Grain selection
Grains are hand-picked with fine-tipped tweezers from mineral concentrates and selected on the basis of
morphology, size, optical clarity, and, ideally, compositional uniformity. Euhedral crystals or grains with
morphologies that can be geometrically approximated as one of several typical morphologies, such as
hexagonal or orthorhombic prisms with pryamidal terminations, are favored, because these simplify alphaejection corrections. Highly irregular or rounded grains can also be dated if such corrections are not needed
or can be applied to prolate spheroids. Especially in the case of apatite, larger grains are generally favored
over smaller ones, because this provides higher analyte contents that often, but not always, lead to better
analytical precision, and minimizes consequences of inaccurate alpha-ejection corrections. Selection of
grains based on these criteria is usually accomplished with a high-powered (~120-160x) stereo-zoom
microscope and/or SEM, often with the aid of digital photographic documentation and measurement.
Finally, grains may be selected on the basis of compositional uniformity, particularly with respect to U and
Th, to minimize complications related to alpha-ejection corrections, perturbations to He diffusion profiles,
and potential intragrain He diffusivity variations arising from radiation damage (Farley et al., 2011). Grain
selection on this basis may be accomplished using electron microbeam or CL imaging, or LA-ICP-MS,
including on grain interiors (e.g., Hourigan et al., 2005).
4
He measurement
Helium is extracted from aliquots using laser heating using long-wavelength lasers such as those with diode
(~800-900 nm), fundamental Nd:YAG (1064 nm), or CO2 (10.6 um) sources. Nb foil tubes are used as
"microfurnaces" to contain analyzed minerals. Crystal-bearing foil tubes (and also empty ones to check 4He
blanks) are individually heated under a laser beam to about 900-1300 °C for several minutes in an ultrahigh vacuum chamber typically pumped to <10-8 Torr using turbomolecular and/or ion pumps. The UHV
line itself is made of polished stainless steel. Valves with Cu seats control gas flow through the line. After
release from the sample, gas is spiked with 3He for measurement by isotope dilution. Released gas (plus or
minus the 3He spike) is purified by heated alloy-metal getters and/or a temperature-cycled cryogenic coldhead in the UHV line (Lott and Jenkins, 1984), to improve ionization and resolution of the 3He, and in some
cases 4He, peaks. Purified gas is released into the ioninzing source of a quadrupole mass spectrometer. 4He
blanks are typically 0.01-0.1 fmol. The intensity of the mass 3 peak is corrected for isobaric interferences
from HD and H3+ based on correlations between mass 1 (inferred to be H+) and mass 3 intensities when no
He is present in the system. Because of the large mass fractionation, measured 4He/3He ratio cannot be
directly converted to 4He contents simply using an estimate of the spike's 3He content and 4He/3He.
Quantification of 4He, therefore, is done by first measuring 4He/3He on a 3He-spiked reference sample of
4
He representing a known amount 4He. Assuming a linear response between He moles and ratios across
some range of confidence, the molar 4He content of an unknown can then be calculated as the ratio of the
4
He/3He measurements of the sample and spiked 4He reference standard, multiplied by the moles of 4He
delivered in the reference standard. This reference standard is analyzed, using the same procedures as
unknown samples, several times during an analytical session, to account for changes in isotopic
fractionation or sensitivity bias with a variety of causes. The molar 4He reference standard is delivered
through a small (0.1-1.0 cm3) pipette attached to a larger volume tank of 4He. With knowledge of the
volumes of the standard tank and pipette, as well as the initial molar 4He content inside the standard volume,
the 4He delivery of the reference standard and its depletion as a function of number of shots taken can be
calculated. Measurement of these reference volumes is performed by gas expansion and absolute pressure
measurement using capacitance manometry, and correcting for thermal transpiration effects (Setina, 1999).
U-Th-Sm-Zr-Ca measurement
Molar contents of parent nuclides in apatite and zircon (and Ca in apatite or Zr in zircon) are measured
following 4He extraction and measurement via isotope-dilution using ICP-MS. Post-heating, crystalbearing metal tubes are removed from the UHV line, placed in plastic containers, and spiked with nitric
acid solutions of U and Th (±Sm±Ca±Zr) isotopes enriched in 233U or 235U and 229Th or 230Th
(±147Sm±42Ca±90Zr). Acid solutions are then added and heat is applied to dissolve the apatite or zircon
contents of the tubes and mix the aliquot with the isotopically enriched spike. Apatite can be dissolved by
relatively dilute HNO3 at temperatures less than ~100 °C for an hour or so. Zircon and some other minerals
require more aggressive techniques, such as serial HF-HNO3 and HCl solutions in high-pressure dissolution
vessels at temperatures above 200 °C. Isotope ratio measurements made by ICP-MS can be converted to
molar contents in aliquots using standard isotope dilution equations. Isotopic compositions and elemental
concentrations of the spike solutions can be determined for conditions similar to those used to measure the
unknowns by measuring "spike blanks" and spiked solutions of natural elements with known
concentrations. Ca and Zr quantified by isotope dilution or peak height standardization are used to
stoichiometrically estimate the mass of crystal matrix (e.g., apatite or zircon) in each aliquot. This is not
necessary for calculating dates, as this is done on a molar basis for parent and daughters. But knowledge of
crystal matrix mass allows for calculation of parent and daughter concentrations in aliquots, which can
provide complementary constraints bearing on provenance, petrogenesis, or radiation damage effects.
Alpha ejection corrections
For samples that are internal fragments of much larger domains with reasonably uniform parent nuclide
concentrations (e.g., Durango apatite), long alpha stopping distances may be ignored. This is because any
4
He produced within the aliquot but "ejected" out of it would presumably be balanced by "implantation" of
4
He from adjacent regions. However, typical apatite or zircon crystals have U and Th concentrations higher
than adjacent minerals in their host rocks. They also have dimensions (~50-200 µm) that are not many times
the lengthscale of alpha stopping distances. Therefore some significant fraction of alpha particles created
within them will have been ejected from the crystal, and any age calculated directly from the measured (UTh)/He will be younger than the "true" formation or cooling age, regardless of whether 4He was lost via
diffusion.
In most situations the most straightforward way to deal with this is to "correct" the measured (U-Th)/He
age in a way that effectively adds back in the 4He that have been lost to extragranular ejection. Because
alpha stopping distances are known or can be calculated relatively precisely for each nuclide and for each
mineral (Zeigler, 2008), the appropriate correction for each nuclide can be calculated as a function of the
size and morphology of the dated aliquot. For parent nuclides with multiple intermediate daughter products,
a mean stopping distance can be assumed (Ketcham et al., 2011).
For an idealized spherical crystal, Farley et al. (1996) showed that the fraction of 4He with stopping distance
S that come to rest within a sphere of radius R (i.e., the fraction of the total produced, hence FT) will be
3
4
𝑆𝑆
𝑅𝑅
1
16
𝑆𝑆 3
𝐹𝐹𝑇𝑇 = 1 − � � � � + � � �𝑅𝑅3 �.
for large spheres (R>>S) this simplifies to
𝑆𝑆
4
𝐹𝐹𝑇𝑇 = 1 − � � 𝛽𝛽,
where β is the surface area to volume ratio of the sphere (because for a sphere β = 3/R). For more complex
and realistic morphologies Monte Carlo models can be used to fit polynomial expressions for FT values for
a given nuclide and a given grain morphology, as a function of β, as shown below. Farley et al.'s (1996)
polynomial expression for Ft for a given nuclide i is
𝐹𝐹𝑇𝑇𝑖𝑖 = 1 + 𝐴𝐴1 𝛽𝛽 + 𝐴𝐴2 𝛽𝛽 2,
where A1 and A2 are coefficients derived from generalized fits to alpha ejection observed in Monte Carlo
models for a grain of a given morphology. Coefficients for apatite (Farley, 2002) and zircon (Hourigan et
al., 2005) commonly used in this approach are shown in Table X, and surface-area and volume formulae
for idealized morphologies commonly used for routine apatite and zircon crystals are given in Table 1.
Table 1. Factors A1 and A2 for calculating fraction of He retained in crystals from the 238U and 232Th decay
series in zircon, for different assumed crystal geometries. In these formulations the 235U and 232Th stopping
distances are considered similar enough to yield equal polynomial coefficients for FT calculations
Tet. prism with pinac.
terminations (Farley 2002)
Hex.
Prism
with
pinac.
terminations (Farley 2002)
Tet.* prism with pyramidal
terminations (Hourigan et al. 2005)
Parent Nuclide
A1
A2
A1
A2
A1
A2
238
U
-4.31
4.92
-5.13
6.78
-4.28
4.37
232
Th
-5.00
6.80
-5.90
8.99
-4.87
5.61
*Tetragonal prism morphology was used for Monte Carlo modeling but use of an orthorhombic prism with
c-axis-perpendicular variation within the range of most zircons produced similar results.
Table 2. Surface areas and volumes of idealized morphologies similar to those commonly encountered in
routine apatite and zircon (U-Th)/He dating.
Geometry
Volume
Surface Area
Orthorhombic
1
SAz = 4(l − h1 − h2 )(r1 + r2 ) + 2r1 a + 2r2 b
V z = 4r1 r2 (l − h1 − h2 )+ h1 h2
prism
with
3
a = h12 + r22 + h22 + r22
pyramidal
b = h12 + r12 + h22 + r12
terminations
Prolate
Spheroid
( )
( )
2 


l
2
π
r
 −1 

2
2
SA ps = 2πr + 
 sin 
2
2
 l

−r 
2



2
V ps = πr 2 l
3
(l 2) − r
(l 2)
2
2
Hexagonal
3√3 2
3√3 2
𝑉𝑉ℎ𝑝𝑝 =
𝑤𝑤 𝑙𝑙
𝑆𝑆𝑆𝑆ℎ𝑝𝑝 = 3𝑙𝑙𝑙𝑙 +
𝑤𝑤
prism
8
4
Note: l = c-axis-parallel length; h1, h2 = pyramidal termination lengths; r1, r2 = mutually-perpendicular
prism half-widths or average equatorial radius, w = distance between mutually opposed apices parallel to
c-axis in hexagonal prism.
The Farley (2002) and Hourigan et al. (2005) approaches described above and in Tables 1, 2 assume a
single set of coefficients for 232Th and 235U, because the mean alpha stopping distances in their decay chains
are the same to within 2% in both apatite and zircon. FT for 147Sm can be approximated by scaling the FTs
for 238U and 147Sm in the Ketcham model (see below) for typical apatite and zircon dimensions:
apatite:
147
𝐹𝐹𝑇𝑇
𝑆𝑆𝑆𝑆
147
𝐹𝐹𝑇𝑇
𝑆𝑆𝑆𝑆
238
= −0.09158 ∙ �𝐹𝐹𝑇𝑇
𝑈𝑈
238
= −0.00779 ∙ �𝐹𝐹𝑇𝑇
2
238
� + 0.46974 ∙ 𝐹𝐹𝑇𝑇
𝑈𝑈
zircon:
𝑈𝑈
2
238
� + 0.45193 ∙ 𝐹𝐹𝑇𝑇
𝑈𝑈
+ 0.61895;
+ 0.62526.
Natural apatite crystals are often cleavage-fractured perpendicular to the c-axis, which may result in loss of
the original c-axis-perpendicular terminations. Assuming this fracturing happened after the thermal history
recorded in the He content of the grain (e.g., during mineral separation), and that it removed more than one





alpha-stopping distance, the surface-area-to-volume of the grain should be modified to reflect the loss of
some of the alpha-ejection affected surface. Farley (2002) showed that the alpha-ejection-related
consequences (but not the diffusion profile consequences) of this can be conveniently done by multiplying
the measured length by 1.5 or 2, for one or two missing tips, respectively, in the equation for the surfacearea-to-volume ratio,
8
𝛽𝛽 =
√3𝑤𝑤
2
𝑤𝑤1
2
𝑤𝑤2
2
𝑙𝑙
+ .
Analogously, zircon crystals approximating orthorhombic prisms that have lost their (usually pyramidal)
terminations, have surface-area-to-volume ratios expressed by:
1
𝑙𝑙
𝛽𝛽1𝑚𝑚 = +
+
; 𝛽𝛽2𝑚𝑚 =
2
𝑤𝑤1
+
2
,
𝑤𝑤2
Where the subscripts 1m and 2m refer to the number of missing terminations, l is the c-axis-parallel length,
and w1 and w2 are the c-axis-perpendicular widths of the prism.
A different but similar adjustment to the surface-area-to-volume ratio must be made to grains that have lost
significant surface area along a face parallel to the c-axis. This may occur naturally or upon mineral
separation, but more commonly it arises with grains plucked from grain mounts as might be made for
fission-track dating or microanalytical techniques such as ion probe or LA-ICP-MS. For apatite, a modified
surface-area-to-volume ratio can be estimated assuming that polishing was subparallel to the c-axis, and
removed more than one alpha-stopping distance and less than half of the entire width of the crystal (Reiners
et al. 2007). Under these conditions, the modified surface-area to volume ratio of the polished grain (for an
assumed pre-polishing cylindrical geometry with pinacoidal terminations and not including the surface area
of the polished face) is
β=
2
2 rω
,
+
l r 2ω + (r − d ) 2rd − d 2
Where d is polishing depth, l is crystal length, r is c-axis perpendicular half-width (radius), and


ω = π − cos −1 1 −
d
.
r
A similar and simpler approach is also possible for zircon, if the pyramidal terminations are ignored, so that
β=
2 1
1
+ +
,
l r1 2r2 − d
where r1 and r2 are the c-axis-perpendicular half-widths of the crystal parallel and normal to the polishing
directions, respectively.
Ketcham et al.'s (2011) more recent and slightly more generalized treatment of alpha ejection correction
casts the general equation for the numerical fits to specific morphologies somewhat differently than Farley
et al. (1996):
3
4
𝑅𝑅
𝑅𝑅𝑆𝑆
𝑅𝑅 2
𝑅𝑅𝑆𝑆
𝑅𝑅 3
𝑅𝑅𝑆𝑆
𝐹𝐹𝑇𝑇 = 1 − � � � � + 𝑏𝑏2 � � + 𝑏𝑏3 � � ,
where in this case R is the stopping distance and Rs is the radius of a sphere with the equivalent β of the
grain of interest. As in the previous cases, the polynomial coefficients are determined from Monte Carlo
models for specific morphologies. The Ketcham et al. approach has the advantage of casting the formulation
as one in which the stopping distance R is explicit in the expression so that it may be refined or adapted to
any nuclide, even if the expressions for the combined coefficients and Rs can become a little complicated.
For example, for apatite with a generalizeable morphology as shown in Fig. 1 (and pyramidal terminations,
if present, constrained to be at 45° angles from the c-axis),
Figure 1. Schematic of idealized crystal morphologies for zircon and apatite, showing dimensional
parameters used in example Ft equations x and x. From Ketcham et al. (2011).
the FT equation is
3 𝑅𝑅
𝐹𝐹𝑇𝑇 = 1 − � � � �
4 𝑅𝑅𝑆𝑆
+ �(0.2093 − 0.0465𝑁𝑁𝑃𝑃 ) �𝑊𝑊 +
𝐿𝐿
√3
�
𝑊𝑊 √3⁄2 + 𝐿𝐿 𝑅𝑅 2
+ �0.1062 +
� �𝐻𝐻 − 𝑁𝑁𝑃𝑃
��
4
𝑉𝑉
𝑅𝑅 + 6�𝑊𝑊√3 − 𝐿𝐿�
0.2234𝑅𝑅
where NP is the number of pyramidal terminations, V is the volume of the crystal, and other variables are
as shown in Fig. 1. For zircon with morphology of an orthorhombic prism with or without pyramidal
terminations (constrained to be at 45° angles from the c-axis), FT can be adequately described as
𝑅𝑅 2
3 𝑅𝑅
𝑎𝑎2 + 𝑏𝑏 2
(𝑎𝑎
𝐹𝐹𝑇𝑇 = 1 − � � � � + �0.2095(𝑎𝑎 + 𝑏𝑏 + 𝑐𝑐) − �0.096 − 0.013
+
𝑏𝑏)𝑁𝑁
�
�
𝑃𝑃
𝑐𝑐 2
𝑉𝑉
4 𝑅𝑅𝑆𝑆
where the variables are as in the previous case and shown in Fig. 1.
Figure 2 shows FT values for apatite and zircon for a range of crystal sizes, calculated using the Farley
(2002), Ketcham et al. (2011), and Hourigan et al. (2005) methods.
Figure 2. FT values for apatite and zircon grains with dimensions on an idealized crystal morphology as
shown on left side. Trend labels are shown in inset; F = Farley (2002); H = Hourigan et al. (2005); K =
Ketcham et al. (2011). These examples show results for grains with L/W = 2.67 (near the midpoint of the
distribution shown in Farley et al. (1996) for typical apatite grains. The x-axis is Rs, the radius of a sphere
with an equivalent surface-area-to-volume ratio as the grain. In these examples, apatite c-axis
perpendicular width is Rs/0.5786; zircon c-axis perpendicular with is Rs/0.6319. Apatite and zircon in the
Ketcham et al. (2011) model require assumptions of pyramidal tips at 45° from prisms, but can
accommodate grains with one or no pyramidal tips. Farley et al. (2002) and Hourigan et al. (2005) models
do not explicitly calculate FTs for147Sm, and assume the same FTs for 235U and 232Th, based on the very
similar average stopping distances of these two nuclides.
A mean FT for a specific aliquot with specific molar proportions of parent nuclides can be calculated from
the weighted FTs for each nuclide specific to the size and shape of the grain, based on the relative alpha
productivities and the concentrations of each element,
𝐹𝐹𝑇𝑇𝑚𝑚 =
∑𝑘𝑘 𝐹𝐹𝑇𝑇𝑖𝑖 𝑀𝑀𝑘𝑘 𝐴𝐴𝑖𝑖𝑘𝑘 𝜆𝜆𝑖𝑖𝑘𝑘
∑𝑘𝑘 𝑀𝑀𝑘𝑘 𝐴𝐴𝑖𝑖𝑘𝑘 𝜆𝜆𝑖𝑖𝑘𝑘
where FTi , Aki, and λki are the mean FT, atomic abundance, and decay constant of the nuclide of interest,
respectively, and Mk is the number of moles or atoms of the element of the nuclide.
As an approximation, the alpha-ejection correction can be applied by simply dividing the raw (U-Th)/He
date or 4He content FTm, but this is not accurate over long timescales, because of the exponential nature of
daughter ingrowth. A more accurate approach is to consider each parent nuclide as capable of only
producing the fraction of 4He daughter that remain in the crystal, by multiplying each measured parent
nuclide content by its nuclide-specific FT, and recasting the age equation as
4
He = 8⋅238 U ⋅ FT238 (e λ8t − 1) + 7⋅235 U ⋅ FT235 (e λ7t − 1) + 6⋅232 Th ⋅ FT232 (e λ2t − 1)+147Sm ⋅ FT147 (e λ147t − 1) .
Complications and considerations related to alpha ejection correction
Several possible complications can arise in attempting to routinely treat the problem of long-alpha stopping
distances in (U-Th)/He dating. Chief among these are the effects of 1) interaction of the alpha ejection and
diffusive concentration profiles in affecting diffusive loss, 2) the presence of strong intragranular parent
nuclide zonation, 3) and from implanation of He into grains, and 4) natural or artificial abrasion grains that
removes some of the alpha-ejection affected profile.