A revised alpha-ejection correction calculation for (U-Th)/He thermochronology dates of broken apatite crystals

. Accurate corrections for the effects of alpha ejection (the loss of daughter He near grain or crystal surfaces due to long alpha-stopping distances) is central to (U-Th)/He thermochronometry. In the case of apatite (U-Th)/He dating, alpha-ejection correction is complicated by the fact that crystals are often broken perpendicular to the c-axis. In such cases the correction should account for the fact that only some parts of the crystal are affected by alpha-ejection. A common current 10 practice to account for such broken crystals is to modify measured lengths of broken crystals missing one termination by a factor of 1.5, and those missing both terminations by a factor of 2. This alpha-ejection “correction correction” systematically overestimates the actual fraction of helium lost to alpha ejection, and thus overcorrects the measured date relative to that determined for an otherwise equivalent unbroken crystal. The alpha-ejection-affected surface-area-to-volume ratio of a fragmented crystal is equivalent to the surface-area-to-volume ratio of an unbroken crystal twice as long (for fragments with 15 one termination), and equivalent to that of an unbroken crystal infinitely long (for fragments with no termination). We suggest it is appropriate to revise the fragmentation correction to multiply the length of crystals missing one c-axis termination by 2, and those missing both c-axis termination by some large number >~20, respectively. We examine the effect of this revised correction and demonstrate the accuracy of the new method using synthetic datasets. Taking into account alpha-ejection, rounding of the He concentration profile due to diffusive loss, and accumulation of radiation damage 20 over a range of thermal histories, we show that the revised fragmentation alpha-ejection correction proposed here accurately approximates the corrected date of an unbroken crystal (“true” date) to within < 0.7% on average (±4.2%, 1 σ ), whereas the former method overcorrects dates to be ~3% older than the “true” date, on average. For individual grains, the former method can result in older dates by a few percent in most cases, and by as much as 12% for grains with aspect ratio of up to 1:1. The revised alpha-ejection correction proposed here is both more accurate and more precise than the previous correction, and 25 does not introduce any significant systematic bias to the apparent dates from a sample.


Introduction
Since the development of modern apatite (U-Th)/He thermochronometry, the technique has become a versatile and powerful tool for a range of geological problems (Zeitler et al. 1987;Farley et al., 1996;Flowers et al., in press a, b).To fully leverage the power of the technique, however, it is necessary both to account for the wide range of possible complications that 30 commonly cause data dispersion greater than analytical errors.Of particular significance is correcting for the loss of daughter nuclides due to the problem of alpha ejection.Apatite He dating uses the accumulation of daughter nuclide 4 He (i.e., alpha particles) from the spontaneous alpha decay of 238 U, 235 U, and 232 Th (as well as a minor contribution from 147 Sm) to constrain possible thermal histories of samples, which is sometimes simplified as providing a date of cooling through some closure temperature (~30-90 °C) at which helium diffusion out of a crystal is sufficiently slow for the system to be 35 considered closed.However, this method is complicated by the fact that the sizes of most typical apatite crystals are only https://doi.org/10.5194/gchron-2022-11Preprint.Discussion started: 2 May 2022 c Author(s) 2022.CC BY 4.0 License.
several times greater than the stopping distance of alpha particles (~20 µm), meaning that the fraction of 4 He ejected from a crystal must be accounted for in most applications (Farley et al., 1996).
Careful measurement of crystal geometries allows accurate approximation of the cumulative alpha-ejection loss of helium from a crystal (Ziegler;1977;Farley et al., 1996;Farley, 2002;Hourigan et al. 2005;Ketcham et al. 2011;Reiners et al., 40 2018).Because the likelihood of an alpha particle being ejected from a crystal is directly related to a parent nuclide's proximity to the crystal surface, the fraction of helium retained in the crystal (F T ) is a function of a crystal's surface area to volume ratio (β) (Farley et al., 1996).In the simplest case of a spherical grain with homogenous parent nuclide distribution, F T is a cubic polynomial function of β (Farley et al., 1996).F T can be estimated for other geometries using a polynomial function calibrated by Monte Carlo alpha-ejection models (Farley, 2002;Ketcham et al, 2011).In practice, a parent-nuclide-45 specific  " # is determined and a corrected date can be calculated by incorporating it into the full decay equation: -./  0 123 4 − 1 + 7( )( " -.9 -.9 )  0 12: 4 − 1 + 6( ℎ)( " -.--.-)  0 121 4 − 1 + ( )( " @&A ) @&A  0 BCD 4 − 1 where t is the unknown variable that must be solved numerically or iteratively; 4 He, 238 U, 235 U, 232 Th, and 147 Sm contents are measured;  # is the decay constant for the given isotope;  " # , the alpha-ejection correction factor for the given isotope calculated from crystal geometry (Ketcham et al., 2011).A more approximate corrected date can also be calculated by simply dividing the measured (raw) date by F T (Farley and Stockli, 2002), though this is less accurate for older dates.
These calculations generally assume the ideal case of a euhedral, prismatic crystal with homogenous parent nuclide 55 distribution, entire original crystal faces, and insignificant parent nuclide concentrations outside and within one alphastopping distance of the exterior of the crystal during the interval in which temperature was low enough to accumulate He.When these assumptions are violated, further adjustments to the standard F T correction are required.
If information about the magnitude and pattern of parent-nuclide zonation is available, an adjusted F T may be applied to account for inhomogeneous parent-nuclide distribution (Hourigan et al., 2005, Farley et al., 2011, Ault and Flowers, 2012, 60 Gautheron et al., 2012).Absent such information, as is the case in most routine analyses, use of the standard unzoned correction assuming homogenous distribution of the parent nuclide would introduce errors that skew a crystal's apparent date to be younger, if its rim is enriched in parent nuclides, and older if it is depleted in parent nuclides (Farley et al., 1996;Hourigan et al., 2005).For apatite, these errors are usually minor (<1.5% for 80% of apatite crystals, and <9.5% for 95%), because apatite crystals in most cases do not typically exhibit extreme zonation of parent nuclides (Ault and Flowers, 2011).65 Furthermore, the errors are usually symmetrically distributed, with apatite populations not exhibiting bias towards either rimenriched or rim-depleted grains (Ault and Flowers, 2011).Accounting for effects of He implantation from sources external to the grains is not typically possible for grains separated from their petrographic context, although in some cases particular date-eU or date-size correlations may be used to interpret such effects (Spiegel et al., 2009;Murray et al., 2014).
The focus of this paper is the adjustment to the F T correction that should be made in the case of crystals that are broken 70 perpendicular to the c-axis, as common for apatite.If errors due to fragmentation are large, they can significantly impede our ability to extract geologically meaningful information from dates calculated from parent-daughter nuclide ratios.Broken and suboptimal crystals are frequently analyzed, particularly when the quality of mineral separates is poor and/or the apatite yield from a sample is low.In addition, imperfect basal (0001) cleavage in apatite (Dana, 1963;Palache et al., 1963) leads to the fact that many dated crystals are broken perpendicular to their c-axis and lack original terminations, even for high-quality 75 https://doi.org/10.5194/gchron-2022-11Preprint.Discussion started: 2 May 2022 c Author(s) 2022.CC BY 4.0 License.
samples.Assuming that fragmentation occurred recently relative the date measured (e.g. during mineral separation, or in the case of detrital samples, during recent transport), a common strategy is to apply a fragmentation correction to the F T calculation, which accounts for the fact that the fracture exposes surface area where alpha ejection did not occur (Farley, 2002).This correction seeks to approximately correct for the originally greater length of the unbroken apatite crystal, by multiplying the length of all broken crystals by 1.5 (if one end is broken) or 2 (if both ends are broken) (Farley, 2002;Farley 80 et al., 1996;Brown et al., 2013;Beucher et al., 2013;Reiners et al., 2018).Though it is not possible to find the original length of broken crystals, Farley (2002) argued that these approximations are sufficient because F T is relatively insensitive to the length of the crystal.
An alternative approach to this problem is that of Brown et al. (2013), who argued that, for interpreting thermal histories, it is best to leave dates uncorrected and instead evaluate the variation in date among crystals with different morphologies and 85 numbers of broken ends.If one assumes that breakage occurred prior to cooling to temperatures of partial He retention, raw (uncorrected) dates of broken crystals can vary by up to 60% for certain t-T histories, and that for sufficiently large datasets of fragmented crystals, considering the patterns of dispersion in uncorrected dates can constrain thermal history (Brown et al. 2013).In practice, however, F T -corrected dates remain widely reported, partly because correcting for alpha ejection and fragmentation is necessary to compare dates to other datasets and dates of geologic significance.A more accurate correction 90 would allow both broken and unbroken crystals within a sample and across samples to be appropriately compared without introducing additional systematic bias.
Although the conventional fragmentation correction has been widely applied since the widespread application of the technique, its accuracy and precision has not been demonstrated.In the first part of this paper, we consider the rationale behind the early approach, then propose a revision and compare the results of both methods.We then test the new method 95 using synthetic data and demonstrate the accuracy of the revised correction.We take into account a range of broken crystal sizes, number of terminations present, and various thermal histories and their associated effects on helium diffusivity, and we quantify the uncertainty that can be attributed to the fragmentation correction alone.Considering the numerous natural sources of uncertainty in apatite He dating, achieving greater confidence in the accuracy of the fragmentation F T correction and minimizing its uncertainty ultimately aids in the interpretation of other possible sources of uncertainty and errors (He et 100 al., 2021).

Revision of F T correction for broken crystals
For an idealized spherical grain, the alpha-ejection correction is a function of the radius of the sphere (R) and the alpha stopping distance for the given parent nuclide (S): 105 [Eq. 2] (Farley et al., 1996).Where R>>S, the function approaches a linear relationship: [Eq. 3] 110 (Farley et al., 1996), where β is the ratio of surface area of a crystal to its volume.In other words, the fraction of helium lost due to alpha ejection near the crystal surface is approximately a function of the ratio of surface area of a crystal to its https://doi.org/10.5194/gchron-2022-11Preprint.Discussion started: 2 May 2022 c Author(s) 2022.CC BY 4.0 License.
volume.Considering more realistic crystal geometries, polynomial equations that define the  " # value as a function of β have been empirically determined using Monte Carlo simulations for each parent nuclide i and their respective alpha stopping distance (Farley et al., 1996;Hourigan et al., 2005).For hexagonal prisms, simply measuring the length (L) and radius or 115 half-width of the cylindrical prism (R) allows the computation of β: The general idea behind modified F T corrections is to modify β, under the assumption that the polynomial functions relating β and F T are nearly identical for similar geometries (e.g., hexagonal prism with bipyramidal or pinacoidal terminations).This 120 was the approach taken to correct for lost crystal surface in the case of crystals polished parallel to the c-axis (Reiners et al., 2007).In the case of c-axis perpendicular breakage, the Farley et al. (1996) approach sought to establish the length of the original, unbroken crystal.Because it was observed that the corrected-length-to-radius ratios of most apatite crystals (5:1) were sufficiently high such that the F T corrections become largely independent of length, it became standard practice at most laboratories to simply modify β by multiplying the lengths of broken crystals by arbitrary factors of 1.5 or 2, thereby 125 modifying F T (Farley et al. 1996;Farley, 2002).Alternatively, in the slightly different context of inverse modelling a large set of uncorrected ages, Beucher et al. (2013) suggested that a rule of thumb for predicting the unknown initial length should be to add the maximum fragment length of a set of fragments and two times the maximum radius.To the first order, guessing the unknown initial length using consistent but arbitrary factors such as these suffices to roughly account for the loss of alpha-ejection-affected surface area at the tips: this is because as L increases, the increase in surface area of a crystal 130 is less than that of the volume, in effect reducing β.
In detail, however, the fraction of helium remaining in a fragmented crystal does not depend on the unknown (and precisely unknowable) initial length.Rather, it should be directly related to the surface area of a broken crystal that was originally affected by alpha-ejection.Assuming that we can identify when crystals have lost one or both terminations, that the breakage generally occurs more than one average-alpha-stopping-distance from the tip, and that diffusion has not significantly 135 modified the daughter concentration profiles, the alpha-ejection-affected-surface-area-to-volume ratio (β α ) could be simply calculated by measuring and then subtracting the surface area of the broken face(s) from the total surface area.For a hexagonal prism, simple geometric calculations demonstrate that the β α of singly and doubly broken crystals are equivalent to the β of an unbroken crystal of the same width that is twice as long, or infinitely long, respectively; this is shown graphically in Fig. 1.Consider that the helium profile in an unbroken crystal is symmetrical such that when broken in half, 140 each half will have the same fraction of helium remaining (F T ); thus, conversely, any broken crystal with one termination (breakage occurring more than one alpha-stopping distance from the tip) has the same F T as a hypothetical unbroken crystal double its length (but not the same as the original unbroken crystal).For crystals with no terminations remaining, any c-axis perpendicular segment or cross section of the crystal will have the same F T , no matter its length or its position along the fragment; therefore fragments with no terminations have the same F T as an infinitely long unbroken crystal, where the 145 terminations, which have a different F T , have a vanishingly small effect on the overall F T of the crystal.155 It follows, then, that a more accurate fragmentation correction that explicitly considers the lost surface area of a broken crystal should be to multiply the length of a broken crystal by 2 or some large number (to simulate the limit as L approaches infinity), respectively, rather than 1.5 or 2, i.e.: for crystals broken on one end; and 160 for crystals broken on both ends where L is the measurement of the crystal dimension perpendicular to the fracture (since crystals commonly break perpendicular to the c-axis in apatite crystals, L should usually be measured parallel to the c-axis, even if it is shorter than the width).This simple correction has the benefit of applying to other geometries, regardless of the shape of its body (e.g.165 cylindrical, tetragonal, or hexagonal) or the shape of its terminations (e.g.pyramidal, flat, or rounded): the β α of any singly broken crystal is equivalent to the β of a whole crystal twice its length, and the β α of any doubly broken crystal is equivalent to the β of a whole crystal infinitely long.The correction can thus be applied to all crystals broken perpendicular to the caxis, regardless of original length, requiring knowledge of only the width and length of the broken crystal, and the number of terminations present.We emphasize that though this fragmentation correction is similar in form to that of Farley (2002) in 170 that it involves length-modifying factors, it differs in that it seeks to approximate the length of a whole crystal with the same fraction of helium remaining as the fraction remaining in the broken fragment, rather than seeking to approximate the unknown length of the original unbroken crystal.

Accuracy of the revised F T correction
The new protocol accurately corrects for the effect of fragmentation, deviating by 0.0% ± 1.4% (1σ) from the corrected date 185 of the unbroken crystal, under the ideal assumptions stated above: that the fragmentation has occurred more than one average-alpha-stopping-distance from the tip, and that there is no significant diffusion-induced modification of the helium concentration profile (Fig. 2a).By comparison, under the same assumptions, the old protocol leads to corrected dates that are almost all too old, on average by 2.6% ± 2.7%.Characterization of the uncertainty for both protocols is based on the application of the protocols to a synthetic dataset of raw (uncorrected) dates of broken prismatic crystals where the corrected 190 dates of the original unbroken crystals are known.Note that as a simplification, a length-modifying factor of 20 is used to approximate the limit to infinity (see Section 4.1).For comparable results, we used the same datasets of raw uncorrected dates from Brown et al. (2013), which are generated from the volume-integrated 4 He concentration in a random set of crystal fragments broken at varying positions along the original crystal (Beucher et al. 2013).These original unbroken crystals from which the fragments were generated have a constant geometry (hexagonal prism that is 400 µm long and 150 µm wide).To 195 test the accuracy of the proposed protocol under the stated ideal assumptions, we excluded any randomly generated crystals that are <20 µm from the tip, and included only fragment sets that experienced the two thermal histories associated with the least amount of diffusive modification of the helium profile.We assume uniform spatial distribution of the parent nuclide, and apply both protocols to all fragments as we would in routine laboratory analyses: i.e. we assume no knowledge of the original length and thermal history of the crystals to compute the corrected age.Only the length and width of the broken 200 crystals, and the number of terminations present, are used for the calculation.
Though the new F T correction for broken crystals is more accurate as a whole than the old correction, the two ideal assumptions of the simple geometric argument above introduce additional uncertainty.In a more realistic scenario, when the two assumptions are relaxed, the proposed fragmentation correction results in a broader range of uncertainty (+0.7% ± 4.2%), but it is nevertheless more accurate and more precise than the old protocol (+2.9% ± 5.0%) (Fig. 2b).Using the new 205 protocol, only 3% of corrected dates deviate from the corrected date of the unbroken crystal by greater than 10%; this represents a 66% reduction relative to the prior protocol.These results are based on the full fragment dataset from Brown et al., 2013, which includes a representative range of thermal histories that are more complicated than simple rapid cooling (i.e.slow, monotonic cooling; prolonged isothermal residence in the partial retention zone followed by rapid cooling; a mix of slow cooling and isothermal holding in the partial retention zone; and gradual reheating (e.g.burial) followed by rapid 210 cooling; cf.Wolf et al. 1998).23% of the randomly generated fragments in that dataset were broken within ~20µm from the tip, accounting for the possibility that in real laboratory analyses, it is not always discernible whether an apatite crystal was broken more than an alpha-stopping distance from the tip.Thus, our application of the two F T corrections to the full dataset in Fig. 2b approximates the actual circumstances under which broken crystals are analyzed.

Difference between corrections for different crystal dimensions
Though multiplying a fragment length by different factors may seem to be a minor revision, the resulting difference in corrected dates is not negligible.This is partly because the presumption that F T corrections does not strongly depend on length, upon which the fragmentation correction was initially based, does not hold true for the smaller crystals commonly 235 analyzed today (e.g., c-axis perpendicular width < 150µm).Fig. 3 shows the effect of crystal length (or modified crystal length, as used in the calculation of β or β α ) on the inverse value of F T , an approximation for the correction's effect on the final reported date (Fig. 3).Since both protocols effectively multiply the length of a broken crystal to compute an adjusted F T value, the inverse F T values of the new and old protocols for any given crystal width all lie on the same curve for F T as a function of length (normalized to width).Particularly when the length of a broken crystal is close to its width, and when the 240 width is small, the F T correction is not independent of the modified length.For example, for a singly-broken crystal that is 60 µm in width and equally long (the minimum dimensions of crystals routinely analyzed in our lab), the difference between the new and old protocols, would be 4%; for a doubly-broken crystal of the same dimensions, the difference would be 12% (Fig. 4).The overcorrection of the previous protocol could be even larger for drum-shaped fragments (i.e.crystals broken on both ends, and shorter in c-axis-parallel length than width).For a broken crystal that is 140 µm in both width and length, the 245 difference would be 2% and 5% (for singly and doubly-broken crystals, respectively).The magnitude of these differences is not negligible, at least relative to other sources of error in F T corrections.By comparison, for example, the updated alphaejection models of Ketcham et al. (2011) based on revised alpha-stopping distances affects dates by approximately 1-5%, and 2D measurement of crystal geometry introduces errors of ~2% (Cooperdock et al., 2019).

Uncertainty in fragmentation correction compared to other sources of date dispersion
The larger uncertainties in the fragmentation correction with the two ideal assumptions relaxed (Fig. 2b) are largely due to the diffusive modification of helium profiles.In our test of this protocol, all cases of corrected fragment dates that deviate by more than >5% from the corrected date of the unbroken crystal can be attributable to thermal histories involving prolonged 260 residence in the partial retention zone (Fig. 5).Without a priori knowledge of a sample's thermal history, this is a problem for the new fragmentation correction just as it is for the old protocol, because the calculation of F T correction only assumes loss of helium due to alpha ejection.The additional uncertainty associated with the fragmentation correction fundamentally relates to the fact that using β α to correct F T implies taking the lost surface area ("skin") affected by alpha ejection as a proxy for the lost volume (the outer "shell") of the crystal affected by alpha ejection.265

270
We emphasize that because the F T -corrected dates of the fragments are compared to the F T -corrected date of a whole crystal, Fig. 2 and Fig. 5 assesses only the effect of brokenness correction alone.Zonations, eU variation, and diffusive helium loss remain important sources of additional error and dispersion (e.g.Meesters and Dunai, 2002;Herman et al., 2007, Gautheron et al 2012, Brown et al., 2013;Beucher et al., 2013).Notwithstanding the effects of date variation due to all these effects, the 275 fragmentation correction proposed in this paper more consistently and accurately reproduces the F T -corrected date of unbroken crystal.An illustrative case is that of a hypothetical date-elevation transect from a crustal block that cooled slowly through the partial retention zone until some point in time, then subsequently experienced very rapid cooling (from Brown et al., 2013).A key observation of Brown et al. (2013) was that the large dispersion of raw uncorrected fragment dates is due to the fact that these dates can be both younger and older than the whole crystal, and that fragments of same length can yield 280 different dates, while conversely, fragments of different lengths can yield the same date.The dispersion is compounded because slow cooling leads to significant diffusive modification of the helium profile in a crystal.Despite this large dispersion of uncorrected fragment dates (up to 60%), and despite variations in eU and grain sizes, applying the new fragmentation correction introduces limited uncertainty relative to the dispersion caused by other effects (Fig. 6).This facilitates interpretation of widely-dispersed data by reducing the number of variables that must be considered, and 285 demonstrates the utility of applying a fragmentation correction when analysis of the pattern of dispersion in >20-30 crystals is not practical.Finally, while both the new and old F T correction for broken crystals reliably approximates the corrected https://doi.org/10.5194/gchron-2022-11Preprint.Discussion started: 2 May 2022 Author(s) 2022.CC BY 4.0 License.
date of an unbroken crystal for a range of eU and crystal sizes, the new correction reduces the systematic bias that is introduced by the old protocol when many broken crystals are analyzed in a sample by ~3-4% (Fig. 6).

300
Finally, the uncertainty due to the application of the fragmentation correction can be propagated with other sources of error (analytical, zonation, etc.), for appropriate comparison of fragment dates and other corrected dates where the correction was not applied.For larger datasets, especially in the case of larger-n analyses, this contributes to our understanding of the expected distribution of corrected dates from any given sample (Fig. 7) (He et al., 2021).Previous work has shown, for 305 example, that (1) size-and eU-dependent diffusivity can cause apparent dates to systematically vary (Reiners and Farley, 2001;Flowers et al. 2009;Whipp et al., 2022).
( contribute helium to the grain but are not in the analyzed aliquot) (Spiegel et al. 2009;Fitzgerald et al. 2006;Murray et al. 2014).
(3) Non-uniform distribution of the parent nuclides can cause dates to be both older or younger by a few percent in most cases.Even if zonations are not accounted for, the probability distribution of errors due to zonations can 315 be approximated by either examining a representative selection of apatite in a sample, or using a reference compilation (e.g.Ault and Flowers, 2011).
(4) Technician-to-technician differences in 2D grain measurement cause date variations that differ from actual 3D geometry by ~2%.(Cooperdock et al., 2019) 320  A future step towards a more rigorous evaluation of the uncertainty of individual-grain analyses or samples as a whole could involve the propagation of each of these uncertainties to account for sample and grain specific information such as the probability of implantation (approximated from the spatial distribution of heavy minerals in a sample, via XRCT), the probability of extreme parent nuclide zonations in a sample (based on fission track mounts) or from a reference compilation, and whether fragments were analyzed.335

Conclusion
Despite the dispersion of raw U-Th/He dates due to fragmentation, it is possible to accurately correct for the effect of fragmentation based on basic measurements routinely recorded during the grain selection process.In compensating for the effects of alpha ejection in broken crystals, the F T correction should be calculated by explicitly taking into account the surface area of the broken face, rather than by assuming the unknown length of the original unbroken crystal.In individual 340 cases, especially crystals with smaller width or whose length is less than or around the same as the width, the difference in apparent dates calculated with the two methods can be 12% or greater.
We further applied both the previous and newly proposed protocol for correction of broken crystals to a synthetic dataset.
Even taking into account the effects of diffusive loss of helium and breakage close to tips of crystals, the proposed protocol more accurately and more precisely approximates the F T -corrected date of an unbroken crystal for a range of complex and 345 simple thermal histories.For a crystal of 150um width, the old calculation leads to apparent dates that are on average 3% older than the corrected dates of unbroken crystals, and in certain cases, up to 20%.
The proposed adjustment allows more accurate comparison of data between samples of varying quality, which is common when a mix of different rock types is sampled.The greatest effect will be for samples where the majority of crystals are broken.Though this adjustment is minor in many cases, when applied to entire datasets, it significantly reduces one common 350 source of error in calculations of individual apparent dates, and removes an easily correctable source of systematic bias towards older dates.
Fig. 1.The alpha-ejection-affected surface-area-to-volume ratio, or β α , of a broken crystal that has lost a basal fragment longer than one alpha-stopping distance is equivalent to the surface-area-to-volume ratio of an unbroken crystal twice as long (for 150

Fig. 3 .
Fig. 3.The inverse of F T value plotted as a function of crystal length L, as used in the calculation of β or β α .For a broken crystal, this is measured parallel to the c-axis (as a multiple of width), and modified by some factor (see legend).The inverse of F T can be 220

Fig. 4 .
Fig. 4. Difference between the corrected date calculated from the standard protocol and revised protocol (Dashed-broken on both ends; Solid-broken on one end), shown for a range width-to-length ratios commonly seen in broken crystals. 255

Fig. 5 .
Fig. 5. Uncertainty associated with the proposed fragmentation correction due to inclusion of fragments broken too close to the tip (dashed) and due to thermal histories that involve significant diffusive modification of the helium profile (dotted).The histogram and curves are stacked and cumulative, such that the dashed probability curve includes both close-to-tip fragments as well as fragments with significant diffusive loss.

290Fig. 6 .
Fig. 6. a) Date-elevation transect of a crustal block that cooled slowly through the partially retention zone, and was subsequently rapidly exhumed (cf.Brown et al. 2013, Fig. 9), showing fragment dates corrected using the new protocol (red), compared to previous protocol (blue), and the expected whole-crystal date (black).Note that the red and black triangles nearly overlap in all cases.Corrected dates are from fragments of varying lengths, both singly and doubly broken, generated from crystals with 295

Fig. 7 .
Fig. 7. Schematic illustration of how uncertainties due to different complications in apatite helium dating, e.g.uncertainty due to zonation (Ault and Flowers, 2011)(a), uncertainty due correction for broken crystals, this study (b), and error due to implantation and other effects (c) could together be combined to form a theoretical apparent date distribution (d) that would inform our interpretation of real data and the choice of the appropriate summary statistic (whether the minimum, mean, median, peak date) 325 330 https://doi.org/10.5194/gchron-2022-11Preprint.Discussion started: 2 May 2022 c Author(s) 2022.CC BY 4.0 License.
) The presence of extraneous daughter nuclides whose parent nuclides are not accounted for can cause outlier dates many multiples older than the true date ( 4 He-rich fluid inclusions; U-or Th-rich inclusions or 310 microinclusions where the inclusions are not fully dissolved; grain boundary phases or adjacent grains that https://doi.org/10.5194/gchron-2022-11Preprint.Discussion started: 2 May 2022 c Author(s) 2022.CC BY 4.0 License.