the Creative Commons Attribution 4.0 License.
the Creative Commons Attribution 4.0 License.
Resolving the effects of 2D versus 3D grain measurements on apatite (U–Th) ∕ He age data and reproducibility
Richard A. Ketcham
Daniel F. Stockli
(U–Th) ∕ He thermochronometry relies on the accurate and precise quantification of individual grain volume and surface area, which are used to calculate mass, alpha ejection (F_{T}) correction, equivalent sphere radius (ESR), and ultimately isotope concentrations and age. The vast majority of studies use 2D or 3D microscope dimension measurements and an idealized grain shape to calculate these parameters, and a longstanding question is how much uncertainty these assumptions contribute to observed intrasample age dispersion and accuracy. Here we compare the results for volume, surface area, grain mass, ESR, and F_{T} correction derived from 2D microscope and 3D Xray computed tomography (CT) length and width data for > 100 apatite grains. We analyzed apatite grains from two samples that exhibited a variety of crystal habits, some with inclusions. We also present 83 new apatite (U–Th) ∕ He ages to assess the influence of 2D versus 3D F_{T} correction on sample age precision and effective uranium (eU). The data illustrate that the 2D approach systematically overestimates grain volumes and surface areas by 20 %–25 %, impacting the estimates for mass, eU, and ESR – important parameters with implications for interpreting age scatter and inverse modeling. F_{T} factors calculated from 2D and 3D measurements differ by ∼2 %. This variation, however, has effectively no impact on reducing intrasample age reproducibility, even on small aliquot samples (e.g., four grains). We also present a grainmounting procedure for Xray CT scanning that can allow hundreds of grains to be scanned in a single session and new software capabilities for 3D F_{T} and F_{T}based ESR calculations that are robust for relatively lowresolution CT data, which together enable efficient and costeffective CTbased characterization.
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(U–Th) ∕ He thermochronometry of accessory phases, such as apatite and zircon, has been widely applied to study tectonic, volcanic, and surface processes (e.g., Zeitler et al., 1987; Stockli et al., 2000; Ehlers and Farley, 2003; Reiners and Brandon, 2006). The method is based on the radiogenic accumulation of He from the alpha decay of U, Th, and Sm isotopes and the diffusive loss of He via thermal processes. In addition, He is lost due to the “long alpha stopping distances” associated with the kinetic energy of alpha decay (∼5 MeV), requiring a shapebased alpha ejection correction (F_{T} correction) (Farley et al., 1996). This correction as traditionally applied includes several simplifications and assumptions, such as an idealized grain geometry and homogenous parent nuclide concentrations (Farley et al., 1996; Farley, 2002; Ketcham et al., 2011). It has been shown that due to uncertainties in grain geometry, stopping distances, and parent nuclide zonation and variability, this correction can contribute > 50 % of the total analytical uncertainty (Farley and Stockli, 2002). Similarly, low error, highly dispersed apatite (U–Th) ∕ He ages are problematic for robust interpretation and time–temperature modeling (e.g., Fox et al., 2019). The observation that the scatter of measured ages in even wellunderstood samples exceeds expectation based on analytical errors, combined with the knowledge that the above simplifications will not always hold, has led to the practice of reporting errors derived from the reproducibility of standards rather than propagated analytical uncertainties in He dating. While the effect and mitigation of parent nuclide zonation in apatite and zircon to improve the accuracy and precision of (U–Th) ∕ He ages have been studied (e.g., Farley et al., 1996; Hourigan et al. 2005; Ketcham et al. 2011; Gautheron et al., 2012; Bargnesi et al., 2016; Danisik et al., 2017; McDannell et al., 2018), the effects of grain morphology and measurement on age, uncertainty, and intrasample variability are less known, with only a few previous studies on improvements to grain measurement (Herman et al., 2007; Evans et al., 2008; Glotzbach et al., 2019).
In practice, for the determination of a correct He age, the grain dimensions and shape must be measured to compute an F_{T} correction factor prior to He and U, Th, and Sm analysis, assuming either parent nuclide homogeneity or prescribing an assumed or measured 1D or 2D parent nuclide zonation (Farley et al., 1996; Farley, 2002). While not directly related to the computation of He ages, these same grain dimensions are also used to calculate grain size parameters for the purpose of calculating isotopic and/or elemental concentrations and for age interpretation and diffusion or thermal history modeling (Shuster et al., 2006; Flowers et al., 2007, 2009; Flowers, 2009; Gautheron et al., 2009; Flowers and Kelley, 2011). For example, the grain mass, which is used to calculate the grain U, Th, Sm, and He concentrations, is often derived from the grain volume and an assumed density. Similarly, correlation between grain size (ESR) and He aliquot age has been used for qualitative and quantitative thermal history reconstruction using He diffusivity models (Reiners and Farley, 2001; Flowers and Kelley, 2011). Thus, the ability to measure accurate and precise grain dimensions, volumes, and surface areas for mineral grains has cascading effects for the determination, reporting, and interpretation of (U–Th) ∕ He data.
Most commonly, F_{T}, volume, and surface area are calculated using two or three grain dimensions (length + width 1± width 2) measured in 2D on an optical microscope using imaging software with a micrometerbased calibration. This approach requires the assumption of an idealized grain shape that most closely matches the mineral habit, such as a hexagonal prism for apatite or tetragonal prism for zircon, while simplifying (or ignoring) the more complex grain terminations (Farley et al., 1996; Farley, 2002). Hence, it has been best practice to select euhedral mineral grains to most closely match assumed, idealized grain shapes and large grains to minimize the amplification of uncertainties related to the F_{T} correction. However, even in felsic magmatic samples with highquality apatite, grains are often characterized by a wide range of grain shapes, variations in grain terminations, and the potential for broken or chipped surfaces that cause deviations from the idealized hexagonal prism. Furthermore, apatite grains often do not represent symmetric or equidimensional hexagonal prisms and are characterized by varying face widths, commonly, but also possibly inconsistently, lying on their largest and flattest face on the microscope slide and thus potentially introducing systematic biases during the selection of the clearest, inclusionfree grains.
Recognizing that this opticalmicroscopy approach is both limiting and may be an important source for error or bias in (U–Th) ∕ He ages and their interpretation, more sophisticated approaches have been proposed to determine grain dimensions, namely methods that do not require assuming a grain shape (Herman et al., 2007; Evans et al., 2008; Glotzbach et al., 2019). One approach presented by Glotzbach and others (2019), called “3DHe”, is an openly available software that uses orthogonal 2D grain photos to model accurate 3D grain shapes. Another approach is to employ Xray computed tomography (CT) to determine accurate grain shapes in an effort to improve precision and accuracy in F_{T} and (U–Th) ∕ He age determinations (Herman et al., 2007; Evans et al., 2008; Glotzbach et al., 2019). Herman et al. (2007) used 3D CT grain dimensions to calculate F_{T} factors and present a production–diffusion model to extract thermal histories for detrital apatite grains. Evans et al. (2008) and Glotzbach et al. (2019) both tested the efficacy of 2D microscope measurements against 3D CT data of zircon and apatite grain shape and size, arriving at quite different estimated discrepancies between microscope measurements and the CT data (1 %–24 % and < 1 %–6 %, respectively).
This new study investigates the effect of 2D versus 3D grain geometry measurement techniques on grain dimension, volume, surface area, ESR, mass, F_{T}, and the corrected age as well as effective uranium (eU) concentrations. In contrast to previous studies, which used 5–24 grains, we characterized > 100 apatite grains from two granitic samples for a more statistically robust comparison and in an effort to more systematically capture variations in apatite morphologies and sizes, as well as to screen for inclusions. We chose samples from crystalline basement that experienced fastcooling histories in order to target the impact of grain measurement techniques and minimize the effects of cooling history and transport on the (U–Th) ∕ He age and dispersion. The apatite grains were picked and measured by a single analyst using 2D optical techniques and then CT scanned. Building on previous work, we present a method for relatively rapid scans of > 100 grains at 4–5 µm resolution, enabling affordable and efficient 3D screening. We introduce the capabilities of an updated version of Blob3D (Ketcham, 2005; freely distributed software) that allows for the efficient batch processing of CTscanned grains and outputs parameters such as grain volume and 3D F_{T}. We further develop an approach for calculating ESR on the basis of equivalent F_{T} rather than an equivalent surfacetovolume ratio as a more direct and accurate means of approximating the diffusional domain as a sphere. Finally, in contrast to previous studies, we use the results of > 80 apatite (U–Th) ∕ He ages to evaluate the reliability of the 2D measurements as well as the impact on the (U–Th) ∕ He age and uncertainty.
Geologic background of the samples
For this study, we selected two plutonic samples from the Cretaceous Cordilleran magmatic arc in the western USA that yielded abundant, highquality apatite and have been part of previous thermochronometric studies. Sample 97BSCR8 is from a granodiorite in the Carson Range in the eastern Sierra Nevada along the Nevada–California border. The sample yielded an apatite fission track age of 68±2 Ma (P(X^{2})=75.4 %, 25 grains, N_{s}=1341) (Surpless et al., 2002). The second sample, 95BS11.3, is from a quartz monzonite exposed in the Wassuk Range in western Nevada, exhumed during Basin and Range normal faulting. The sample has a reported apatite fission track age of 16.3±1.4 Ma (P(X^{2})=76.1 %, 30 grains, N_{s}=158) and apatite (U–Th) ∕ He age of 9.9±1.9 Ma (Stockli et al., 2002). These samples were chosen for their abundant apatite and relatively simple cooling histories. Their geologic histories are relevant to the present study in that the apatite grains derive from plutonic rocks and did not experience complex metamorphic or magmatic histories, nor natural abrasion during sedimentary transport. Furthermore, both are plutonic samples that experienced rapid postmagmatic cooling or faultrelated exhumation and are expected to have spent little time in the apatite He partial retention zone.
2.1 Grain selection and 2D measurements
Apatite grains were picked from two samples, 97BSCR8 (n=50) and BS9511.3 (n=62), using a Nikon SMZU/100 optical microscope at a total magnification of 180×. Apatite grains were selected to include the range of grain morphologies present in the sample (e.g., broken, flat, and prismatic ends). Intentionally, several grains with visible inclusions were also selected to evaluate how well these inclusions showed up in the CT scans. All apatite grains were photographed using a Nikon digital ColorView camera connected to the microscope. The short and long axes were measured manually using AnalySIS^{®} imaging software (Figs. 1 and 3). We chose to measure a single width and did not flip the apatite 90^{∘} because this is still common practice in many labs and would allow us to compare the “simplest” 2D measurement approach with the 3D CT data. For sample BS9511.3, grains were imaged and measured on doublesided sticky tape in preparation for the CT mount (Fig. 1). However, we determined that this can cause grains to sit in upright orientations, which is fine for CT scanning but not for 2D measurements. For sample 97BSCR8 each apatite grain was placed on a glass slide for 2D measurements and then transferred to the sticky tape for the CT mount to remedy this issue (Fig. 1).
2.2 Grainmounting procedure for CT
Once the grains were measured optically in 2D, they were mounted for CT scanning by orienting several tens of grains on a plastic disk and stacking multiple disks (Fig. 2). The procedure to create a singlelayer mount for multigrain scanning entails covering a flat top of a pushpin with doublesided sticky tape that can be precut using a standard hole punch. Apatite grains are then picked directly onto the tape in a gridlike pattern. The pushpin surface is ∼5 mm in diameter, which easily allows ≥50 apatite grains to be mounted in one layer, tightly spaced, without touching. Grains could be packed more densely as long as they can be reliably identified after scanning; they can even be touching, although this leads to a small increase in processing time to separate them using functions in the Blob3D software.
To utilize the total scanned volume, at least five multigrain layers can be stacked for a single scan (up to 5 mm tall). To create stackable layers, sturdy plastic disks are made using a standard hole punch, with one side of the disk covered with doublesided sticky tape and apatite grains mounted in the procedure outlined above. Once all the layers are mounted and all excess tape is trimmed, the disks are stacked on top of the push pin. The arrangement is secured by a thin wrap of parafilm. The parafilm and sticky tape are critical to ensure the crystals and layers do not move during scanning. This mount can be easily disassembled after scanning to retrieve the grains for further analysis.
2.3 Xray CT scanning
The multigrain mounts were scanned with a Zeiss Xradia MicroXCT scanner at the University of Texas HighResolution Xray CT Facility (Ketcham and Cooperdock, 2019). Optimal scanning parameters will vary with the instrument being used, with top priorities being to minimize scanning artifacts and noise, while also minimizing time and cost. Lower Xray energies are more sensitive to compositional variations but more prone to beamhardening artifacts. We experimented with various settings in this study. The grain mount for sample 97BSCR8 was scanned with Xrays set at 100 kV and 10 W, with a 1.0 mm SiO_{2} filter. 1153 views were gathered at 1.5 s per view, for an acquisition time of 28.9 min. Source–mount distance was 37.7 mm, and mount–detector distance was 12.8 mm. The 2048×2048 camera data were binned by two, and the lowerenergy Xrays and weaker filtering necessitated the application of a beamhardening correction during reconstruction. The reconstructed data had a voxel (3D pixel) size of 5.03 µm.
The grain mount for sample BS9511.3 was scanned with Xrays set at 150 kV and 10 W with a 1.6 mm CaF_{2} beam filter, acquiring 571 views at 1.5 s per view, for an acquisition time of 14.3 min, not including calibration. Source–mount distance was 37.7 mm, and mount–detector distance was 17.8 mm. The camera data were binned by 2, and no beamhardening correction was applied during reconstruction. The resulting data had a voxel size of 4.58 µm.
Example images from the two datasets are shown in Fig. 3, illustrating some of the tradeoffs. The scan data for BS95 are noisier, primarily due to the faster acquisition, higher Xray energy, and more severe filtering. Even with this level of noise, highattenuation inclusions are evident. The scan data for 97BS are less noisy, allowing for the detection of a fluid inclusion, but beam hardening due to the lowerenergy Xray spectrum has caused faint streaks to emanate from or connect some grains. These subtle artifacts have a negligible effect on measurements but may be expected to increase in severity with more or higherdensity grains.
2.4 Grain size and shape, F_{T}, mass calculations
2.4.1 2D measurement calculations
The microscope length and width measurements are used to calculate volume and surface area, which are then used to calculate mass, ESR, and F_{T,U} and F_{T,Th} for each apatite grain, following methods laid out in Farley et al. (1996), Farley (2002), and Farley and Stockli (2002) (Fig. 4). An equidimensional hexagonal prism geometry was assumed with the length (L) measurement for height of the prism and the halfwidth (r) for the radius of the prism. All equations used for calculating these parameters are included below or in the Appendix.
Volume (V):
where L is height and r is the halfwidth.
Surface area (SA):
where L is height and r is the halfwidth.
Equivalent spherical radius (ESR):
Mass:
F_{T,U} and F_{T,Th} (2D case; e.g., Farley, 2002):
Mean F_{T} (see Appendix for explanation) from Farley et al. (1996) (used here for 2D calculations):
where ${a}_{\mathrm{238}}={\left(\mathrm{1.04}+\mathrm{0.245}\times \frac{\mathrm{Th}}{\mathrm{U}}\right)}^{\mathrm{1}}$.
From Blob3D for this study (used here for 3D calculations):
where ${A}_{\mathrm{238}}={\left(\mathrm{1.04}+\mathrm{0.247}\left[\frac{\mathrm{Th}}{\mathrm{U}}\right]\right)}^{\mathrm{1}}$ and ${A}_{\mathrm{232}}={\left(\mathrm{1}+\mathrm{4.21}/\left[\frac{\mathrm{Th}}{\mathrm{U}}\right]\right)}^{\mathrm{1}}$.
Effective uranium concentration (eU) (see Appendix for explanation):
2.4.2 3D calculations
Our principal 3D calculations were implemented in Blob3D (Ketcham, 2005), a program written in the IDL programming environment for efficient measurement of the dimensions, shape, and orientation of discrete features in volumetric datasets. The typical Blob3D method for calculating volume is to segment the grains based on a threshold set at 50 % of the CT number (grayscale) difference between apatite and the surrounding air. If grains are touching, or close enough to touching that their selected regions are connected, the software provides several separation methods, the simplest being an erode–dilate procedure. Volume is calculated as the number of voxels in a grain multiplied by the voxel volume, and surface area is calculated by summing the areas of the triangular facets of an isosurface surrounding the grain, which is smoothed to reduce excess roughness from the cubic voxel edges. The shape parameters BoxA, BoxB, and BoxC are respectively the length (L), width (W), and height corresponding to the dimensions of the smallest rectangular box that will enclose the grain (Ketcham and Mote, 2019). BoxC is calculated as the shortest 3D caliper length, BoxB is the shortest caliper length orthogonal to BoxC, and BoxA is the caliper length perpendicular to BoxC and BoxB (Fig. 4; Appendix C).
A Monte Carlo method was implemented to measure F_{T}, probably similar in many, but not all, respects to previous work (Herman et al., 2007; Glotzbach et al., 2019). Stopping distances for ^{238}U, ^{235}U, ^{232}Th, and ^{147}Sm for the set of minerals reported in Ketcham et al. (2011) are included in the software. Taking the set of selected voxels for a grain, the origin point for each alpha particle is selected by first randomizing from which voxel to start and then randomizing an ($x,y,z$) location within that voxel. The direction for each particle is obtained by sequentially stepping through a list of nearuniformly distributed orientations calculated by starting with an octahedron and subdividing each triangular face four times until there are 1026 vertices, which are then scaled to lie on a unit sphere (Ketcham and Ryan, 2004). This approach provides slightly better precision than randomizing orientations, and 200 000 Monte Carlo samples are sufficient to get precision to within 0.1 % in all tests reported below. Separate F_{T} factors for each decay chain (F_{T,238}, F_{T,235}, F_{T,232}, F_{T,147}) are calculated, and a revised method for calculating mean F_{T} that more precisely accounts for ^{235}U is provided in Eq. (6) (explanation in Appendix A).
If the resolution of the scan is low with respect to the stopping distance (^{238}U stopping distance ∕ voxel size < 4), excess surface roughness effects from voxelation are reduced by supersampling. The voxels for each grain and the surrounding voxels are subdivided into 27 (3^{3}) elements, and the supersampled volume is smoothed with a 5voxelwide cubic kernel. The result is then thresholded using a value that maintains the original volume as closely as possible.
These methods were tested on ideal spheres and cylinders, with radii of 63 and 31.5 µm and the latter with an aspect ratio of 4 (Appendix B). At voxel sizes up to 8 and 4 µm for the respective radii, mean F_{T,238} values averaged within 0.2 % of the idealshape values for spheres; further doubling the voxel sizes raised the mean error to 0.5 %. Cylinders performed better, with a mean error of 0.3 % when voxel sizes were 1∕4 of the radius.
In their Monte Carlo F_{T} implementation, Herman et al. (2007) report poor precision for small spheres when their centers are not centered in a voxel, with errors rising to several percent for a 40 µm radius sphere with 6.3 µm voxels across a range of center locations (calculated F_{T} range ∼0.58–0.67). Errors of this magnitude correspond to the effect of getting the radius wrong by plus or minus almost an entire voxel (∼15 % of the radius), too large to be reasonable and probably caused by a problem with their test. We tested our segmentation method by running 100 000 trials randomizing the location of the sphere center using the same radius and voxel size and got maximum radius errors of +0.8 and −1.1 % and a standard deviation of 0.2 % (Appendix B). We are thus confident that our implementation provides a high degree of accuracy and precision on even very small grains at low resolutions with voxel sizes up to 25 % of the radius.
We took three approaches to calculating ESR from the 3D data. The first two are based on the equivalent surfacetovolume ratio (SV) approach (Meesters and Dunai, 2002). The modelbased value ESR_{SVm} uses the BoxA and BoxB caliper dimensions as L and W for Eqs. (1) through (3), while the 3D CTbased value ESR_{SV3D} uses the 3Dmeasured volume and surface area for Eq. (3). Because of the unsupported assumptions of the model approach and the shortcomings of surface area measurements, both discussed below, neither of these solutions is ideal. An alternative ESR is based on the equivalent F_{T} approach; Ketcham et al. (2011) demonstrated than an equivalent F_{T} sphere provides a more accurate conversion for diffusion calculations than an equivalent SV one. The set of calculations to determine the F_{T}equivalent sphere radius ESR${}_{{F}_{T}}$ are provided in Appendix A.
2.5 (U–Th) ∕ He procedure
The apatite (U–Th) ∕ He ages were analyzed in the UTChron Thermochronology Laboratory at the University of Texas at Austin. Individual grains were measured, wrapped into platinum tubes, loaded into a 42hole sample holder, and pumped to ultrahigh vacuum. Each aliquot was heated to ∼1070 ^{∘}C for 5 min using a Fusions Diode laser system. The released gas was spiked with a ^{3}He tracer and purified by a Janis cryogenic cold trap at 40 K and SAES NP10 getter prior to measurement of the ^{4}He∕^{3}He on a Blazers Prisma QMS200 quadrupole mass spectrometer. Final ^{4}He contents were calculated using a manometrically calibrated ^{4}He standard of known concentration measured during the analytical run. All apatite aliquots were reheated once under the same conditions to ensure full gas release.
After degassing, the platinum packets containing the apatite grains were placed into plastic vials and dissolved in a 100 µL 30 % HNO_{3} ${}^{\mathrm{235}}\mathrm{U}{}^{\mathrm{230}}\mathrm{Th}{}^{\mathrm{149}}\mathrm{Sm}$ spike solution for 90 min at 90 ^{∘}C. After acid digestion, 500 µL of MilliQ ultrapure H_{2}O was added to dilute the solutions to ∼5 % HNO_{3} and equilibrated for ≥ 24 h prior to analysis. The solutions were analyzed using a Thermo Element2 highresolution inductively coupled plasma–mass spectrometer (HRICPMS) equipped with a 50 µL min^{−1} microconcentric nebulizer. Final ^{238}U, ^{232}Th, and ^{147}Sm values were blankcorrected and calibrated using a spiked, gravimetrically calibrated ∼1 ppb standard solution. Final (uncorrected) ages were calculated by solving the He age equation by means of Taylor series expansion and reported with a 6 % standard error based on longterm intralaboratory analysis of apatite age standards. Corrected final ages are determined by dividing the uncorrected age by the mean F_{T} factor (Eq. 5). U, Th, and Sm concentrations, although not used in the age calculations, were determined for reporting purposes using the grain volumes and a nominal apatite density (e.g., Fig. 4, Eq. 4).
CT scanning combined with Blob3D analysis provides 3D grainspecific volume, surface area, dimensions, and F_{T} factors for each decay chain. The 2D optical measurements provide dimension information, which is used to calculate volume, surface area, F_{T,U}, and F_{T,Th} based on an assumed grain geometry of an equidimensional hexagonal prism (all results are reported in the Appendix). We assume that the 3Dmeasured volume and F_{T} values are sufficiently accurate to benchmark the 2D data (all comparisons reported in Table 1 and Fig. 5). Surface area is more problematic to benchmark due to a number of factors, such as fractal roughness, CT data blurring, and voxelation effects, as discussed below, and thus 2D and 3D results can only be compared in a relative sense for surface area.
ESR(SV_{m}): BoxA and BoxB assuming hexagonal prism shape, ESR(SV_{3D}): Blob3D volume and SA measurements, ESR(F_{T}): F_{T}equivalent sphere.
2D and 3D data are compared for each sample and as an entire population in Tables 1 and 2. The average 3D ∕ 2D ratio of each parameter is reported with its 1σ standard deviation. This average ratio shows whether the 2D measurements on average overestimate (ratio < 1) or underestimate (ratio > 1) the 3D measurements. Also reported is the absolute percent difference between the 2D and 3D measurements to illustrate the magnitude of deviation between the measurements. While comparing 2D and 3D results, it became apparent that one 2D grain measurement was made at an incorrect microscope magnification setting, causing the length and width to be off by 2 times, far greater than every other grain measured. Hence, this grain measurement (97BSCR81) was not included when calculating the average differences between 3D and 2D measuring techniques.
3.1 Grain factors
Grains from both samples display a range of habits typical for apatite, including two flat ends, two prismatic ends, one flat and one prismatic end, and one or two broken or chipped ends (Figs. 1 and 4). The grain morphology and the presence of any visible inclusions were recorded during handpicking (Table 2). Surprisingly, there are no clear systematic relationships between the presence of inclusions and grain age or grain shape and ESR, volume, or surface area. The 2D length measurements are on average ∼2 % smaller than the 3D BoxA dimension. On the other hand, the 2D width dimension is on average ∼3 % greater than the 3D BoxB dimension (Table 1).
One inevitable source of uncertainty in 2D length and width measurements is analyst judgment and error. For example, if a grain has uneven terminations, it is at the analyst's discretion to measure the longest axis or split the difference, whereas the CT analysis always reflects the longest axis. Similarly, CT scanning is also not subject to any user error introduced by measuring the apatite grain not lying on its widest face or at an incorrect magnification. In our dataset, a couple of grains have very large deviations from the CTderived volume, which may be caused by the microscope magnification setting being slightly off during measuring. Of course, the degree of analyst error is subject to many factors (e.g., experience of the analyst, the age and type of microscope, measuring software, etc.) and must be addressed on a labbylab basis. In this study we found that human error may lead to “outliers” in the results, and therefore it is a factor that we consider.
3.2 Volume and surface area
Volumes and surface areas calculated using the 2D microscope dimensions both average ∼20 % larger than the 3D calculations (3D ∕ 2D_{VOL}=0.82, 3D ∕ 2D_{SA}=0.81) (Table 1, Fig. 5). Specifically, 2D volumes and surface areas calculated from length and width data assuming a hexagonal prism shape have an absolute average difference of 23±32 % (2σ) and 22±18 % (2σ), respectively, from 3D Blob3Dcalculated volumes and surface areas.
3.3 ESR and mass
The 2D ESR is calculated using the surfaceareatovolume ratio (SA∕V), which is derived assuming a hexagonal prism with the length and width dimensions measured on the microscope (Eq. 2, Fig. 6). The 3D data had the ESR calculated based on SA∕V in three ways. First, the SA∕V for ESR_{SVm} is calculated using the BoxA and BoxB values provided by Blob3D and assuming a hexagonal prism, mimicking the 2D approach. The variation between 2D and 3D ESR_{SVm} measurements has a 2σ spread of ±12 %, but the variability is fairly evenly split in overestimating and underestimating the ESR such that the average 3D ∕ 2D ratio is 1.02. Second, the 3D SA∕V is calculated using the surface area and volume measurements output by Blob3D (ESR_{SV3D}). The variation between 2D and 3D ESR_{SV3D} is even larger at ±18 % (2σ), with an average 3D ∕ 2D ratio of 1.01 (Table 1, Fig. 5).
The F_{T}based ESR was on average similar to the SVbased one (ESR${}_{{F}_{T}}/$ ESR_{SVm}=1.0), but the variation was ±9 % for the two samples, and extreme values were 9 % higher and 21 % lower. The relative variation of the ESR${}_{{F}_{T}}$ value with the 2D data is ±14 %, similar to that for the other 3D ESR calculations (Table 1, Fig. 5).
The grain mass is calculated from the volume data using a nominal apatite density, and therefore 2D and 3D mass determination directly reflect the variability in the 2D and 3D volume data. The 2D approach consistently overestimates the mass, with a high degree of scatter (3D ∕ 2D $=\mathrm{0.82}\pm \mathrm{0.44}$ (2σ)) (Table 1, Fig. 5).
3.4 F_{T} corrections
F_{T,U} and F_{T,Th} correction factors calculated from the 2D data are generally 1 %–2 % lower than the Blob3D U and Th F_{T} factors. To combine the F_{T} factors into a single term that is applied to the (U–Th) ∕ He age, a mean F_{T} was calculated in two ways using Eq. (6) (see Methods). This results in mean F_{T} factors that vary by an average of 2 % between the 2D and 3D datasets. The 1σ scatter in 3D ∕ 2D F_{T} factors is 1.8 %, though individual differences can reach up to 9 % (Table 1, Fig. 5).
3.5 (U–Th) ∕ He age and effective uranium
We calculated the apatite (U–Th) ∕ Hecorrected age by dividing the raw (U–Th) ∕ He age by the mean F_{T} factor. The 2D F_{T} (U–Th) ∕ He ages tend to be slightly older than the 3D F_{T} (U–Th) ∕ He ages (3D ∕ 2D =0.99) owing to the fact that the 2D F_{T} values are slightly lower, leading to a larger correction (Table 1, Fig. 5). The average difference between the 2D and 3D F_{T}corrected ages is 2 %, mimicking that of the variation between 2D and 3D F_{T} (full range is < 1 % to 9 %). This has an insignificant impact on the mean age and uncertainty for both samples. Sample 97BSCR8 has a 2D F_{T} mean age of 56.8±2.9 Ma and a 3D F_{T} mean age of 56.0±2.9 Ma (Table 2, Fig. 5). Sample BS9511.3 has a 2D F_{T} age of 12.2±4.0 Ma and a 3D F_{T} mean age of 12.1±4.0 Ma (Table 2, Fig. 5).
The effective uranium concentrations (eU = [U] + [Th] $\times \mathrm{0.238}+$ [Sm] ×0.0012) for the apatite are normalized to the mass of the grain. Since 2D and 3D grain mass calculations varied by ∼25 %, the eU concentration measurements vary by a similar degree (3D ∕ 2D $=\mathrm{1.29}\pm \mathrm{0.24}$ (2σ)) (Table 1, Fig. 5). Note that not all grains were analyzed for U, Th, and Sm, so there are less data for eU comparison than mass.
4.1 Accuracy of 2D vs. 3D grain measurements
4.1.1 Volume and surface area
One of the main motivations behind this study was to assess the accuracy of 2D grain measurements and using an assumed grain geometry for calculating grain parameters (volume, ESR, mass, F_{T}) and the impact on the accuracy of the final (U–Th) ∕ He age and eU. For this reason, we selected two samples from crystalline basement rocks that experienced relatively fast exhumation and no significant subsequent reheating in order to reduce the impact of geologic or kinetic factors that could lead to age dispersion.
The most striking deviations between 2D and 3D measurements are in the volume and surface area, which 2D measurements consistently overestimated by 20 %–25 % in our study, with a large degree of scatter (1σ=22 % and 14 %, respectively). These results are in line with previous work. Evans et al. (2008) observed a similar discrepancy in the five apatite grains they measured: their 2Dbased volumes were 30 % greater than the 3D volumes (Table 3). Our dataset contains > 100 apatite grains, implying that the 2D overestimation of volume (and therefore mass) may be systematic in the 2D measurement approach. In contrast, Glotzbach et al. (2019) analyzed 24 apatite grains and found that the 2D volume measurements varied by a similar magnitude (∼15 %) but did not systematically overestimate the volume as in our study and Evans et al. (2008) (Table 3). This is likely due in large part to their procedure of measuring three dimensions and selecting the appropriate shape model on a grainbygrain basis, including ellipsoids for anhedral grains and accounting for terminations using the functions provided in Ketcham et al. (2011), rather than assuming exclusively flatterminated hexagonal prisms.
There are multiple factors that can contribute to overestimating the volume of a given apatite crystal. First, the assumption of a hexagonal prism crystal shape with flat terminations, in which the length of the grain is used as the height of the prism, has the potential to overestimate the volume if the crystal has tapered ends (Fig. 4). However, our data suggest this can only account for about a third of the volume difference because even crystals with two flat (or broken) ends still had an average volume difference of 13 %. Second, the ideal prism model also presumes a perfect, equalsided hexagonal cross section perpendicular to the c axis, for which the ratio of width to height should be $\mathrm{2}/\sqrt{\mathrm{3}}$, or 1.1547. The 3D shape measurements give mean ratios of 1.25(02) and 1.23(01) for our two samples, indicating that the cross sections are on average flatter than ideal hexagonal prisms. The nonideality of this cross section was also noted by Glotzbach et al. (2019) and can result in either an underestimate or overestimate of volume, depending on which face the grain is lying on when measured in 2D. The systematic bias we observe is not surprising as apatites commonly come to rest on their flatter side, whereas some of our observed scatter comes from this not always being the case. We estimate that this shape divergence explains about a quarter of the departure between 2D and 3D volume in our data. The remaining deviation may be due to chipped crystals, surface roughness, or other deviations from a perfect prism that the 2D calculation cannot account for.
A number of factors will directly impact surface area calculations. Surface area is calculated from the 2D measurements by assuming a perfectly smooth prism. CT has the potential to capture irregular surfaces present in natural apatite grains, which if present and resolution is sufficient, should lead to higher surface area calculations in the 3D data. However, surface area is problematic to measure in CT data, regardless of resolution. Irregular surfaces are to some degree fractal entities, making their measured areas dependent on measurement scale, and the “correct” answer is not straightforward to define. All CT images are naturally blurry to some extent, smoothing out both irregularities and also sharp corners and edges. Conversely, the 3D measurement process of segmentation by thresholding can lead to artificial enhancement of surface area due to voxelation effects (the 3D equivalent of pixilation).
In our data, the 2D measurements consistently result in a higher surface area than the 3D measurements. This is probably partly due to the ∼5 µm resolution of our CT data and also to the flatterminated hexagonal prism model leading to an overestimate. Evans et al. (2008) observe a similar discrepancy in surface area measurements between 2D and 3D data (2D ∼23 % higher) with a 3.77 µm resolution scan (Table 3). On the other hand, Glotzbach et al. (2019) scanned their grains at a 1.2 µm resolution and their 2D measurements gave surface areas on average 8 % lower than 3D (Table 3). As with volume, a large part of the difference is probably due to their using a more accurate shape model than an ideal equalsided hexagonal prism. The overshoot may be in part due to their higher CT data resolution capturing roughness better, but their 3D images also show voxelation effects such as ridge sets on flat surfaces that likely increased their surface areas to an unknown extent.
We note that the nature of the alpha stopping process, both in reality and as simulated, makes it essentially a ∼20 µm smoothing filter, so shortlengthscale roughness has a negligible effect on alpha particle retention and F_{T} calculation. This point is demonstrated by our sensitivity analysis (Appendix B), which shows that a bumpy, voxelated sphere has the same F_{T} correction as a perfect, smooth one. Thus, while surface area is difficult to measure precisely in general, it is unimportant to measure precisely for this application.
4.1.2 Mass and eU
The discrepancy in volume between 2D and 3D measurements directly impacts the mass calculation, causing the grain masses derived from the 2D measurements to be ∼25 % higher than the 3D grain mass determinations (Fig. 6). Evans and others (2008) found similar deviations, with their masses calculated from 2D volumes ∼30 % greater than their masses for 3D volumes (Table 3). Both of these divergences stem from using the assumption of a flatended hexagonal prism, whereas an approach that takes grain shape into account when choosing the F_{T} formula (Ketcham et al., 2011; Glotzbach et al., 2019) avoids this systematic bias. However, in all cases that use perfect shape models, the relative scatter is on the order of 20 % (1σ), which is high enough to be worth fixing.
Although the age equation does not require knowledge of the grain volume or mass, both are necessary to calculate reported concentrations for U, Th, Sm, and He (Fig. 6). The U, Th, and Sm concentrations, often combined into a single term, “effective uranium” (eU), have been used a proxy for radiation damage within a crystal, and age versus eU correlations are commonly used for interpretation of age scatter and thermal history inverse modeling (e.g., Flowers et al., 2009; Guenthner et al., 2013; Fox et al., 2019). Therefore, accurate knowledge of volume has cascading effects from mass to eU concentration and age interpretation (Fig. 6). Comparison between eU calculated for the 3D mass data and 2D mass data shows that the 2D masses underestimate the bulk eU concentrations by ∼20 %–30 %. This is consistent with the 2D mass data being ∼25 % higher than the 3D mass data, which would have the effect of “diluting” any eU signal; moreover, the much higher degree of scatter in the mass data caused by 2D analysis (±44 % (2σ)) can be expected to muddy any age–eU correlation that may be present.
4.1.3 ESR
The various ESR calculations all yielded similar results on average but high degrees of variation between measurement and calculation modes (5 %–6 %). In addition to being more accurate for simplifying complex shapes to spheres for diffusion calculations, the ESR${}_{{F}_{T}}$ method is also likely more robust than others that presume or measure surface area. Surface area, beyond being difficult to define and measure for irregular natural objects in a resolutionresistant way, has only secondary importance for diffusion and F_{T} calculations when it varies on a fine scale compared to the grain (i.e., micrometerscale roughness). Analogously with mass, excess variation in ESR (±14 % (2σ)) can degrade age–size correlations.
4.1.4 F_{T}
A somewhat surprising result of our study is that, despite volume and surface areas being very different between the 2D and 3D methods, these differences largely canceled each other out in S∕Vbased F_{T} calculations. This is in large part because volume and surface area covary, both in the assumed models and the actual measurements, so an error in one leads to a similar magnitude of error in the other (Fig. 6).
A result that more closely conformed to expectation is that, as grain size fell, dispersion between 2D and 3D F_{T} values increased, although it remained modest. The standard deviation of 3D ∕ 2D F_{T,U} was 2.7 % for grains with F_{T,U} values from 0.6 to 0.7, 2.4 % from 0.7 to 0.8, and 1.3 % for grains above 0.8.
While the above comparison takes into account 24 to 53 grains per sample, most applications of (U–Th) ∕ He analyze 3–5 grains per sample. As a more practical comparison of the difference between 2D and 3D Mean F_{T}, we randomly subsampled the average of four grains from our results 1000 times (Fig. 7). We found that even when subsampling four grains, ∼90 % of runs had a mean deviation in 3D ∕ 2D F_{T} less than 3 %.
4.2 Reproducibility of (U–Th) ∕ He ages
In addition to assessing the accuracy of using the 2D measurements, this study aimed to quantify the uncertainties that may be introduced by such measurements, particularly in F_{T}, as a means to potentially improve age accuracy, precision, and intrasample dispersion. Previous studies have estimated that uncertainties in F_{T} calculation can account for 1 %–5 % of sample age uncertainty (Evans et al., 2008; Glotzbach et al., 2019). Our results are consistent with this range and suggest that uncertainties in the U and Th F_{T} calculation are on the order of 1 %–3 %, and mean F_{T} varies by 2 % (Table 1). We find the greatest deviations are likely caused by user error for our samples and not the assumed grain geometry. In samples with less euhedral apatite grains, the effects of F_{T} and an assumed grain geometry can increase.
Our data also show that the 3D F_{T} correction does not increase the overall sample age precision for the samples in this study. For sample 97BSCR8, 24 apatite grains were analyzed, two of which are outliers. Of the two outliers, one (97BSCR81) was clearly caused by a user error during microscope measurement, leading to an incorrect F_{T} correction (0.55) and old age (78.8 Ma). This was discovered during 3D image processing, in which the same grain was identified, measured correctly, and produced an F_{T} of 0.76 and a more congruent corrected age of 57.2 Ma. In contrast, for a second outlier (97BSCR824), the 2D and 3D F_{T}corrected ages both produced anomalous ages of 101.2 and 98.4 Ma, respectively. An unusually high He concentration is the likely culprit for the old age for this grain, but its cause is not evident from our data. Excluding these two outliers, the average age and uncertainty for the sample population (n=22 grains) calculated based on the 2D and 3D measurements are indistinguishable (56.8±2.9 and 56.0±2.9 Ma); relative errors are 5.1 % in both cases.
Similarly, the sample ages calculated with 3D and 2D data for 95BS11.3 (n=59 aliquots) are indistinguishable at 12.2±4.0 and 12.1±4.0 Ma, respectively. Unlike sample 97BSCR8, there was no clearcut evidence of user error, and the relatively high age uncertainty (33 %) is reproducible between the 2D and 3D F_{T}corrected ages. Five aliquots produced ages > 20 Ma, which skews the mean age older (the median age is 10.2 Ma, within the error of the previous reported age in Stockli et al., 2002). The apatite ages do not correlate with factors such as ESR (grain size) or eU. The > 20 Ma aliquots all have high He concentrations (nmol g^{−1}) compared with the bulk of the sample, suggesting excess He, possibly due to implantation from high U–Th neighbors, or the presence of undetected and insoluble high eU inclusions.
In addition to the above calculations, we randomly subsampled four grains 1000 times to assess the variability in F_{T}corrected age for a number of grains that is more comparable to other studies. The results are plotted in Fig. 7 and reported in Table 2. The mean of the 1000 trials is indistinguishable from the entire analyzed population.
Overall, these data suggest that although the 3D F_{T} can provide a more accurate F_{T} correction and varies from 2D estimations by ∼2 %, it has a minimal effect on the calculated sample age (1 %–2 %) and no effect on the reproducibility for these two samples. This is not surprising, as a ∼2 % error would constitute a negligible proportion of the oftencited 6 % dispersion derived from analyzing age standards; error propagation indicates that removing a source of 2 % error would only reduce an overall 6 % error to 5.7 %. This points to the importance of other factors in intrasample dispersion, such as U–Th zonation, and/or excess He from nanoinclusions or high U–Th neighbors.
4.3 Effects of inclusions or broken grains
It is widely accepted that inclusions and broken grains are both contributors to intrasample dispersion and inaccurate He ages, particularly anomalously old ages. Inclusions in apatite can act as He traps or a source for excess He, particularly mineral inclusions that do not dissolve during apatite HNO_{3} digestion (e.g., Ehlers and Farley, 2003). Both apatite samples had multiple grains with highdensity and lowdensity inclusions detectable by microscope during picking and/or the CT scan (Fig. 2). In both samples, the presence of inclusions did not have any discernable effect on the (U–Th) ∕ He age (Table 2). While inclusions are certainly a source for error and dispersion in many samples and should be avoided, at least the easily visible ones do not appear to be relevant in these samples, which suggests they are likely also not U–Thbearing inclusions. For future studies, an added benefit of CT is the detection of high and lowdensity mineral and fluid inclusions.
Similarly, broken grains can be a source of dispersion if they were broken after the sample passed through the He partial retention zone, e.g., after the grain began to accumulate He (see Beucher et al., 2013; Brown et al., 2013). Typically, this may occur during erosional transport or during mineral separations. Brown et al. (2013) estimate that broken grains can contribute 7 to > 50 % dispersion from the sample age, depending on cooling history. In our samples, grain terminations varied from doubly prismatic to flat and in some cases appeared chipped or broken. However, there is no clear correlation between the chipped or broken grains and He age (see Table 2). One possibility is that the grains broke prior to cooling through the He retention zone. This seems somewhat unlikely, given that both samples come from crystalline rocks. Alternatively, and perhaps more plausibly, the variety of crystal habits may reflect how the crystals grew in the host rock. In any case, the grains in these samples that appear to be chipped or broken are not obvious sources for the age dispersion observed in the samples.
4.4 Benefits and limitations of Xray CT over microscope measurements
This study purposefully selected “highquality” apatite from fastcooled plutonic samples to quantify the base uncertainty introduced by 2D measurements and grain shape assumptions on F_{T} and (U–Th) ∕ He age factors. Although we found that 3D grain characterization techniques did not reduce intrasample age dispersion in our samples, it is still highly probable that the 3D approach can improve dispersion in samples with less euhedral apatite and more complicated geologic histories. Furthermore, CT scanning mineral grains for (U–Th) ∕ He chronometry has both analytical and practical benefits that go beyond grain measurement. CT provides more accurate grain volume measurements, which becomes increasingly important as grain shapes deviate from idealized forms (e.g., abraded or broken grains). CT data are able to highlight inclusions or other internal heterogeneities based on contrasts in density in the Xray data, which may not be visible by the naked eye. Furthermore, the CTmounting method and scanning conditions outlined in this study allow for the scanning of up to 250 grains in a single session, and potentially many more, making it cost and time effective. Different mineral phases can be scanned together, and data can be processed in a batch so that from a single scan, one can gather volume, surface area, caliper dimensions, F_{T}, mass, and ESR at once for several samples and phases. Furthermore, the 3D F_{T} and F_{T}based ESR capabilities of the Blob3D software introduced in this study make batch processing the CT data straightforward. Thus, an analyst will be able to image, characterize, and quantify hundreds of mineral grains in significantly less time than conventional microscope measuring. We anticipate that more volumebased shape measurements can and will be developed to automatically and quantitatively evaluate grains for euhedrality, rounding, broken faces, and a wealth of other potentially informative data.
CT scanning mineral grains used for (U–Th) ∕ He dating also has the benefit of removing many possible sources of user error during the grain measurement step. Unlike with microscope measurements, the orientation of the apatite grain on the CT mount does not matter, and there is no need to set a magnification or trace the dimensions of the grain by hand, reducing the potential for mistakes. CT also eliminates variability that may arise from different microscopes, lighting conditions, and imaging software, and it creates a digital archive of 3D grain shapes, densities, and internal structures that a microscope photo cannot capture.
The one required user input to our method is specifying the threshold CT number for grain measurement, for which we recommend using the midpoint value between the mineral and the surrounding medium (e.g., air, epoxy). When scan resolution is low in terms of both voxel size and sharpness, additional care is required; if edge blurring approaches the center of a grain, an alternative thresholding or segmentation procedure may be necessary to obtain accurate volumes (Ketcham and Mote, 2019). We thus do not recommend pushing resolution limits too far; voxel sizes generally should not exceed 1∕8 to 1∕10 of the shortest dimension of a grain. CT measurement accuracy also requires that the scans be as free as possible from artifacts that cause local changes in CT numbers, such as beam hardening, photon starvation, or rings. We further note that software artifact corrections can sometimes introduce secondary artifacts that may be harder to recognize but still affect calculations (Ketcham and Carlson, 2001), so care is required in the scanning process.
The main limitation of using CT is access to the instrumentation and cost for sample analysis. However, CT scanners are becoming more common as desktop instruments in earth science departments, and many universities have imaging facilities that include microCT. As CT instruments continue to proliferate and costs continue to fall, we anticipate that measuring, screening, and documenting grains used for thermogeochronology will become a widely used practice.
The shape and size of 109 apatite grains from two rapidly cooled plutonic samples were analyzed by 2D and 3D methods. 2D length and width measurements made on an optical microscope were used to calculate surface area, volume, ESR, mass, and F_{T} assuming an ideal equalsided, flatterminated hexagonal prism grain shape. The same apatite crystals were scanned using Xray computed tomography at a 4–5 µm resolution, and the same factors were calculated using Blob3D software, which does not require assuming a grain shape. A total of 83 new apatite (U–Th) ∕ He ages were collected to resolve the influence of 2D versus 3D F_{T} correction factors on final (U–Th) ∕ He age and reproducibility. With these data, we derive the following conclusions.

Deviations between 2D and 3D measurements were greatest in volume and surface area (∼25 %), which caused mass and eU calculations to deviate by a similar magnitude. Volume and surface area measurements also showed high dispersion of 44 % and 28 % (2σ), respectively. These sources of scatter weaken the ability to use age–eU and age–size correlations to help interpret age distributions.

2D F_{T} measurements only contribute ∼2 % error on average, even with the erroneous assumption of an ideal grain shape.

Inclusions and broken or chipped ends did not have a discernible impact on the (U–Th) ∕ He age dispersion in these samples.

The combined (U–Th) ∕ He ages for each sample were indistinguishable for 2D and 3D F_{T} corrections. Similarly, the amount of intrasample dispersion was identical (both > 5 %). This implies that factors other than F_{T} dominate the intrasample age uncertainty.
In addition, we present a bulk scanning method that easily allows for the analysis of > 250 grains in a single session, new Blob3D software 3D F_{T} and shape measurement functions, and new calculations for eU and ESR${}_{{F}_{T}}$.
The code and data are available in the Supplement to this paper. CT data are archived at https://doi.org/10.17612/CZYHKC13 (Ketcham and Cooperdock, 2019).
A1 ESR${}_{{F}_{T}}$ and mean F_{T}
The starting point for calculating the equivalent F_{T} sphere radius (ESR${}_{{F}_{T}})$ when F_{T} values are provided for each decay chain is the F_{T} equation for a sphere (Farley et al., 1996; Ketcham et al., 2011):
where R is the sphere radius, S is stopping distance, and B is an adjustment factor for the 3rddegree polynomial term to account for S being the weighted mean of stopping distances along branching decay chains rather than a single stopping distance. For U and Th decay chains B should be 1.31, and for single stopping distances it should be 1 (Ketcham et al., 2011).
Solving this equation for S∕R over the F_{T} range from 0.5 to 1 using a 3rddegree polynomial to match the effect of the cubic term gives
The polynomial in Eq. (A2a) is the appropriate one to use for data to be reported in age tables; Eq. (A2b) is provided for completeness and may be useful for comparing to other calculations that use mean S values to represent chains.
The F_{T} value to use is the weighted mean incorporating the separate factors F_{T,238}, F_{T,235}, and F_{T,232}, accounting for different alpha productivity along each chain. Expanding the approach of Farley (2002) to account precisely for ^{235}U, we calculate
so that the weighted mean, $\stackrel{\mathrm{\u203e}}{{F}_{T}}$, is
Solving the result of Eq. (A2) for ESR${}_{{F}_{T}}$ requires the analogous calculation to determine the weighted mean stopping distance, $\stackrel{\mathrm{\u203e}}{S}$:
where S_{238}, S_{235}, and S_{232} are the weighted mean stopping distances for each decay chain (18.81, 21.80, and 22.25 µm, respectively, for apatite, but the calculation applies to any mineral). Then, combining Eqs. (A2) and (A5) gives
A2 eU
The earliest mention of eU, or effective uranium with respect to He production, we are aware of is in Shuster et al. (2006), who put forward the formula
where brackets indicate composition in parts per million without a detailed description of its derivation. Converting from elemental or isotopic compositions in parts per million to an equivalent alpha particle production rate requires accounting for decay constants, isotopic proportions, alpha particle production, and atomic mass. We calculate the presentday alpha production rate R_{α} (here: α g^{−1} yr^{−1}) as
where A is Avogadro's number, λ is the decay constant, p is isotopic proportion, N is the number of alpha particles produced in the decay chain, and m_{a} is atomic mass. The eU factor is then calculated by dividing the Th and Sm R_{α} by the combined U R_{α} utilizing the values in Table A1; we find the eU equation to be slightly different:
We do not know the reason for the small discrepancy with Eq. (A7), but the ∼1 % difference in the effect of Th is not likely to be important for current uses of eU. The 0.238 factor has a likely uncertainty of ±0.002; the ^{232}Th halflife currently recommended by the nuclear chemistry community has only three significant figures based on a weighted average of several determinations using different methodologies (Browne, 2006; Holden, 1990), whereas the geological community has adopted the value from the single study with the highest reported precision (Le Roux and Glendenin, 1963; Steiger and Jäger, 1977).
We include Sm for completeness, but as its alpha decay has a relatively low recoil energy it is not clear whether simply counting the particle is the most appropriate way to include its potential contribution to damage that affects helium diffusivity. An alternative formulation can be posed in terms of energy deposition (kerma; Shuster and Farley, 2009):
where E is the mean alpha particle recoil energy for the decay chain. The revised kermabased quantity, eU_{k}, is then
This relation predicts that Sm will have an even lower relative contribution to diffusivity than indicated in Eq. (A9), but Th will be 11 % more potent due to its higher mean recoil energy compared to ^{238}U. We do not currently recommend this approach, but it does pose a potentially testable hypothesis.
This Appendix describes a series of tests that demonstrate the accuracy and precision of the methods for F_{T} calculations implemented in Blob3D (Ketcham, 2005). All calculations are performed in Blob3D or with scripts in IDL, the computer language in which Blob3D is written.
B1 Centered spheres
In the first set of tests, we use spheres, which Herman et al. (2007) recognized as a good test shape because its surface is poorly approximated by coarse stacked cubes. We begin with a 128^{3}voxel field, and select all voxels with centers within 63 voxel widths of the center of the volume, creating a 63 µm radius sphere with a 1voxelthick black boundary on all sides. Four additional lowerresolution versions were then created by rebinning the original dataset to make volumes with 64^{3}, 32^{3}, 16^{3}, and 8^{3} voxels; these datasets were then padded with an additional layer of black (nonselected) voxels on three sides to ensure the spheres had a black boundary on all sides for Blob3D processing. In the 8bit data volumes, selected voxels have a value of 255 (white) and nonselected ones a value of 0 (black).
If the voxel width is 1 µm in the 128^{3} dataset, the resulting ideal sphere radius is 63 µm, which has an F_{T,238} correction of 0.7777 (stopping distance 18.81 µm). Because of voxelation effects, the actual volume selected will be slightly different than the ideal case; for example, the volume in the 128^{3} dataset corresponds to an equivalent sphere radius (ESR) of 63.02 µm. With each rebinning step, doubling the voxel size roughly maintains the original volume, simulating lower resolution; i.e., 2 µm voxels for the 65^{3}voxel dataset, 4 µm for 33^{3}, 8 µm for 17^{3}, and 16 µm for 9^{3}. We ran an initial set of tests using these voxel sizes and an additional set with the voxel size halved, corresponding to a 31.5 µm radius crystal, close to the lower end of the practical limit (${F}_{T,\mathrm{238}}=\mathrm{0.5655}$).
Because the calculation employs a Monte Carlo algorithm, answers change slightly from run to run, so for each dataset and resolution results from five Blob3D runs were used to gauge precision. Results are provided in Table B1 and shown in Fig. B1 as the mean measured (calculated) F_{T} divided by the ideal value for the ESR of the volume actually selected at each resolution, with bars showing 1 standard error.
Results for the 63 µm sphere test are in Table B1a and Fig. B1a. Solid symbols show the result of the normal Monte Carlo analysis, with results accurate to within 0.1 % at up to a 4 µm voxel size, but mean errors rise to approach 1 % with 8 µm voxels. Halftone symbols show the result of altering the processing by first supersampling the volume, subdividing each voxel into a 3^{3} set, and then smoothing the expanded data volume with a 5voxelwide filter, followed by rebinarizing the data with a threshold (value 127) prior to the Monte Carlo analysis. This step improves accuracy at 8 µm resolution to within 0.4 % on average and also further reduces the sub0.1 % error at the 4 µm level. However, the 127 rethreshold value is not the optimal one, as it slightly shrinks the volume due to the overall convex shape of the grain, so the algorithm finds the optimal threshold that reproduces as closely as possible the presupersampled grain volume. The result improves the 8 µm calculation yet more, reducing the mean error to just over 0.2 %, and even with 16 µm voxels the error is only just over 0.5 %. This improvement also demonstrates that getting the volume correct is a primary control on the accuracy of the F_{T} calculation; this principle is used to examine the case of noncentered spheres later in this Appendix.
Remaining tests use the convention that when voxel sizes are 4 µm or higher the constantvolume supersampled approach is used; the only cost of supersampling is slightly more computing time, which is still less than 1 s per grain (but could rise above this level if employed with smaller voxels and larger grains). The 31.5 µm sphere test (Table B1b, Fig. B1b) shows similar results as the larger case; mean errors are less than 0.5 % up to voxel sizes of 8 µm.
B2 Cylinders
As most apatite (and zircon) grains are elongate, we also tested cylinders as a closetoworstcase endmember, again because a round outline is more poorly approximated by cubes than a hexagonal or tetragonal one. We created the cylinders by stacking 510 63voxelradius circles with blank slices at each end to achieve an aspect ratio close to 4 and downsampled as with the sphere test four times by powers of 2. Results are shown for the 63 and 31.5 µm cases, with respective ideal F_{T,238} values of 0.8350 and 0.6772, in Table B1c–d and Fig. B1c–d. Even in the coarsestresolution cases, the mean calculated F_{T,238} values are only off the ideal by 0.3 %.
B3 Noncentered spheres
In their Monte Carlo F_{T} implementation, Herman et al. (2007) report poor precision for small spheres when their centers are not centered in a voxel, with errors rising to several percent for a 40 µm radius sphere with 6.3 µm voxels across a range of center locations (calculated F_{T} range ∼0.58–0.67). Errors of this magnitude correspond to the effect of getting the radius wrong by plus or minus almost an entire voxel.
We tested for voxelation effects on dimensional measurements by running 100 000 trials randomizing the location of the sphere center in a voxel grid using the same radius and voxel size, once again selecting all voxels with centers within the radius of the randomized center. Converting the resulting volumes to sphereequivalent radii, we got a mean radius error of 0 %, maximum radius errors of +0.8 and −1.1 %, and a standard deviation of 0.2 %. At 40 µm (a severe case) a 1 % change in radius leads to a ±0.5 % change in F_{T,238} (range 0.6494–0.6561). Together, these results indicate that the degree to which a sphere is offcenter to the CT voxel grid has only a very small effect on its measured size and a correspondingly smaller effect on the F_{T} determination.
There is a case in which resolution is a concern, however, which is when the grain size approaches the “true” data resolution. All CT data are blurry to some extent due to the finite size of the Xray focal spot and detector elements, among other factors (ASTM, 2011). This blurring can be characterized as a pointspread function (PSF), which can be considered as a smoothing kernel that “blurs” reality as the CT process translates it into a voxel grid. If the smoothing function width, which can be roughly estimated as the number of voxels it takes to fully transition from one material into another across a flat interface (Ketcham et al., 2010), approaches the grain radius, it can affect grain size and shape measurement (Ketcham and Mote, 2019). Typical PSF widths are on the order of 3–5 voxels in most CT data, so as a rule of thumb the voxel size should be limited to less than 20 % of the grain's shortest dimension. Even in this case accurate grain measurements are possible but require additional steps and calibrations, as described by Ketcham and Mote (2019).
We are thus confident that our implementation provides a high degree of accuracy and precision on even very small grains at low resolutions at which voxel sizes are up to 20 % of the radius.
^{1} Sampling is either normal, supersampled, or supersampled maintaining constant volume (cv). ^{2} ESR_{m}: measured equivalent sphere radius, as the voxelated spheres had slightly different volumes than ideal ones. ^{3} ${F}_{T,\mathrm{238},\mathrm{ideal}}$: F_{T,238} value (for the ^{238}U stopping distance for apatite) for the given shape with the voxelated volume and, for cylinders, aspect ratio. ^{4} F_{T,238}: mean measured F_{T,238} value over five trials, with estimated precision in parentheses.
IDL code for conducting offcenter sphere volume test.
This Appendix briefly describes how 3D shape calculations are conducted in Blob3D software (Ketcham, 2005; Ketcham and Mote, 2019), as they apply to measuring grain shape for apatite (or any mineral grain for which a shape analysis is conducted).
The measurement process is illustrated in animation 97BSCR8C.mp4 in the Supplement, which illustrates the shape calculation on several apatite grains in sample 97BSCR8. The measurement process consists of generating a 3D shape and measuring the area of its projection (i.e., outline or shadow) over various angles. The procedure first finds the mean projected area by projecting the shape over a uniform distribution of orientations. It then uses the minimum and maximum projected area found in that sampling as starting points to find the true minimum and maximum projected areas via an optimization algorithm (which looks like “jiggling” the shape in the animation). It then calculates the circularity as the ratio of the maximum projection perimeter to a circle with the same area. The routine then finds the longest caliper dimension (ShapeA) or, in other words, the longest dimension that would be measured in 3D using a caliper. After finding the projection with the longest caliper dimension, the object is rotated around the long axis to find the longest caliber dimension orthogonal to it (ShapeB). The third shape parameter (ShapeC) is the caliper dimension orthogonal to the first two, which is found by rotating the object 90^{∘}. Finally, the procedure uses the same method but in the opposite order, finding the shortest caliper dimension (BoxC), the shortest dimension orthogonal to it (BoxB), and the caliper dimension orthogonal to those (BoxA).
The ShapeABC parameters correspond to the longstanding traditional shape measurement method for rounded or irregular particles (Sneed and Folk, 1958; Wilson and Huang, 1979), but the BoxABC parameters (Blott and Pye, 2008) are more appropriate for regular shapes. For example, for a perfect cube, ShapeA is the longest cornertocorner distance, which will be longer than ShapeB and ShapeC, while BoxA, BoxB, and BoxC will all have the same value: the cube edge length. When measuring an apatite grain, BoxC will usually be the “flattest” part of the hexagonal cross section, BoxB will be the orthogonal cornertocorner distance of the hexagon, and BoxA will be the length in the prismatic direction unless it is fragmented or has a very low aspect ratio.
The supplement related to this article is available online at: https://doi.org/10.5194/gchron1172019supplement.
EHGC collected and processed data, made figures, and contributed to the writing of the paper. RAK initiated the study, processed data, and contributed to the writing of the paper. DFS initiated the study with RAK and contributed to the writing of the paper.
The authors declare that they have no conflict of interest.
We thank Jessie Maisano for acquiring and reconstructing the CT data. These data were collected at the UTCT NSF MultiUser Facility. This paper was improved by helpful reviews from Christoph Glotzbach and two anonymous reviewers.
This research has been supported by the National Science Foundation, Division of Earth Sciences (grant no. 1762458).
This paper was edited by Cecile Gautheron and reviewed by Christoph Glotzbach and two anonymous referees. This work was conducted through Jackson School of Geosciences funds to Daniel F. Stockli and an NSF GRF and WHOI postdoc scholarship to Emily H. G. Cooperdock.
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 Abstract
 Introduction
 Methods
 Results
 Discussion
 Conclusions
 Code and data availability
 Appendix A: Calculating ESR${}_{{F}_{T}}$, mean F_{T}, and eU
 Appendix B: Evaluation of accuracy and precision in Blob3D F_{T} calculations
 Appendix C: Blob3D shape calculations
 Author contributions
 Competing interests
 Acknowledgements
 Financial support
 Review statement
 References
 Supplement
assumed2D versus
true3D grain shapes measured by a microscope and Xray computed tomography, respectively, we find that volume and surface area both differ by ~ 25 % between the two techniques and directly affect mass and concentration measurements. But we found a very small effect on the FT correction (2 %) and no discernible impact on mean sample age or dispersion.
 Abstract
 Introduction
 Methods
 Results
 Discussion
 Conclusions
 Code and data availability
 Appendix A: Calculating ESR${}_{{F}_{T}}$, mean F_{T}, and eU
 Appendix B: Evaluation of accuracy and precision in Blob3D F_{T} calculations
 Appendix C: Blob3D shape calculations
 Author contributions
 Competing interests
 Acknowledgements
 Financial support
 Review statement
 References
 Supplement