can there be a single solution that is faithful to the uncertainty in both the length measurement (both from the measurement and from natural variation) and the time intervals (which are certainly not even)? This paragraph seems to drift off-topic.
[line 258-261] A strange statement; you’ve already corroborated that anisotropy is removed well at varying levels of annealing. The lengths measured for a given annealing experiment project to a narrow range, but the means are significantly different at different levels of annealing. The original length and orientations are what the projection is based on – they must matter, or else the distributions of high-angle tracks at each annealing level would not be so narrow.
[line 270-271] An alternative possibility could be that the section designated l₃ may not always be completely etched, due to the delay.
[line 275] It would be worth comparing this result with the recent report by Li et al. (2003, EPSL) of a gap at the center of each track upon formation. It’s worth pondering whether there is really only one gap, and it’s always at site of the original fissioning nucleus.
[line 317-324] Another apparent oddity is that some many tracks at 0-20° and 85-90° apparently need to etch for the full 20 seconds, which should be rare to impossible.
[line 321-324] It’s not clear whether this explanation makes sense; because etching along the penetrating channel is fast, there should be plenty of time for deeper tracks to grow wide and not be affected by the surface. This might be tested with the author’s data – is there a correlation between track depth and width, or track depth and angle? This explanation predicts that tracks at 60-75° should be deeper, on average.
[Figure 5, lines 327-328] The y-axis in these figures is frequency; it’s not clear what they have to do with delta-w. Were the wrong plots put into this figure?
[line 342-343] Do longer tracks attain a greater width before they reach their ends? Figure 5a-c seems to contradict this – tracks are shorter at all angles as annealing progresses, but average widths seem to increase.
[line 366] It seems like Tamer’s step etch data can be used to test some of the implied assertions of the model in Fig. 6a-d. Is it really tenable that tracks at ~70 degrees are not measurable at effective times >13s when you can measure them after a 10s step, with some as long as 13 um (after only 3s of etching)? The model described here predicts that there should be a large deficit of tracks at 60-75° if searching for them at 10s and verifying that they are still present after 20s, but that is not the observation in Tamer and Ketcham (2020). Also, again, are tracks with effective etch times of over 18s realistic given the need to penetrate the polished surface (how deep are these >18s tracks)? This is another case where it could be interesting to check track depth varies with c-axis angle.
[line 432] It seems worth asking whether track rates really vary by a full order of magnitude, or the uncertainty of the rate estimation has something to do with it, or possibly because track etch rate varies and they are trying to measure etch rates along different sections of tracks. For example, a prediction of the Ketcham and Temer (2021) model is that the etch rate for the s₁ track sections should be faster than for the s₂’s; this seems to be the case in Fig. 1g at least. Plotting these against each other seems like an easy test to try.
[Figure 8] This figure demonstrates a mathematical error in how the author has tried to reproduce the model of Ketcham and Tamer (2021). The graph at the top of the figure has the track end at vT=0, rather than vT=vB. The result of this error is seen most obviously in Fig. 8c; the contours indicate that the along-track rate is less than the bulk rate, which contradicts the Ketcham and Tamer model.
[line 450-451] “... except for perhaps 1 µm at either end” seems to admit that track etch rate does vary, as the thinning of the track toward its tip is impossible to miss. Furthermore, although Fig. 8 has mathematical problems, it’s notable that the only place where the variation in etch rate is similarly impossible to miss is in the last µm or so before each tip.
[line 468-469] Is such curvature unobserved, or just unobservable? What curvature there is predicted by the model along the midsection of the track is extremely subtle, and arguably beyond the resolution of optical microscopy, with its diffuse track edges…
[line 478-480] … for example, the white outlines in Fig 9a,b are sometimes on the inside edge of the blurred region, sometimes on the outside edge. One could draw a curved line on the left side of Fig 9b that is a scaled version of 8c.
[line 480] Excess compared to what? I note that in Fig 9e there are several shorter, under-etched tracks at the edge of visibility, some marked with white arrows and some without. What baseline is the author comparing to?
[line 490-492] A bizarre but clarifying assertion. In the Ketcham and Tamer (2021) model, vT approaches vB at the ends, not zero. Only in the author’s attempt to reproduce it does vT approach 0.
[line 554] Accelerated length reduction was posed as (and still is) an intentionally generic, non-interpretive term that encompasses gap formation but leaves open the option for other possibilities.
[line 591-592] And yet such an increase is very clearly present in Tamer’s data (10s experiments in Tamer and Ketcham 2020b, Fig. 1). It’s not terribly scientific to simply ignore the data the contradicts one’s conclusion. Can the author provide at least a hypothesis for the incompatibility between the step etch data and those presented here? Which data are more reliable, confined length or inferences from circles on blurry outlines?
[line 593-595] It’s not clear what an “excess” of confined tracks has to do with the projected lengths of surface-intersecting tracks, the vast majority of which start etching immediately.
[Figure 10, Figure A3, line 510] These figures also show the author’s mathematical error. Calculated track etch times are much too long for the linear model, because the author assumes a vT of zero at the track tip, rather than vB. There is no explanation for how 6s was determined or estimated to be the time required to start etching the first track. And, I have to say, the track segment next to the alpha sure looks like it has curved walls to me, and not just in the last micrometer…
It’s very worth noting that this is a beautiful track-in-track-in-track (TINTINT) image, the first I’m aware of in the literature, and poses an excellent test for the Ketcham and Tamer (2021) variable along-track etching model – can all that etching occur in the time given? I’ve attached a spreadsheet that demonstrates that it can, though not if it really took 6 seconds to start the first etch. If one drops that time to 2 seconds, the model predicts the position of all track tips to within 1 s of the end of etching at 20s. One can also increase the start time a bit by making some other assumptions that are within the measurements. It also bears mentioning that Tamer and Ketcham (2020) may not have measured a track such as beta, because the top tip is very indistinct. In any event, it would be good to know the basis of that 6s determination, short of which I’ll assume for now that my model passed this test.
[Appendix A] Here the author has attempted to derive a simpler set of equations than those listed by Ketcham and Tamer (2021) for their model. There could well be a more elegant formulation than mine (to paraphrase Dutch grandmaster J. van der Weil, “I am a butcher, not an artist”), but these are not yet it owing to assuming that vT=0 at the tip. One obvious place things are wrong is the case marked (1) va = c a , which should be va = c a + vB, and similarly (2) should be va = c(l/2 – a) + vB. How the author calculated c, though not spelled out, was probably also a bit off (should be (vmax – vB)/(l/2)).
[Figure A2] The distances between the dots are not consistent with each other; if one measures them in pixels, there seems to be about a vertical distortion of about 5%, in both Figures A2 and 10, assuming the markers correspond. The measurements are provided on another tab in the attached sheet. It’s not clear if this was because the picture was subtly and inadvertently scaled unevenly at some point before or when it was pasted into this document; since both images are distorted by a similar amount, I’d guess it was at some earlier stage in the process.

Citation: https://doi.org/10.5194/gchron-2023-13-CC1
AC1:
'Comment on gchron-2023-13', Raymond Jonkheere, 26 Jul 2023

Please find my replies to the comments of Dr. Ketcham in the attached pdf.
Sincerely, Raymond Jonckheere

Citation: https://doi.org/10.5194/gchron-2023-13-AC1
- CC2: 'Reply on AC1', Richard A. Ketcham, 09 Aug 2023
  
  Publisher’s note: the content of this comment was adjusted and a new supplement was uploaded on 15 November 2023. This was approved by the GChron editors since some formulations were inappropriate.
  
  Citation: https://doi.org/10.5194/gchron-2023-13-CC2
RC1:
'Comment on gchron-2023-13', Anonymous Referee #1, 01 Aug 2023

This is a very interesting and important paper, in principle suitable for publication in Geochronology. The author examined the confined fission tracks (FTs) in apatite in detail, with a particular focus on the shape and width of etched tracks. He analyzed four samples of Durango apatite annealed at different temperatures, and measured geometrical parameters (length, width, angle to c-axis, etc.) of horizontal confined FTs by utilizing induced FTs in pre-annealed apatite. This is an original investigation with unique geometrical analysis of confined FTs at different stages of annealing, and thus has an important impact on the FT thermochronology.
However, I found following important issues/pitfalls in the present manuscript that should be treated appropriately before publication:
(1) First of all, there is no description about the assessment of uncertainty of individual data in the geometrical analysis, particularly the width of FTs, which is likely the key parameter for reliable confined FT length analysis. I agree with the author’s point of view that the assessment of track width is the key, but then, the author should explicitly describe in the text the uncertainty (i.e., accuracy and precision) of track width measurement. (The author merely gives in Table 1A an “Error” of 0.01 micrometer (= 10 nm) which is amazingly small for optical microscopic observation.) Otherwise, it may result in the overinterpretation of the obtained data within the range of uncertainty, and lead to total misunderstanding of the phenomena. Note that this issue involves the propagation of analytical errors in calculating model parameters, such as track etch rate.
(2) The assumption should be more explicitly documented for calculating the etching rate (and other parameters) from the observed geometrical information (i.e., length and width of a part of confined FTs). The documentation needs to be given in the relevant part of the text, not only giving a series of equations. Otherwise, it may be difficult for readers to follow the logic of the study. For example, the author gives the variation of effective etch time versus angle to c-axis (Fig. 6), calculated from track etch rate values (Fig.7; constant etch velocity is assumed without explicit documentation). Then later in the text, he discusses the validity of assuming constant etch velocity throughout an entire track length (Fig. 8). Such a framework of the paper is just confusing and I suggest reorganizing the text in a more appropriate logical flow. Concerned with this, the author should better document/discriminate between physical theory, experimental observation, model calculation, and interpretation. These appear to be confused/contaminated from each other in places in the current text. This makes it difficult to understand the significance of new findings in the study.
(3) We see many typos of the experimental parameters in the text and figures that are similar to each other. This makes it further difficult to read the paper correctly.

Because of these issues, I judge that it is not appropriate to accept the paper at its present form of data presentation and interpretation. Therefore, I regret to suggest rejecting the paper, with a strong encouragement for the author to resubmit the material as a more carefully reorganized and rewritten manuscript.

Citation: https://doi.org/10.5194/gchron-2023-13-RC1
- AC2: 'Reply on RC1', Raymond Jonkheere, 03 Aug 2023
  
  Please find my replies in the attached pdf. Sincerely, R. Jonckheere.
  
  Citation: https://doi.org/10.5194/gchron-2023-13-AC2
RC2:
'Comment on gchron-2023-13', Raymond Donelick, 01 Aug 2023

General Comments
It is an honor and pleasure to review this excellent paper. Well done Dr Jonckheere! You have quantified the mental picture in my head of the many nuances of confined fission track lengths in apatite.
I highly recommend this paper be accepted for publication and I leave it to the author to choose to make the minor clarifications I request below.
I have not read the comments that have been posted for this paper (as of 31 July 2023) and may not do so.
Specific Comments
Introduction: The biases discussed do not include mention of other decisions facing an analyst such as: 1) Is that a naturally etched fission track? Sample TI of Carlson et al. (1999) exhibits many naturally etched fission tracks near natural grain surfaces. 2) Is that feature a fission track at all or some outlier that should be removed from the dataset? Like potentially naturally etched fission tracks, some potentially non-fission track features can etch like fission tracks but be too long (I use 19 microns as a cutoff for low-temperature natural and laboratory samples). Or perhaps an etched feature looks like an etched fission track but it is too short for the current fission track population(s) under study. Human analysts are smart enough to make these decisions and these decisions should be done openly.
Lines 46-50: I can never produce such a plot as discussed here. The notion of measuring sufficiently-etched fission tracks (apples) and contaminating those data by also measuring under-etched fission tracks (oranges) is not appealing to me. My plots would instead have no data (shorter etch times), then very little data, then sufficient data (longer etch times). I do not mix apples and oranges when it comes to measuring confined fission tracks in apatite.
Equations: It is these equations that can be used to show that non-elliptical, polar coordinate plots (length, angle to c-axis) of horizontal confined fission track lengths from a single population represent a mixture of apples (sufficiently etched tracks defining the ellipse, usually at higher angles to c-axis) and oranges (under-etched tracks falling short of the ellipse, usually at lower angles to c-axis).
Lines 172-177: Thank you for reproducing my work and that of Dr. Bill Carlson. The agreement between our works is truly independent and it is not accidental, but instead demonstrates that we as analysts can behave like machines with the ability to maintain biases within ourselves and share those biases with other machines.
Lines 184-187: My first submission of Donelick (1991) was to EPSL and a reviewer rejected the paper on the basis that "they [polar-coordinate plots of fission track lengths] are not ellipses", because his/her published mixtures of apples and oranges did not match my apples-only data.
Lines 199-209: Excellent discussion of this interesting issue. As we are after the mean and the mean ranges vary accordingly, it seems it should be possible to prove mathematically this is precisely how it should be.
Lines 248-261: In Donelick et al. (1999) I proposed a surface energy model for fission track annealing in apatite. In this model, converging track sides (during annealing) parallel to c-axis are flat and perpendicular to c-axis are rough with pyramidal faces. In this model, at high angles to c-axis and high degrees of annealing you get the results of Donelick et al (1999) – systematic accelerated length reductions - with the occasional Green et al. (1986) – segmentation where one or more opposing and ‘rouge’ pyramids intersect – along with the tips pinching off and appearing to diffuse into the bulk crystal as in Paul and Fitzgerald (1992). Surface energy minimization explains anisotropy and all experimental observations.
Lines 270-271: This statement is totally consistent with the surface energy minimization model above.
Lines 346-348: I hazard to say that only analysts willing to mix apples and oranges have this problem as your (and my) experiments demonstrate. This is especially true for samples with low confined fission track densities such as highly annealed experiments or the many, many, many low-fission-track-density natural samples studied without using 252Cf/particle accelerator ion implantation.
Line 424: I see this minor but significant effect for la0 and lc0 and a correlation between these and Dpar and Dper, respectively. I presented these data in Amsterdam (2004) but do not have the reference.
Lines 461-463: Track shapes are due to surface energy ratios, the surfaces being different apatite crystal planes in contact with the etchant of some molarity and at some temperature. For a given apatite, we expect surface energy ratios to change – thus changing track shape - with changing acid strength and temperature (and possibly pressure).
Lines 510-511: I have observed huge differences among unidirectional, low angle-of-incidence 252Cf tracks in apatite. The differences are much more than those observed for confined fission tracks from 235U or 238U but keep in mind that confined tracks include the other half of the track that is missing when using 252Cf. It seems likely that 252Cf length correlates with nucleus energy.
Lines 544-546: Thank you for showing this. Thank you also for making available your imagery for posterity. In the current state-of-the-art, measurements for fission track experiments should be performed using such imagery and such imagery should be made available to anyone freely who requests it. The days of “just trust me” should end.
Clarifications Requested
Lines 193-195: Do you mean the standard deviations of c-axis projected fission track lengths? Please clarify.
Lines 237-239: I assume you mean “...shorter projected mean lengths” “...longer projected mean lengths” “...order of projected mean lengths”. Please clarify.
Lines 241-242: It might be useful to cite here Jensen and Hansen (2021) and related comments https://gchron.copernicus.org/preprints/gchron-2021-8/#discussion.
Lines 427-421: It is worth noting here that I measured the Carlson et al. (1999) data using transmitted light only

Citation: https://doi.org/10.5194/gchron-2023-13-RC2
- AC3: 'Reply on RC2', Raymond Jonkheere, 03 Aug 2023
  
  Please find my replies in the attached pdf. Sincerely, R. Jonckheere.
  
  Citation: https://doi.org/10.5194/gchron-2023-13-AC3
AC4: 'Comment on gchron-2023-13', Raymond Jonkheere, 12 Aug 2023

Please find my comments attached as pdf.
Sincerely, Raymond Jonckheere

Citation: https://doi.org/10.5194/gchron-2023-13-AC4

Status: closed

CC1: 'Comment on gchron-2023-13', Richard A. Ketcham, 13 Jul 2023

Comments on “On etching, selection and measurement of confined fission tracks in apatite”, by Raymond Jonckheere
By Richard Ketcham, July 12 2023
This is a very interesting study that builds upon previous innovative work by the author and his student. Utilizing track shape is an extremely promising way to wrest more information out of fission-track data. As the technology of data acquisition and image analysis continues to progress, some shape data will undoubtedly become routinely available with little effort, and can certainly be utilized. That time is not quite here yet, but the author’s group has invested considerable effort in try to build the foundations for what is to come, both observational (Aslanian et al. 2021) and theoretical (Jonckheere et al. 2022). The future of fission-track analysis will certainly include aspects of this work.
That said, there are some critical errors in the present manuscript, which in my opinion make it unsuitable for publication at this time. First, the author attempts to compare his results to the (possibly competing) predictions of the variable along-track etching rate model of Ketcham and Tamer (2021), but rather than using the equations therein attempts to rederive them anew. In doing so, he made a critical error by taking the etch rate at the track tip to be zero, rather than the bulk etch rate. Simply put, his equations do not correctly reflect my model, rendering a number of plots and assertions incorrect. Second, he fails to “remove the log in one’s own eye before removing the mote from his brother’s” in not investigating the limitations of his own data. The wavelength of visible light (0.4-0.7 µm) induces an unavoidable limit to the precision of optical microscopy data, which is not too influential for track length measurements but is a much larger issue for track width, and larger still for measuring angles from which to infer etch rates. Thus far I’m not aware of any attempt to quantitatively investigate the uncertainties of these data, either by mathematical analysis or brute force repeated measurements. This makes it very difficult to critically think through the implications of these data. In addition, it appears that each half-track is measured once, and so it’s unclear how the author would detect a change in along-track etch rate. Third, the author neglects to discuss data that appear plainly contradictory to his assertions. Tamer and Ketcham (2020) present clear indications of changes of track etch rate, such as annealed tracks being longer than non-annealed ones after 10s of etching, as well as the overall sequence of lengthening over a step etch experiment. As discussed in the detailed comments below, the model proposed here makes a number of further predictions about minimum or maximum track etch times that appear well tested and definitively excluded by the Tamer and Ketcham (2020) results. Given that the same individual (M.T. Tamer) produced both data sets, the scientific process suggests that this evident disparity in results at least be discussed, and credible ideas offered on where the problem might be, or what alternatives are available. For example, it may be plausible that the huge variation in track etch rate inferred from these measurements could be from measuring different sections of tracks – arguably a more parsimonious explanation than asserting that etch rates vary from track to track by a factor of 10. Or, if Tamer’s previous data are unreliable, where might that have stemmed from?
The Ketcham and Tamer (2021) variable along-track etching model is very simplified because it is based on mean confined length data from step-etch experiments, and with few data points to fit one can only test a simple model. That said, the constant-core etch rate model preferred by the author was the first one I derived and tested because I thought going into it that it would be the correct answer. It was the exercise of actually trying to fit the Tamer and Ketcham (2020) data (replicated in Tamer and Ketcham 2023, Chem. Geol.) that pointed to the linear model as adequate for most (but not necessarily all) levels of annealing. Shape data promises to be an excellent independent or complementary data source for deciphering track structure amidst the uncertainty stemming from not knowing when any given track starts etching, and it may well allow development of a more detailed and comprehensive model closer to physical reality. The author may be repeating my mistake in believing he knows the answer ahead of time, and thus limiting his field of consideration.
Finally, I note that the acknowledgements state that M.T. Tamer “made a substantial contribution to the measurements but desires not to be listed as a co-author.” I gather that this was because he was being asked to be on a manuscript that both contradicted and ignored his own measurements, putting him in an impossible place. This is a shame, and should be remedied if possible.
Detailed comments:
[line 10] The statement that the widest tracks “must also be the shallowest” is incorrect in two ways. First, width depends heavily on crystallographic orientation, and so one needs to consider this within angular bins of some sort. Second, within a given angular bin, width depends on time of intersection by the etchant channel (i.e. effective etching time). A shallower track is more likely to be intersected early, but given a limited number tracks within a bin, one or more deeper tracks easily could be intersected earlier, and thus be wider.
[line 18] It’s unclear whether the measurements described here actually tested for variation along the track length; only one rate was measured per half-track in most cases.
[line 102-103] Indeed, this is a significant bias. It might be clearer to just compare this to a 16-micron track (4.4 degree dip). It probably only has a small effect on this study, but is not something one would want to do when measuring unknowns.
[Figure 1] Needs scale bar or statement of image width.
[line 120] How do you know if there was a gap if you did not pierce it? It might eb better to state this as an interpretation, rather than an observation.
[line 130] Again, if only one pair of measurements is made for a half-track, it’s not clear whether or how that permits a change in etch rate along the track to be detected. It’s also not clear how reproducible these measurements are with respect to (a) where one places the two circles along the track, and (b) how precise the circle margins are given the limited resolution of these measurements imposed by the wavelength of light. For example, some circles in Figure 1 go out to the edges of the blurry (resolution-limited) track boundaries, while others are set noticeably within those boundaries. How does this affect the rate measurement? Has a multiple-measurement, come-back-to-it-later study been done?
[line 154] It’s unclear what is meant by one “participant” versus the other, and what they both did. Did two people make all of these measurements (thus providing a repeatability analysis it would eb good to report), or did one make all of the measurements and the other just check to make sure they looked OK?
[line 204] Unclear what is meant by “both projections”, and which is the former versus latter.
[line 242] Not really; one must impose additional assumptions concerning monotonicity, or let the uncertainty going back in time propagate to very large values.
[line 245] The reference is an abstract; I guess one can use it to claim that someone once said uncertainties can be taken into account, but it’s not a source of information for how to go about it. In any event, how can there be a single solution that is faithful to the uncertainty in both the length measurement (both from the measurement and from natural variation) and the time intervals (which are certainly not even)? This paragraph seems to drift off-topic.
[line 258-261] A strange statement; you’ve already corroborated that anisotropy is removed well at varying levels of annealing. The lengths measured for a given annealing experiment project to a narrow range, but the means are significantly different at different levels of annealing. The original length and orientations are what the projection is based on – they must matter, or else the distributions of high-angle tracks at each annealing level would not be so narrow.
[line 270-271] An alternative possibility could be that the section designated l₃ may not always be completely etched, due to the delay.
[line 275] It would be worth comparing this result with the recent report by Li et al. (2003, EPSL) of a gap at the center of each track upon formation. It’s worth pondering whether there is really only one gap, and it’s always at site of the original fissioning nucleus.
[line 317-324] Another apparent oddity is that some many tracks at 0-20° and 85-90° apparently need to etch for the full 20 seconds, which should be rare to impossible.
[line 321-324] It’s not clear whether this explanation makes sense; because etching along the penetrating channel is fast, there should be plenty of time for deeper tracks to grow wide and not be affected by the surface. This might be tested with the author’s data – is there a correlation between track depth and width, or track depth and angle? This explanation predicts that tracks at 60-75° should be deeper, on average.
[Figure 5, lines 327-328] The y-axis in these figures is frequency; it’s not clear what they have to do with delta-w. Were the wrong plots put into this figure?
[line 342-343] Do longer tracks attain a greater width before they reach their ends? Figure 5a-c seems to contradict this – tracks are shorter at all angles as annealing progresses, but average widths seem to increase.
[line 366] It seems like Tamer’s step etch data can be used to test some of the implied assertions of the model in Fig. 6a-d. Is it really tenable that tracks at ~70 degrees are not measurable at effective times >13s when you can measure them after a 10s step, with some as long as 13 um (after only 3s of etching)? The model described here predicts that there should be a large deficit of tracks at 60-75° if searching for them at 10s and verifying that they are still present after 20s, but that is not the observation in Tamer and Ketcham (2020). Also, again, are tracks with effective etch times of over 18s realistic given the need to penetrate the polished surface (how deep are these >18s tracks)? This is another case where it could be interesting to check track depth varies with c-axis angle.
[line 432] It seems worth asking whether track rates really vary by a full order of magnitude, or the uncertainty of the rate estimation has something to do with it, or possibly because track etch rate varies and they are trying to measure etch rates along different sections of tracks. For example, a prediction of the Ketcham and Temer (2021) model is that the etch rate for the s₁ track sections should be faster than for the s₂’s; this seems to be the case in Fig. 1g at least. Plotting these against each other seems like an easy test to try.
[Figure 8] This figure demonstrates a mathematical error in how the author has tried to reproduce the model of Ketcham and Tamer (2021). The graph at the top of the figure has the track end at vT=0, rather than vT=vB. The result of this error is seen most obviously in Fig. 8c; the contours indicate that the along-track rate is less than the bulk rate, which contradicts the Ketcham and Tamer model.
[line 450-451] “... except for perhaps 1 µm at either end” seems to admit that track etch rate does vary, as the thinning of the track toward its tip is impossible to miss. Furthermore, although Fig. 8 has mathematical problems, it’s notable that the only place where the variation in etch rate is similarly impossible to miss is in the last µm or so before each tip.
[line 468-469] Is such curvature unobserved, or just unobservable? What curvature there is predicted by the model along the midsection of the track is extremely subtle, and arguably beyond the resolution of optical microscopy, with its diffuse track edges…
[line 478-480] … for example, the white outlines in Fig 9a,b are sometimes on the inside edge of the blurred region, sometimes on the outside edge. One could draw a curved line on the left side of Fig 9b that is a scaled version of 8c.
[line 480] Excess compared to what? I note that in Fig 9e there are several shorter, under-etched tracks at the edge of visibility, some marked with white arrows and some without. What baseline is the author comparing to?
[line 490-492] A bizarre but clarifying assertion. In the Ketcham and Tamer (2021) model, vT approaches vB at the ends, not zero. Only in the author’s attempt to reproduce it does vT approach 0.
[line 554] Accelerated length reduction was posed as (and still is) an intentionally generic, non-interpretive term that encompasses gap formation but leaves open the option for other possibilities.
[line 591-592] And yet such an increase is very clearly present in Tamer’s data (10s experiments in Tamer and Ketcham 2020b, Fig. 1). It’s not terribly scientific to simply ignore the data the contradicts one’s conclusion. Can the author provide at least a hypothesis for the incompatibility between the step etch data and those presented here? Which data are more reliable, confined length or inferences from circles on blurry outlines?
[line 593-595] It’s not clear what an “excess” of confined tracks has to do with the projected lengths of surface-intersecting tracks, the vast majority of which start etching immediately.
[Figure 10, Figure A3, line 510] These figures also show the author’s mathematical error. Calculated track etch times are much too long for the linear model, because the author assumes a vT of zero at the track tip, rather than vB. There is no explanation for how 6s was determined or estimated to be the time required to start etching the first track. And, I have to say, the track segment next to the alpha sure looks like it has curved walls to me, and not just in the last micrometer…
It’s very worth noting that this is a beautiful track-in-track-in-track (TINTINT) image, the first I’m aware of in the literature, and poses an excellent test for the Ketcham and Tamer (2021) variable along-track etching model – can all that etching occur in the time given? I’ve attached a spreadsheet that demonstrates that it can, though not if it really took 6 seconds to start the first etch. If one drops that time to 2 seconds, the model predicts the position of all track tips to within 1 s of the end of etching at 20s. One can also increase the start time a bit by making some other assumptions that are within the measurements. It also bears mentioning that Tamer and Ketcham (2020) may not have measured a track such as beta, because the top tip is very indistinct. In any event, it would be good to know the basis of that 6s determination, short of which I’ll assume for now that my model passed this test.
[Appendix A] Here the author has attempted to derive a simpler set of equations than those listed by Ketcham and Tamer (2021) for their model. There could well be a more elegant formulation than mine (to paraphrase Dutch grandmaster J. van der Weil, “I am a butcher, not an artist”), but these are not yet it owing to assuming that vT=0 at the tip. One obvious place things are wrong is the case marked (1) va = c a , which should be va = c a + vB, and similarly (2) should be va = c(l/2 – a) + vB. How the author calculated c, though not spelled out, was probably also a bit off (should be (vmax – vB)/(l/2)).
[Figure A2] The distances between the dots are not consistent with each other; if one measures them in pixels, there seems to be about a vertical distortion of about 5%, in both Figures A2 and 10, assuming the markers correspond. The measurements are provided on another tab in the attached sheet. It’s not clear if this was because the picture was subtly and inadvertently scaled unevenly at some point before or when it was pasted into this document; since both images are distorted by a similar amount, I’d guess it was at some earlier stage in the process.

Citation: https://doi.org/10.5194/gchron-2023-13-CC1
AC1:
'Comment on gchron-2023-13', Raymond Jonkheere, 26 Jul 2023

Please find my replies to the comments of Dr. Ketcham in the attached pdf.
Sincerely, Raymond Jonckheere

Citation: https://doi.org/10.5194/gchron-2023-13-AC1
- CC2: 'Reply on AC1', Richard A. Ketcham, 09 Aug 2023
  
  Publisher’s note: the content of this comment was adjusted and a new supplement was uploaded on 15 November 2023. This was approved by the GChron editors since some formulations were inappropriate.
  
  Citation: https://doi.org/10.5194/gchron-2023-13-CC2
RC1:
'Comment on gchron-2023-13', Anonymous Referee #1, 01 Aug 2023

This is a very interesting and important paper, in principle suitable for publication in Geochronology. The author examined the confined fission tracks (FTs) in apatite in detail, with a particular focus on the shape and width of etched tracks. He analyzed four samples of Durango apatite annealed at different temperatures, and measured geometrical parameters (length, width, angle to c-axis, etc.) of horizontal confined FTs by utilizing induced FTs in pre-annealed apatite. This is an original investigation with unique geometrical analysis of confined FTs at different stages of annealing, and thus has an important impact on the FT thermochronology.
However, I found following important issues/pitfalls in the present manuscript that should be treated appropriately before publication:
(1) First of all, there is no description about the assessment of uncertainty of individual data in the geometrical analysis, particularly the width of FTs, which is likely the key parameter for reliable confined FT length analysis. I agree with the author’s point of view that the assessment of track width is the key, but then, the author should explicitly describe in the text the uncertainty (i.e., accuracy and precision) of track width measurement. (The author merely gives in Table 1A an “Error” of 0.01 micrometer (= 10 nm) which is amazingly small for optical microscopic observation.) Otherwise, it may result in the overinterpretation of the obtained data within the range of uncertainty, and lead to total misunderstanding of the phenomena. Note that this issue involves the propagation of analytical errors in calculating model parameters, such as track etch rate.
(2) The assumption should be more explicitly documented for calculating the etching rate (and other parameters) from the observed geometrical information (i.e., length and width of a part of confined FTs). The documentation needs to be given in the relevant part of the text, not only giving a series of equations. Otherwise, it may be difficult for readers to follow the logic of the study. For example, the author gives the variation of effective etch time versus angle to c-axis (Fig. 6), calculated from track etch rate values (Fig.7; constant etch velocity is assumed without explicit documentation). Then later in the text, he discusses the validity of assuming constant etch velocity throughout an entire track length (Fig. 8). Such a framework of the paper is just confusing and I suggest reorganizing the text in a more appropriate logical flow. Concerned with this, the author should better document/discriminate between physical theory, experimental observation, model calculation, and interpretation. These appear to be confused/contaminated from each other in places in the current text. This makes it difficult to understand the significance of new findings in the study.
(3) We see many typos of the experimental parameters in the text and figures that are similar to each other. This makes it further difficult to read the paper correctly.

Because of these issues, I judge that it is not appropriate to accept the paper at its present form of data presentation and interpretation. Therefore, I regret to suggest rejecting the paper, with a strong encouragement for the author to resubmit the material as a more carefully reorganized and rewritten manuscript.

Citation: https://doi.org/10.5194/gchron-2023-13-RC1
- AC2: 'Reply on RC1', Raymond Jonkheere, 03 Aug 2023
  
  Please find my replies in the attached pdf. Sincerely, R. Jonckheere.
  
  Citation: https://doi.org/10.5194/gchron-2023-13-AC2
RC2:
'Comment on gchron-2023-13', Raymond Donelick, 01 Aug 2023

General Comments
It is an honor and pleasure to review this excellent paper. Well done Dr Jonckheere! You have quantified the mental picture in my head of the many nuances of confined fission track lengths in apatite.
I highly recommend this paper be accepted for publication and I leave it to the author to choose to make the minor clarifications I request below.
I have not read the comments that have been posted for this paper (as of 31 July 2023) and may not do so.
Specific Comments
Introduction: The biases discussed do not include mention of other decisions facing an analyst such as: 1) Is that a naturally etched fission track? Sample TI of Carlson et al. (1999) exhibits many naturally etched fission tracks near natural grain surfaces. 2) Is that feature a fission track at all or some outlier that should be removed from the dataset? Like potentially naturally etched fission tracks, some potentially non-fission track features can etch like fission tracks but be too long (I use 19 microns as a cutoff for low-temperature natural and laboratory samples). Or perhaps an etched feature looks like an etched fission track but it is too short for the current fission track population(s) under study. Human analysts are smart enough to make these decisions and these decisions should be done openly.
Lines 46-50: I can never produce such a plot as discussed here. The notion of measuring sufficiently-etched fission tracks (apples) and contaminating those data by also measuring under-etched fission tracks (oranges) is not appealing to me. My plots would instead have no data (shorter etch times), then very little data, then sufficient data (longer etch times). I do not mix apples and oranges when it comes to measuring confined fission tracks in apatite.
Equations: It is these equations that can be used to show that non-elliptical, polar coordinate plots (length, angle to c-axis) of horizontal confined fission track lengths from a single population represent a mixture of apples (sufficiently etched tracks defining the ellipse, usually at higher angles to c-axis) and oranges (under-etched tracks falling short of the ellipse, usually at lower angles to c-axis).
Lines 172-177: Thank you for reproducing my work and that of Dr. Bill Carlson. The agreement between our works is truly independent and it is not accidental, but instead demonstrates that we as analysts can behave like machines with the ability to maintain biases within ourselves and share those biases with other machines.
Lines 184-187: My first submission of Donelick (1991) was to EPSL and a reviewer rejected the paper on the basis that "they [polar-coordinate plots of fission track lengths] are not ellipses", because his/her published mixtures of apples and oranges did not match my apples-only data.
Lines 199-209: Excellent discussion of this interesting issue. As we are after the mean and the mean ranges vary accordingly, it seems it should be possible to prove mathematically this is precisely how it should be.
Lines 248-261: In Donelick et al. (1999) I proposed a surface energy model for fission track annealing in apatite. In this model, converging track sides (during annealing) parallel to c-axis are flat and perpendicular to c-axis are rough with pyramidal faces. In this model, at high angles to c-axis and high degrees of annealing you get the results of Donelick et al (1999) – systematic accelerated length reductions - with the occasional Green et al. (1986) – segmentation where one or more opposing and ‘rouge’ pyramids intersect – along with the tips pinching off and appearing to diffuse into the bulk crystal as in Paul and Fitzgerald (1992). Surface energy minimization explains anisotropy and all experimental observations.
Lines 270-271: This statement is totally consistent with the surface energy minimization model above.
Lines 346-348: I hazard to say that only analysts willing to mix apples and oranges have this problem as your (and my) experiments demonstrate. This is especially true for samples with low confined fission track densities such as highly annealed experiments or the many, many, many low-fission-track-density natural samples studied without using 252Cf/particle accelerator ion implantation.
Line 424: I see this minor but significant effect for la0 and lc0 and a correlation between these and Dpar and Dper, respectively. I presented these data in Amsterdam (2004) but do not have the reference.
Lines 461-463: Track shapes are due to surface energy ratios, the surfaces being different apatite crystal planes in contact with the etchant of some molarity and at some temperature. For a given apatite, we expect surface energy ratios to change – thus changing track shape - with changing acid strength and temperature (and possibly pressure).
Lines 510-511: I have observed huge differences among unidirectional, low angle-of-incidence 252Cf tracks in apatite. The differences are much more than those observed for confined fission tracks from 235U or 238U but keep in mind that confined tracks include the other half of the track that is missing when using 252Cf. It seems likely that 252Cf length correlates with nucleus energy.
Lines 544-546: Thank you for showing this. Thank you also for making available your imagery for posterity. In the current state-of-the-art, measurements for fission track experiments should be performed using such imagery and such imagery should be made available to anyone freely who requests it. The days of “just trust me” should end.
Clarifications Requested
Lines 193-195: Do you mean the standard deviations of c-axis projected fission track lengths? Please clarify.
Lines 237-239: I assume you mean “...shorter projected mean lengths” “...longer projected mean lengths” “...order of projected mean lengths”. Please clarify.
Lines 241-242: It might be useful to cite here Jensen and Hansen (2021) and related comments https://gchron.copernicus.org/preprints/gchron-2021-8/#discussion.
Lines 427-421: It is worth noting here that I measured the Carlson et al. (1999) data using transmitted light only

Citation: https://doi.org/10.5194/gchron-2023-13-RC2
- AC3: 'Reply on RC2', Raymond Jonkheere, 03 Aug 2023
  
  Please find my replies in the attached pdf. Sincerely, R. Jonckheere.
  
  Citation: https://doi.org/10.5194/gchron-2023-13-AC3
AC4: 'Comment on gchron-2023-13', Raymond Jonkheere, 12 Aug 2023

Please find my comments attached as pdf.
Sincerely, Raymond Jonckheere

Citation: https://doi.org/10.5194/gchron-2023-13-AC4

Raymond Charles Jonckheere

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Raymond Charles Jonckheere

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Short summary

We investigate the relationship between damage trails from uranium fission in apatite and the etched tracks measured with an optical microscope. We aim to understand sampling biases and to improve etching, selection and measurement protocols. The track orientation and length, apatite and track etch rate, and effective etch time co-determine whether a track is suitable for measurement. Reliable data are attainable because the specific sample properties have but a limited effect on these factors.


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On etching, selection and measurement of confined fission tracks in apatite

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