(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 (FT) correction, equivalent
sphere radius (ESR), and ultimately isotope concentrations and age. The vast
majority of studies use 2-D or 3-D microscope dimension measurements and an
idealized grain shape to calculate these parameters, and a long-standing
question is how much uncertainty these assumptions contribute to observed
intra-sample age dispersion and accuracy. Here we compare the results for
volume, surface area, grain mass, ESR, and FT correction derived from
2-D microscope and 3-D X-ray 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 2-D versus 3-D FT correction on sample age precision and effective uranium
(eU). The data illustrate that the 2-D 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. FT factors calculated from 2-D and 3-D
measurements differ by ∼2 %. This variation, however, has
effectively no impact on reducing intra-sample age reproducibility, even on
small aliquot samples (e.g., four grains). We also present a grain-mounting
procedure for X-ray CT scanning that can allow hundreds of grains to be scanned
in a single session and new software capabilities for 3-D FT and
FT-based ESR calculations that are robust for relatively low-resolution
CT data, which together enable efficient and cost-effective CT-based
characterization.
Introduction
(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 shape-based alpha ejection
correction (FT 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
well-understood 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 intra-sample 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 FT correction factor prior to
He and U, Th, and Sm analysis, assuming either parent nuclide homogeneity or
prescribing an assumed or measured 1-D or 2-D 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, FT, volume, and surface area are calculated using two or three
grain dimensions (length + width 1± width 2) measured in 2-D on an
optical microscope using imaging software with a micrometer-based
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 FT correction.
However, even in felsic magmatic samples with high-quality 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, inclusion-free grains.
Recognizing that this optical-microscopy 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 2-D grain
photos to model accurate 3-D grain shapes. Another approach is to employ
X-ray computed tomography (CT) to determine accurate grain shapes in an
effort to improve precision and accuracy in FT and (U–Th) / He age
determinations (Herman et al., 2007; Evans et al., 2008; Glotzbach et al.,
2019). Herman et al. (2007) used 3-D CT grain dimensions to calculate
FT 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 2-D microscope measurements
against 3-D 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 2-D versus 3-D grain geometry
measurement techniques on grain dimension, volume, surface area, ESR, mass,
FT, 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
fast-cooling 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 2-D 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 3-D screening. We introduce the capabilities of an updated version
of Blob3D (Ketcham, 2005; freely distributed software) that allows for the efficient
batch processing of CT-scanned grains and outputs parameters such as grain
volume and 3-D FT. We further develop an approach for calculating ESR on
the basis of equivalent FT rather than an equivalent surface-to-volume
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 2-D 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,
high-quality apatite and have been part of previous thermochronometric
studies. Sample 97BS-CR8 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(X2)=75.4 %, 25
grains, Ns=1341) (Surpless et al., 2002). The second sample,
95BS-11.3, 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(X2)=76.1 %, 30 grains, Ns=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 post-magmatic
cooling or fault-related exhumation and are expected to have spent little
time in the apatite He partial retention zone.
MethodsGrain selection and 2-D measurements
Apatite grains were picked from two samples, 97BS-CR8 (n=50) and BS95-11.3
(n=62), using a Nikon SMZ-U/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” 2-D measurement approach with the 3-D CT data. For sample
BS95-11.3, grains were imaged and measured on double-sided 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 2-D measurements. For sample 97BS-CR8 each apatite
grain was placed on a glass slide for 2-D measurements and then transferred
to the sticky tape for the CT mount to remedy this issue (Fig. 1).
Apatite grain photos with 2-D measurements taken on an optical
microscope. Dimensions are reported in micrometers and the grain aliquot name
is in the top left corner of each photo. The top row is photographed on
double-sided sticky tape, and the bottom row is photographed on a glass
slide.
Grain-mounting procedure for CT
Once the grains were measured optically in 2-D, 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 single-layer mount for
multi-grain scanning entails covering a flat top of a pushpin with
double-sided sticky tape that can be precut using a standard hole punch.
Apatite grains are then picked directly onto the tape in a grid-like
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 multi-grain 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 double-sided 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.
Schematic rendering of the CT-mounting procedure. Grains are adhered
to the top of a plastic disk using double-sided sticky tape, with multiple
grains placed onto a 5×5 mm surface. Multiple plastic disk layers with
grains may be assembled and then stacked to take full advantage of the
height of the scan. These layers are held together using parafilm, and a
hash mark on the pushpin enables further orientation of the scan in order to
retrieve the grains afterwards for further analysis.
X-ray CT scanning
The multi-grain mounts were scanned with a Zeiss Xradia MicroXCT scanner at
the University of Texas High-Resolution X-ray 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 X-ray energies are more sensitive to compositional
variations but more prone to beam-hardening artifacts. We experimented with
various settings in this study. The grain mount for sample 97BS-CR8 was
scanned with X-rays set at 100 kV and 10 W, with a 1.0 mm SiO2 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 lower-energy
X-rays and weaker filtering necessitated the application of a beam-hardening
correction during reconstruction. The reconstructed data had a voxel (3-D
pixel) size of 5.03 µm.
The grain mount for sample BS95-11.3 was scanned with X-rays set at 150 kV
and 10 W with a 1.6 mm CaF2 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 beam-hardening
correction was applied during reconstruction. The resulting data had a voxel
size of 4.58 µm.
Example CT slices (a, b) and 3-D renderings (c, d) of
apatite grain mounts for BS95 (a, c) and 97BS (b, d). Arrows indicate two
grains with high-attenuation mineral inclusions in BS95 and a fluid
inclusion in 97BS. The CT slice for 97BS is actually an oblique slice through
the original data to allow all grains to appear in the same image.
Example images from the two datasets are shown in Fig. 3, illustrating
some of the trade-offs. The scan data for BS95 are noisier, primarily due to
the faster acquisition, higher X-ray energy, and more severe filtering. Even
with this level of noise, high-attenuation 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 lower-energy X-ray 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 higher-density grains.
Grain size and shape, FT, mass calculations2-D 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 FT,U
and FT,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 half-width (r) for the radius
of the prism. All equations used for calculating these parameters are
included below or in the Appendix.
Volume (V):
V=3×√32×r2×L,
where L is height and r is the half-width.
Surface area (SA):
SA=6×r×L+3√3×r2,
where L is height and r is the half-width.
Equivalent spherical radius (ESR):
ESR=3×VSA.
Mass:
mass=3.2gcm3×V(mm3)×1000.
FT,U and FT,Th (2-D case; e.g., Farley, 2002):
FT,U=1-5.13×SAV+6.78×SAV2,FT,Th=1-5.9×SAV+8.99×SAV2.
Mean FT (see Appendix for explanation)
from Farley et al. (1996) (used here for 2-D calculations):
FT=a238×FT,U+(1-a238)×FT,Th,
where a238=1.04+0.245×ThU-1.
From Blob3D for this study (used here for 3-D calculations):
FT‾=A238FT,238+A232FT,232+1-A238-A232FT,235,
where A238=1.04+0.247ThU-1 and A232=1+4.21/ThU-1.
Effective uranium concentration (eU) (see Appendix for explanation):
eU=U+0.238Th+0.0012Sm.
(a, b) Rendering of dimension data collected by 2-D and 3-D methods.
Length and width are measured in 2-D using an optical microscope measuring
the long and wide axis of a grain. Blob3D reports the length, width, and
height (BoxA, BoxB, and BoxC) based on the best-fit rectangle for the grain
dimensions. (c) Rendering of the full range of variations in grain
terminations exhibited by the apatite in this study. Highlighted in gray are
potential areas of overestimated volume if an ideal hexagonal prism is
assumed and calculated with 2-D length and width data.
3-D calculations
Our principal 3-D 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 3-D 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 FT, probably similar in
many, but not all, respects to previous work (Herman et al., 2007; Glotzbach
et al., 2019). Stopping distances for 238U, 235U, 232Th, and
147Sm 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 near-uniformly 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 FT factors for each decay chain (FT,238, FT,235,
FT,232, FT,147) are calculated, and a revised method for
calculating mean FT that more precisely accounts for 235U is
provided in Eq. (6) (explanation in Appendix A).
If the resolution of the scan is low with respect to the stopping distance
(238U stopping distance / voxel size < 4), excess surface
roughness effects from voxelation are reduced by super-sampling. The voxels
for each grain and the surrounding voxels are subdivided into 27 (33)
elements, and the super-sampled volume is smoothed with a 5-voxel-wide 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
FT,238 values averaged within 0.2 % of the ideal-shape 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 FT 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 FT 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 3-D data. The first two
are based on the equivalent surface-to-volume ratio (SV) approach (Meesters
and Dunai, 2002). The model-based value ESRSVm uses the BoxA and BoxB
caliper dimensions as L and W for Eqs. (1) through (3), while the 3-D CT-based value ESRSV3D uses the 3-D-measured 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 FT approach; Ketcham et al. (2011) demonstrated than an
equivalent FT sphere provides a more accurate conversion for diffusion
calculations than an equivalent SV one. The set of calculations to determine
the FT-equivalent sphere radius ESRFT are provided in Appendix A.
(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 42-hole 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 3He tracer and purified by a Janis
cryogenic cold trap at 40 K and SAES NP-10 getter prior to measurement of the
4He/3He on a Blazers Prisma QMS-200 quadrupole mass spectrometer.
Final 4He contents were calculated using a manometrically calibrated
4He 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 % HNO3235U-230Th-149Sm spike solution for 90 min at
90 ∘C. After acid digestion, 500 µL of Milli-Q ultrapure
H2O was added to dilute the solutions to ∼5 %
HNO3 and equilibrated for ≥ 24 h prior to analysis. The
solutions were analyzed using a Thermo Element2 high-resolution inductively coupled plasma–mass spectrometer (HR-ICP-MS) equipped with a
50 µL min-1 micro-concentric nebulizer. Final 238U, 232Th, and
147Sm values were blank-corrected 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 long-term intra-laboratory analysis of apatite age standards. Corrected
final ages are determined by dividing the uncorrected age by the mean
FT 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).
Scatter plots for volume, surface area, eU,
mass, mean FT, ESR, and age for both samples. Both
samples are plotted together unless otherwise noted. Each
data point represents a single apatite aliquot. Dashed lines
represent the percent difference from the 1:1 line (black line). 3-D
data measurements are plotted on the y axis in all plots. 2-D
measurements overestimate volume, surface area, and
mass and underestimate eU and mean FT.
Results
CT scanning combined with Blob3D analysis provides 3-D grain-specific volume,
surface area, dimensions, and FT factors for each decay chain. The 2-D optical measurements provide dimension information, which is used to
calculate volume, surface area, FT,U, and FT,Th based on an assumed
grain geometry of an equidimensional hexagonal prism (all results are
reported in the Appendix). We assume that the 3-D-measured volume and FT
values are sufficiently accurate to benchmark the 2-D 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 2-D and 3-D results can only be compared in a relative sense for surface area.
ESR(SVm): BoxA and BoxB assuming hexagonal prism shape, ESR(SV3-D): Blob3D volume and SA measurements, ESR(FT): FT-equivalent sphere.
2-D and 3-D data are compared for each sample and as an entire population in
Tables 1 and 2. The average 3-D / 2-D ratio of each parameter is reported with
its 1σ standard deviation. This average ratio shows whether the 2-D measurements on average overestimate (ratio < 1) or underestimate
(ratio > 1) the 3-D measurements. Also reported is the absolute
percent difference between the 2-D and 3-D measurements to illustrate the
magnitude of deviation between the measurements. While comparing 2-D and 3-D results, it became apparent that one 2-D 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 (97BS-CR8-1) was not included when calculating the average
differences between 3-D and 2-D measuring techniques.
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 2-D length measurements are on
average ∼2 % smaller than the 3-D BoxA dimension. On the
other hand, the 2-D width dimension is on average ∼3 %
greater than the 3-D BoxB dimension (Table 1).
One inevitable source of uncertainty in 2-D 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 CT-derived 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 lab-by-lab 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.
Volume and surface area
Volumes and surface areas calculated using the 2-D microscope dimensions both
average ∼20 % larger than the 3-D calculations (3-D / 2-DVOL=0.82, 3-D / 2-DSA=0.81)
(Table 1, Fig. 5). Specifically, 2-D 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 3-D Blob3D-calculated volumes and surface
areas.
Workflow diagram showing the effect of volume and surface area
measurements on other parameters used for (U–Th) / He age calculation and
interpretation. The average absolute differences between 2-D and 3-D measurements for each of the parameters are reported with their 1σ
uncertainties (reported in Table 1). Note that the greatest deviations are in
volume and surface area, as well as parameters that rely on volume alone.
ESR, FT, and ages deviate less because they use SA/V, which is
∼1 between 2-D and 3-D measurements.
ESR and mass
The 2-D ESR is calculated using the surface-area-to-volume 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 3-D data had the
ESR calculated based on SA/V in three ways. First, the SA/V for ESRSVm
is calculated using the BoxA and BoxB values provided by Blob3D and assuming
a hexagonal prism, mimicking the 2-D approach. The variation between 2-D and
3-D ESRSVm measurements has a 2σ spread of ±12 %, but
the variability is fairly evenly split in overestimating and underestimating the
ESR such that the average 3-D / 2-D ratio is 1.02. Second, the 3-D SA/V is
calculated using the surface area and volume measurements output by Blob3D
(ESRSV3D). The variation between 2-D and 3-D ESRSV3D is even larger
at ±18 % (2σ), with an average 3-D / 2-D ratio of 1.01 (Table 1, Fig. 5).
The FT-based ESR was on average similar to the SV-based one
(ESRFT/ ESRSVm=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 ESRFT value with the 2-D data is ±14 %, similar to that for the other 3-D ESR calculations (Table 1, Fig. 5).
The grain mass is calculated from the volume data using a nominal apatite
density, and therefore 2-D and 3-D mass determination directly reflect the
variability in the 2-D and 3-D volume data. The 2-D approach consistently
overestimates the mass, with a high degree of scatter (3-D / 2-D =0.82±0.44 (2σ)) (Table 1, Fig. 5).
FT corrections
FT,U and FT,Th correction factors calculated from the 2-D data are
generally 1 %–2 % lower than the Blob3D U and Th FT factors. To combine
the FT factors into a single term that is applied to the (U–Th) / He age,
a mean FT was calculated in two ways using Eq. (6) (see Methods). This
results in mean FT factors that vary by an average of 2 % between the
2-D and 3-D datasets. The 1σ scatter in 3-D / 2-D FT factors is 1.8 %,
though individual differences can reach up to 9 % (Table 1, Fig. 5).
(U–Th) / He age and effective uranium
We calculated the apatite (U–Th) / He-corrected age by dividing the raw
(U–Th) / He age by the mean FT factor. The 2-D FT (U–Th) / He ages tend
to be slightly older than the 3-D FT (U–Th) / He ages (3-D / 2-D =0.99)
owing to the fact that the 2-D FT values are slightly lower, leading to a
larger correction (Table 1, Fig. 5). The average difference between the 2-D
and 3-D FT-corrected ages is 2 %, mimicking that of the variation
between 2-D and 3-D FT (full range is < 1 % to 9 %). This has an
insignificant impact on the mean age and uncertainty for both samples.
Sample 97BS-CR8 has a 2-D FT mean age of 56.8±2.9 Ma and a
3-D FT mean age of 56.0±2.9 Ma (Table 2, Fig. 5). Sample
BS95-11.3 has a 2-D FT age of 12.2±4.0 Ma and a 3-D FT mean
age of 12.1±4.0 Ma (Table 2, Fig. 5).
The effective uranium concentrations (eU = [U] + [Th] ×0.238+
[Sm] ×0.0012) for the apatite are normalized to the mass of the grain. Since
2-D and 3-D grain mass calculations varied by ∼25 %, the eU
concentration measurements vary by a similar degree (3-D / 2-D =1.29±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.
DiscussionAccuracy of 2-D vs. 3-D grain measurementsVolume and surface area
One of the main motivations behind this study was to assess the accuracy of
2-D grain measurements and using an assumed grain geometry for calculating
grain parameters (volume, ESR, mass, FT) 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 2-D and 3-D measurements are in the
volume and surface area, which 2-D 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 2-D-based volumes were 30 %
greater than the 3-D volumes (Table 3). Our dataset contains > 100
apatite grains, implying that the 2-D overestimation of volume (and therefore
mass) may be systematic in the 2-D measurement approach. In contrast,
Glotzbach et al. (2019) analyzed 24 apatite grains and found that the 2-D
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
grain-by-grain basis, including ellipsoids for anhedral grains and
accounting for terminations using the functions provided in Ketcham et al. (2011), rather than assuming exclusively flat-terminated 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,
equal-sided hexagonal cross section perpendicular to the c axis, for which
the ratio of width to height should be 2/3, or 1.1547. The 3-D 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 2-D. 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 2-D and 3-D volume in our data. The remaining deviation may be due to chipped crystals,
surface roughness, or other deviations from a perfect prism that the 2-D calculation cannot account for.
A number of factors will directly impact surface area calculations. Surface
area is calculated from the 2-D 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 3-D 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 3-D measurement process of segmentation by thresholding can
lead to artificial enhancement of surface area due to voxelation effects
(the 3-D equivalent of pixilation).
In our data, the 2-D measurements consistently result in a higher surface
area than the 3-D measurements. This is probably partly due to the
∼5µm resolution of our CT data and also to the
flat-terminated hexagonal prism model leading to an overestimate. Evans et
al. (2008) observe a similar discrepancy in surface area measurements
between 2-D and 3-D data (2-D ∼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 2-D measurements gave surface areas on average 8 % lower than 3-D (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 equal-sided hexagonal prism. The
overshoot may be in part due to their higher CT data resolution capturing
roughness better, but their 3-D 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 ∼20µm smoothing
filter, so short-length-scale roughness has a negligible effect on alpha
particle retention and FT calculation. This point is demonstrated by
our sensitivity analysis (Appendix B), which shows that a bumpy, voxelated
sphere has the same FT 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.
Mass and eU
The discrepancy in volume between 2-D and 3-D measurements directly impacts
the mass calculation, causing the grain masses derived from the 2-D measurements to be ∼25 % higher than the 3-D grain mass
determinations (Fig. 6). Evans and others (2008) found similar deviations,
with their masses calculated from 2-D volumes ∼30 % greater
than their masses for 3-D volumes (Table 3). Both of these divergences stem
from using the assumption of a flat-ended hexagonal prism, whereas an
approach that takes grain shape into account when choosing the FT
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 3-D mass data and 2-D mass data shows that the 2-D masses
underestimate the bulk eU concentrations by ∼20 %–30 %. This
is consistent with the 2-D mass data being ∼25 % higher than
the 3-D 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 2-D analysis (±44 % (2σ)) can be expected to muddy any age–eU
correlation that may be present.
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 ESRFT 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 resolution-resistant way, has only secondary importance for diffusion
and FT calculations when it varies on a fine scale compared to the
grain (i.e., micrometer-scale roughness). Analogously with mass, excess
variation in ESR (±14 % (2σ)) can degrade age–size
correlations.
FT
A somewhat surprising result of our study is that, despite volume and
surface areas being very different between the 2-D and 3-D methods, these
differences largely canceled each other out in S/V-based FT
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 2-D and 3-D FT values increased, although it
remained modest. The standard deviation of 3-D / 2-D FT,U was 2.7 % for
grains with FT,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 2-D and 3-D Mean FT, 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 3-D / 2-D FT less than 3 %.
Reproducibility of (U–Th) / He ages
In addition to assessing the accuracy of using the 2-D measurements, this
study aimed to quantify the uncertainties that may be introduced by such
measurements, particularly in FT, as a means to potentially improve age
accuracy, precision, and intra-sample dispersion. Previous studies have
estimated that uncertainties in FT 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 FT calculation are on the order of 1 %–3 %, and mean
FT 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
FT and an assumed grain geometry can increase.
Our data also show that the 3-D FT correction does not increase the
overall sample age precision for the samples in this study. For sample
97BS-CR8, 24 apatite grains were analyzed, two of which are outliers. Of the
two outliers, one (97BS-CR8-1) was clearly caused by a user error during
microscope measurement, leading to an incorrect FT correction (0.55)
and old age (78.8 Ma). This was discovered during 3-D image processing, in
which the same grain was identified, measured correctly, and produced an
FT of 0.76 and a more congruent corrected age of 57.2 Ma. In contrast,
for a second outlier (97BS-CR8-24), the 2-D and 3-D FT-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 2-D and 3-D 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 3-D and 2-D data for 95BS-11.3
(n=59 aliquots) are indistinguishable at 12.2±4.0 and 12.1±4.0 Ma, respectively. Unlike sample 97BS-CR8, there was no clear-cut
evidence of user error, and the relatively high age uncertainty (33 %) is
reproducible between the 2-D and 3-D FT-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.
Histograms showing the 3-D / 2-D mean
FT and 2-D and 3-D (U–Th) / He age. Histograms show the
results of randomly subsampling four grains from each sample 1000 times. For
mean FT and sample age, the subsampling is
indistinguishable from the mean of the whole population analyzed. Note:
numbers on x axis refer to the bars (bins) and not the tick marks.
In addition to the above calculations, we randomly subsampled four grains
1000 times to assess the variability in FT-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 3-D FT can provide a more
accurate FT correction and varies from 2-D 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 often-cited 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 intra-sample dispersion, such
as U–Th zonation, and/or excess He from nano-inclusions or high U–Th
neighbors.
Effects of inclusions or broken grains
It is widely accepted that inclusions and broken grains are both
contributors to intra-sample 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 HNO3 digestion (e.g., Ehlers and Farley, 2003). Both apatite
samples had multiple grains with high-density and low-density 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–Th-bearing inclusions. For future
studies, an added benefit of CT is the detection of high- and low-density
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.
Benefits and limitations of X-ray CT over microscope measurements
This study purposefully selected “high-quality” apatite from fast-cooled
plutonic samples to quantify the base uncertainty introduced by 2-D measurements and grain shape assumptions on FT and (U–Th) / He age
factors. Although we found that 3-D grain characterization techniques did not
reduce intra-sample age dispersion in our samples, it is still highly
probable that the 3-D 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 X-ray data, which may not be visible by
the naked eye. Furthermore, the CT-mounting 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, FT, mass, and ESR at once for several
samples and phases. Furthermore, the 3-D FT and FT-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
volume-based 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 3-D 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 micro-CT. 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.
Conclusions
The shape and size of 109 apatite grains from two rapidly cooled plutonic
samples were analyzed by 2-D and 3-D methods. 2-D length and width measurements
made on an optical microscope were used to calculate surface area, volume,
ESR, mass, and FT assuming an ideal equal-sided, flat-terminated
hexagonal prism grain shape. The same apatite crystals were scanned using
X-ray 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 2-D versus 3-D FT correction factors on final
(U–Th) / He age and reproducibility. With these data, we derive the following
conclusions.
Deviations between 2-D and 3-D 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.
2-D FT 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 2-D and 3-D FT corrections. Similarly, the amount of intra-sample dispersion
was identical (both > 5 %). This implies that factors other
than FT dominate the intra-sample 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 3-D FT and shape measurement functions, and new calculations for eU and
ESRFT.
Code and data availability
The code and data are available in the
Supplement to this paper. CT data are archived at 10.17612/CZYH-KC13 (Ketcham and Cooperdock, 2019).
Calculating ESRFT, mean
FT, and eUESRFT and mean FT
The starting point for calculating the equivalent FT sphere radius
(ESRFT) when FT values are provided for each decay chain is the FT equation for a sphere (Farley et al., 1996; Ketcham et al., 2011):
FT=1-34SR+B16SR3,
where R is the sphere radius, S is stopping distance, and B is an adjustment factor
for the 3rd-degree 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 FT range from 0.5 to 1 using a
3rd-degree 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 FT value to use is the weighted mean incorporating the separate
factors FT,238, FT,235, and FT,232, accounting for different
alpha productivity along each chain. Expanding the approach of Farley (2002)
to account precisely for 235U, we calculate
A3aA238=(1.04+0.247Th/U)-1,A3bA232=(1+4.21/Th/U)-1,
so that the weighted mean, FT‾, is
FT‾=A238FT,238+A232FT,232+1-A238-A232FT,235.
Solving the result of Eq. (A2) for ESRFT requires the analogous
calculation to determine the weighted mean stopping distance, S‾:
S‾=A238S238+A232S232+1-A238-A232S235,
where S238, S235, and S232 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
ESRFT=S‾/SR.
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
eU=U+0.235[Th],
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 present-day alpha production rate Rα (here: α g-1 yr-1) as
Rα=AλpN/ma,
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
ma is atomic mass. The eU factor is then calculated by dividing the Th
and Sm Rα by the combined U Rα utilizing the values in
Table A1; we find the eU equation to be slightly different:
eU=[U]+0.238[Th]+0.0012[Sm](or0.0083[147Sm]).
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 232Th half-life 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):
Rk=AλpNE/ma,
where E is the mean alpha particle recoil energy for the decay chain. The
revised kerma-based quantity, eUk, is then
eUk=U+0.264Th+0.0005Sm(or0.0034[147Sm]).
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 238U. We
do not currently recommend this approach, but it does pose a potentially
testable hypothesis.
* Values for U and Th from Steiger and Jäger (1977).
Evaluation of accuracy and precision in Blob3D
FT calculations
This Appendix describes a series of tests that demonstrate the accuracy and
precision of the methods for FT 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.
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
1283-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 1-voxel-thick black boundary on all sides. Four additional
lower-resolution versions were then created by rebinning the original dataset to make volumes with 643, 323, 163, and 83 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 8-bit 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 1283 dataset, the resulting
ideal sphere radius is 63 µm, which has an FT,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 1283 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 653-voxel dataset,
4 µm for 333, 8 µm for 173, and 16 µm for
93. 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 (FT,238=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) FT 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. Half-tone
symbols show the result of altering the processing by first super-sampling
the volume, subdividing each voxel into a 33 set, and then smoothing the
expanded data volume with a 5-voxel-wide filter, followed by re-binarizing
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 sub-0.1 % error at the 4 µm
level. However, the 127 re-threshold 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 pre-super-sampled 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 FT calculation; this
principle is used to examine the case of non-centered spheres later in this
Appendix.
Remaining tests use the convention that when voxel sizes are 4 µm or
higher the constant-volume super-sampled approach is used; the only cost of
super-sampling 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.
Cylinders
As most apatite (and zircon) grains are elongate, we also tested cylinders
as a close-to-worst-case end-member, again because a round outline is more
poorly approximated by cubes than a hexagonal or tetragonal one. We created
the cylinders by stacking 510 63-voxel-radius circles with blank slices at
each end to achieve an aspect ratio close to 4 and down-sampled 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 FT,238 values of
0.8350 and 0.6772, in Table B1c–d and Fig. B1c–d. Even in the
coarsest-resolution cases, the mean calculated FT,238 values are only
off the ideal by 0.3 %.
Non-centered spheres
In their Monte Carlo FT 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 FT 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 sphere-equivalent 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 FT,238 (range 0.6494–0.6561). Together, these
results indicate that the degree to which a sphere is off-center to the CT
voxel grid has only a very small effect on its measured size and a
correspondingly smaller effect on the FT 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 X-ray focal spot and detector
elements, among other factors (ASTM, 2011). This blurring can be
characterized as a point-spread 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.
Results of Blob3D measurement of synthetic spheres and
cylinders.
1 Sampling is either normal, super-sampled, or super-sampled maintaining
constant volume (cv).
2 ESRm: measured equivalent sphere radius, as the voxelated spheres
had slightly different volumes than ideal ones.
3FT,238,ideal: FT,238 value (for the 238U stopping distance for
apatite) for the given shape with the voxelated volume and, for cylinders,
aspect ratio.
4FT,238: mean measured FT,238 value over five trials, with
estimated precision in parentheses.
Results of Blob3D measurement of synthetic spheres and
cylinders.
IDL code for conducting off-center sphere volume test.
Blob3D shape calculations
This Appendix briefly describes how 3-D 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 97BS-CR8C.mp4 in the
Supplement, which illustrates the shape calculation on several
apatite grains in sample 97BS-CR8. The measurement process consists of
generating a 3-D 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 3-D 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 long-standing 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
corner-to-corner 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 corner-to-corner 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/gchron-1-17-2019-supplement.
Author contributions
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.
Competing interests
The authors declare that they have no conflict of interest.
Acknowledgements
We thank Jessie Maisano for acquiring and reconstructing the CT data. These
data were collected at the UTCT NSF Multi-User Facility. This paper was improved by helpful reviews
from Christoph Glotzbach and two anonymous reviewers.
Financial support
This research has been supported by the National Science Foundation, Division of Earth Sciences (grant no. 1762458).
Review statement
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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