Nathan Cogné
email: nathan.cogne@univ-rennes1.fr

David M Chew

Raymond A Donelick

Claire Ansberque

LA-ICP-MS apatite

Keywords: Fission track dating, LA-ICP-MS, zeta method, trace element analysis, apatite 1-Introduction

published or not. The documents may come

minerals contain sufficient uranium (typically > 10 ppm) to generate a statistically useful quantity of spontaneous fission tracks over geological time. Pioneering development of the fission track method was undertaken in the 1960s and 1970s (summarised in the seminal book on nuclear tracks in solids by [START_REF] Fleischer | Nuclear tracks in solids: principles and applications[END_REF], and established the basis for a new thermochronometer utilizing the age equation below:

𝑡 = 1 𝜆 𝑑 𝑙𝑛 (1 + 𝜆 𝑑 𝜆 𝑓 𝜌 𝑠 [ 𝑈 238 ]𝑅𝜂 ) (1)
where d is the total decay constant of 238 U, f is the spontaneous fission decay constant of 238 U, s is the spontaneous track density on an internal prismatic crystal surface (i.e. the number of tracks per unit area), [ 238 U] is the present-day abundance of 238 U atoms per unit volume, R is the etchable length, and  is the etch efficiency factor. The etch efficiency factor accounts for not all of the tracks intersecting the etched surface being effectively made countable by the etching process; it can vary from one mineral phase to another, one crystal plane to another, and from one direction to another within an etched surface.

Up until the end of the last century, the only practical method way to estimate [ 238 U] was by neutron irradiation of 235 U within the target mineral phase to induce new fission tracks. The fission track community experimented with several methods to relate this induced track density to [ 238 U] during the 1970s and early 1980s (see [START_REF] Gleadow | Fission-track dating methods: What are the real alternatives?[END_REF] for a review of the existing approaches at that time). By the early 1990s, the fission track community had largely adopted a  (zeta) calibration approach [START_REF] Hurford | The zeta age calibration of fission-track dating[END_REF][START_REF] Hurford | Standardization of fission track dating calibration: Recommendation by the Fission Track Working Group of the I.U.G.S. Subcommission on Geochronology[END_REF], with the majority of workers employing the external detector method (EDM). The EDM has several advantages over other based calibration methods (such as the population method), including its ability to

Journal Pre-proof J o u r n a l P r e -p r o o f

produce single grain ages. The EDM involves placing a detector (a low U dielectric material such as a muscovite sheet) in intimate contact with the previously-etched fission track mount or dosimeter glass prior to irradiation; after irradiation the detector is then etched and its induced track density determined. The  method has the advantage of eliminating the need to determine the values of certain parameters which are difficult to establish experimentally (e.g. f and the thermal neutron fluence, 

This can be achieved by recasting the age equation below (eq. 2) in terms of the  calibration factor by dating samples of known age:

𝑡 = 1 𝜆 𝑑 𝑙𝑛 (1 + 𝜆 𝑑 𝜁𝜌 𝑑 𝜌 𝑠 𝜌 𝑖 ) (2)
where d, s and i are the densities, respectively, of countable induced tracks in a dosimeter glass of known U concentration, spontaneous tracks in the sample, and induced tracks in the detector.

Despite these inherent advantages, the EDM has drawbacks as it requires (i)

the irradiation of the samples, unknowns and dosimeter glasses (and their associated muscovite external detectors) by thermal neutrons in a nuclear reactor which is both time consuming and logistically complicated as it involves the production, transport and handling of radioactive samples, (ii) the assumption of intimate contact between the external detector and the grain surfaces being dated, and (iii) the use of hazardous hydrofluoric acid to etch the muscovite external detectors.

2-LA-ICP-MS fission track dating

LA-ICP-MS (laser ablation inductively coupled plasma mass spectrometry) has been used for more than 30 years for in situ elemental analysis of solid samples, with limits of detection now approaching the ppb level at the high mass end of the periodic Journal Pre-proof J o u r n a l P r e -p r o o f table. Following the pioneering work of [START_REF] Cox | Apatite fission-track (FT) dating by LAM-ICP-MS analysis[END_REF], [START_REF] Hasebe | Apatite fissiontrack chronometry using laser ablation ICP-MS[END_REF] demonstrated the feasibility of the LA-ICP-MS method for U concentration determinations in fission track dating. [START_REF] Hasebe | Apatite fissiontrack chronometry using laser ablation ICP-MS[END_REF] used an absolute fission track dating approach that calibrated the U content in the unknown grains against two standard glasses of known U concentration (NIST 610 and 612, Jochum et al., 2011). 44 Ca was used as an internal elemental standard (assuming Ca is essentially stoichiometric in the targeted apatites) to correct for variations in ablation volume.

Although LA-ICP-MS is typically a very forgiving analytical method for matrix differences between apatite and NIST glasses when internal standardization is used (e.g. [START_REF] Chew | LA,Q)-ICPMS trace-element analyses of Durango and McClure Mountain apatite and implications for making natural LA-ICPMS mineral standards[END_REF], minor matrix effects at the percent range are still present for a variety of elements. The absolute fission track dating approach is thus susceptible to minor (< 5-10%) systematic deviations in apatite U contents which propagate through to minor systematic deviations in the accuracy of absolute fission track age determinations by LA-ICP-MS (see section 5.2 for further discussion). [START_REF] Hasebe | Apatite fissiontrack chronometry using laser ablation ICP-MS[END_REF] also suggested that a -based approach analogous to the EDM was possible. [START_REF] Donelick | Apatite Fission-Track Analysis[END_REF] adopted such a -based LA-ICP-MS approach, with 43 Ca employed as an internal elemental standard and thus the single grain apatite fission track age equation becomes:

𝑡 𝑖 = 1 𝜆 𝑑 𝑙𝑛 (1 + 𝜆 𝑑 𝜁 𝐼𝐶𝑃 𝑁 𝑠,𝑖 Ρ 𝑖 Ω 𝑖 ) ( 3
)
where ICP is the zeta calibration factor based on LA-ICP-MS age standard, s,i is the number of counted spontaneous fission tracks for grain i, i is the area over which tracks were counted on grain i and i (uppercase Greek letter rho) is the 238 U/ 43 Ca ratio of grain i.

The associated standard error on a single grain age is given by:

(4)

where s ICP is the standard error on ICP and s i is the analytical error of the LA-ICP-MS measurement of i.

3-The ICP factor determination and use

As for the EDM, the primary zeta factor ICP is determined empirically by employing an apatite fission track reference material of known age and rearranging the age equation as follows [START_REF] Donelick | Apatite Fission-Track Analysis[END_REF]:

𝜁 𝐼𝐶𝑃 = 𝑒 𝜆 𝑑 𝑡 𝑠𝑡𝑑 -1 𝜆 𝑑 ∑𝑁 𝑠,𝑖 ∑ Ρ 𝑖 Ω 𝑖 ⁄ (5)
where tstd is the accepted age of the reference material. The associated standard error is:

𝑠 𝜁 𝐼𝐶𝑃 = [ 𝜁 𝐼𝐶𝑃 2 ∑ 𝑁 𝑠,𝑖 + 𝜁 𝐼𝐶𝑃 2 ∑(𝑠 Ρ 𝑖 Ω 𝑖 ) 2 (∑ Ρ 𝑖 Ω 𝑖 ) 2 + ( 𝜎 𝑡 𝑠𝑡𝑑 (𝑒 𝜆 𝑑 𝑡 𝑠𝑡𝑑 ) ∑𝑁 𝑠,𝑖 ∑ Ρ 𝑖 Ω 𝑖 ⁄ ) 2 ] 1 2 (6)
where tstd is the uncertainty on the age standard.

However, due to changes in plasma tuning conditions between different LA-ICP-MS analytical sessions, the ICP factor must be determined anew for every session.

Given that the uncertainty on the ICP factor is primarily dependant on the number of counted spontaneous fission tracks (s) in the age standard, an unreasonably large s (>2500) would have to be counted at every LA-ICP-MS session to achieve a reasonable uncertainty (below 2% for example). Therefore ICP can become a large source of uncertainty. We present here a methodology that builds on [START_REF] Donelick | Apatite Fission-Track Analysis[END_REF] and [START_REF] Chew | Combined apatite fission track and U-Pb dating by LA-ICP-MS and its application in apatite provenance analysis[END_REF] and that allows for a determination of a precise

Journal Pre-proof J o u r n a l P r e -p r o o f ICP factor during one large primary LA-ICP-MS session. This ICP factor is 're-used' during subsequent secondary LA-ICP-MS sessions with unknowns by revisiting a subset of the primary zeta grains that were analysed during the large primary LA-ICP-MS session.

Data acquisition

The analytical protocol described below was developed in 2012 in Trinity

College Dublin (TCD). Although the analytical conditions and instrumentation have changed with time, there has been no noticeable impact on the accuracy or precision of the method, and most LA-ICP-MS laboratories with modern quadrupole ICP-MS and UV laser systems can employ their own analytical conditions and instrumentation yet still follow our general analytical protocol. In both our laboratories (TCD and Geoscience Rennes) the ablated sample aerosol is carried by He and then mixed with Ar make-up gas and a small volume of N2 to enhance signal sensitivity and reduce oxide formation. Instrument tuning is performed before each primary and secondary LA-ICP-MS session on NIST 612 glass to minimize fractionation for U/Pb geochronology, with the aim of producing Th/U ratios close to the atomic ratios of the glass (ca. 1.01), low oxide production rates (ThO + /Th + < 0.15%) while optimising the Journal Pre-proof J o u r n a l P r e -p r o o f

The element suite selected by the analyst is dependent on the application and can be varied accordingly. For example, for detrital apatite fission track analysis employing simultaneous U-Pb and trace element analysis, between 25 -30 isotopes may be acquired including 238 U, 232 Th and 208,207,206,204 Pb, the REE + Y, 88 Sr, 55 Mn, 43 Ca and 35 Cl. 35 Cl is always analysed (analytical protocol following that of [START_REF] Chew | Apatite chlorine concentration measurements by LA-ICP-MS[END_REF] as this element is a kinetic parameter in many fission track annealing models (e.g. [START_REF] Green | Thermal annealing of fission tracks in apatite, 1, A qualitative description[END_REF]. For a large element suite employing between 25 -30 isotopes the total sweep cycle is typically ca. 450ms.

We employ a single spot ablation per counted area, with parameters that have changed with time both within and between our two laboratories; initial experiments employed a laser repetition rate of 5Hz with an analysis time of 45s, a 30s washout and a fluence of 3J/cm 2 . With improvements to the laser cell washout, the laser repetition rate was increased to 7Hz with 30s acquisition time and a 15s washout.

These parameters are still employed at Géosciences Rennes, while in TCD a repetition rate of 15Hz with 18s acquisition time and a 7s second washout is now typically employed. The laser repetition rate, total shot count and fluence were optimised to yield a pit depth of about 15 µm, with a typical laser ablation spot size of 30 or 35 µm which is always kept constant within a given LA-ICP-MS session. The ablation depth of 15 µm yields sufficient signal for U-Pb geochronology while not yielding a too high pit aspect ratio which would exacerbate downhole fractionation in U-Pb geochronology (cf [START_REF] Woodhead | Zircon Hf-isotope analysis with an excimer laser, depth profiling, ablation of complex geometries, and concomitant age estimation[END_REF][START_REF] Donelick | A discordia-based method of zircon U-Pb dating from LA-ICP-MS analysis of single spots[END_REF]. Issues that can arise from siting a single spot analysis on an apatite grain with potential U zoning are discussed in section 5.5.

The raw isotope data in both laboratories are reduced with Iolite [START_REF] Paton | Iolite: Freeware for the visualisation and processing of mass spectrometric data[END_REF]. The U/Ca ratio determination for fission track dating employs a slightly modified [START_REF] Chew | Combined apatite fission track and U-Pb dating by LA-ICP-MS and its application in apatite provenance analysis[END_REF], where a spherical depth-weighting function is applied so that U concentration data close to the grain surface is weighted more heavily than U concentrations at depth down to a distance of 8 µm (broadly half a fission track length) below the apatite grain surface.

Journal Pre-proof J o u r n a l P r e -p r o o f

Primary LA-ICP-MS ('zeta') session

We employ a Durango crystal which was crushed and sieved to 200 -300 µm as our primary zeta reference material; it is also possible to use a large c-axis parallel slab from a single Durango crystal and analyse individual sub-areas on it. Each c-axis parallel zeta shard is counted at 500x magnification with a 50x dry objective and 10x

oculars to maximise the size of the counted area (i) and thus the number of counted spontaneous fission tracks (Ns). As Durango apatite is gem quality and largely devoid of defects counting using a 50x objective is possible, although we do zoom in using a 100x objective to discriminate between small defects and vertical tracks. A 100x

objective is employed for all unknowns. Both Durango and unknowns are counted using TrackWorks® (Autoscan Systems) which records precise coordinates and either 2D images (or 3D stacks) of the grains. Precise coordination and high-resolution images are required to ensure the position of the laser ablation spot overlaps the counted area. Typically 80 to 100 shards of Durango are analysed during a large primary LA-ICP-MS session. Each counted area on a shard is typically large enough (c. 2×10 -4 cm 2 ) to subsequently accommodate between 15 -25 laser ablation spots.

These include three laser ablation spots for the primary LA-ICP-MS session (i.e. the zeta factor determination) and the shard can then be revisited during subsequent secondary LA-ICP-MS session with unknowns.

It is assumed that the 238 U/ 43 Ca ratio on the scale of the counted shard is constant; laser ablation mapping of our Durango zeta standard (Dur-DCa)

demonstrates that it is relatively homogenous and any minor U zoning present is on a significantly larger scale than that of an individual shard [START_REF] Chew | LA,Q)-ICPMS trace-element analyses of Durango and McClure Mountain apatite and implications for making natural LA-ICPMS mineral standards[END_REF]. During the primary LA-ICP-MS session NIST 612 standard glass analyses are interspersed

Journal Pre-proof J o u r n a l P r e -p r o o f

with Durango shard analyses (typically three NIST 612 analyses followed by 10

Durango shards) to calculate background-and drift-corrected U/Ca ratios. The data are then exported and an in-house excel spreadsheet (Tables S2 andS3 in the Supplementary Material) is used to calculate the primary zeta factor ICP and its error s ICP . This spreadsheet can employ a linear regression through the background-and drift-corrected U/Ca ratios to flatten any residual drift that is not corrected by normalisation to NIST 612. This step is typically not necessary and also assumes that a large suite of shards is analysed during the primary LA-ICP-MS session, as intrashard U/Ca variations in a small dataset would otherwise influence the linear regression. The U/Ca ratio is then converted to a 238 U/ 43 Ca ratio using the natural isotopic abundances of U and Ca, while the Pi value (the 238 U/ 43 Ca ratio from the primary LA-ICP-MS session) for each counted shard is the weighted mean of three different primary laser ablation spots (equation 7 below). This enables potential 238 U/ 43 Ca zoning in each shard to be assessed.

Ρ Pi = ∑Ρ Pi,x s Pi,x 2 ⁄ ∑ s Pi,x -2 (7)
where Pi,x is the 238 U/ 43 Ca ratio of an individual Durango zeta shard, sPi,x is the associated standard error and x corresponds to primary laser ablation spots 1, 2, and



The associated standard error is:

s Pi = [ 1 ∑ s Pi,x -2 ] 1 2 (8)
Then the primary zeta factor ICP and its error s ICP are calculated using equations 5 and 6, respectively. Fig. 1 shows the results for one primary LA-ICP-MS session on three sets of analyses of 94 shards. The 238 U/ 43 Ca ratio is variable from Journal Pre-proof J o u r n a l P r e -p r o o f one shard to another but each shard shows good internal homogeneity with mean and median standard deviations of c. 2.0% (maximum c. 5%), while the analytical uncertainty on 238 U/ 43 Ca ratio of a single analyses is typically around 1.2%, validating our assumption of minimal 238 U/ 43 Ca zoning within each shard.

Secondary LA-ICP-MS sessions with unknowns

The main purpose of the Durango zeta approach detailed here is that the primary zeta factor ICP can be employed in all subsequent secondary LA-ICP-MS sessions involving the analysis of unknowns. This avoids the time-consuming counting and analysis of many Durango shards for every secondary LA-ICP-MS session which would be otherwise required to yield a relatively precise zeta factor.

In this study, secondary LA-ICP-MS sessions with unknowns typically employ four Durango shard analyses and three NIST 612 analyses (for a drift correction) for every 20 unknowns, with each grain sampled by a single spot ablation. Following data reduction (which in this study employed the Trace Elements FTD DRS in Iolite) the LA-ICP-MS session data are exported and we use an in-house excel spreadsheet (provided in the Supplementary Material) to calculate a session-specific zeta fractionation factor (s) that is subsequently applied in the age equation.

Χ 𝑠 = Ρ 𝑃 ̅̅̅̅ Ρ 𝑆 ̅̅̅̅ (9)
whereP is the arithmetic mean of Pi (the weighted mean 238 U/ 43 Ca ratio of each Durango shard from the primary LA-ICP-MS session) andS is the arithmetic mean of the U/Ca ratio of the same Durango shards determined during the secondary LA-ICP-MS session. The zeta fractionation factor corrects for systematic variations in 238 U/ 43 Ca ratios, as it can differ by up to 10% between sessions (section 5.2). This is related to Journal Pre-proof J o u r n a l P r e -p r o o f variations in the analytical conditions (primarily related to the tuning of the ICP-MS instrument) between the primary LA-ICP-MS session and a given secondary LA-ICP-MS session and converts the U/Ca values exported from the Iolite DRS to a 238 U/ 43 Ca ratio. Critically this also means that our analytical protocol is not susceptible to minor (< 5-10%) systematic deviations in apatite U contents which hinder the absolute fission track dating LA-ICPMS approach (section 5.2), our approach simply assumes that the relative differences in 238 U/ 43 Ca ratios between Durango primary zeta grains are constant between sessions. Additionally, employing this zeta fractionation factor allows the Durango primary zeta grains and their associated primary LA-ICP-MS session 238 U/ 43 Ca ratios to be used subsequently on a different instrumental setup.

The zeta fractionation factor s is applied to the measured U/Ca ratio of each Durango shard in a secondary LA-ICP-MS session to obtain Ci (the converted 238 U/ 43 Ca ratio for Durango grain i).

Ρ 𝐶𝑖 = Χ 𝑠 (𝑈 𝐶𝑎 ⁄ ) 𝑖 (10)
The 238 U/ 43 Ca ratios of each Durango shard i can then be directly compared with their corresponding values from the primary LA-ICP-MS session using equation ( 11) below and the variation in this ratio for each shard is plotted (Fig. 2).

𝑅 𝑖 = Ρ 𝑃𝑖 Ρ 𝐶𝑖 ⁄ (11)
This ratio should approximate unity for every Durango shard, and any variation in this ratio should not be related to its laser ablation spot number during a secondary LA-ICP-MS session with unknowns. If this ratio shows systematic variation during such a session it is likely that the drift correction was inappropriate (Fig. 2 shows more significant deviation, it may imply U zonation in this part of the shard. This shard therefore is excluded from the calculation of the fractionation factor (s) and

would not be reused afterwards in subsequent sessions. The associated standard error of s is also calculated (equation 12) and is propagated through to the final age calculation of unknowns.

𝑠 Χ 𝑠 = Χ 𝑠 [ ∑(𝑅 𝑖 -𝑅 ̅) 2 𝑁-1 ] 1 2 (12)
whereR is the arithmetic mean of Ri and N is the number of analysed Durango shards.

Age calculation of unknowns

The zeta calibration factor s calculated using the Durango primary zeta grains is then applied to the depth-weighted U/Ca ratio of the unknown grains to get the i (converted 238 U/ 43 Ca ratio) of each unknown grain:

Ρ 𝑖 = Χ 𝑠 ( 𝑈 𝐶𝑎 ) 𝑤,𝑖 (13)
The standard error is given by

𝑠 Ρ 𝑖 = Ρ 𝑖 [( 𝑠 (𝑈 𝐶𝑎 ⁄ ) 𝑖 (𝑈 𝐶𝑎 ⁄ ) 𝑖 ) 2 + ( 𝑠 Χ 𝑠 Χ 𝑠 ) 2 ] 1 2 (14)
The single grain ages are then calculated using equations 3, and the pooled age of the sample is calculated following [START_REF] Donelick | Apatite Fission-Track Analysis[END_REF]:

𝑡 𝑝𝑜𝑜𝑙𝑒𝑑 = 1 𝜆 𝑑 𝑙𝑛 (1 + 𝜆 𝑑 𝜁 𝐼𝐶𝑃 ∑ 𝑁 𝑠,𝑖 ∑ Ρ 𝑖 Ω 𝑖 ) (15)
The associated standard error on the pooled age is given by: Journal Pre-proof J o u r n a l P r e -p r o o f

𝑠 𝑝𝑜𝑜𝑙𝑒𝑑 = 𝑡 𝑝𝑜𝑜𝑙𝑒𝑑 [ 1 ∑ 𝑁 𝑠,𝑖 + ∑(𝑠 Ρ 𝑖 Ω 𝑖 ) 2 (∑ Ρ 𝑖 Ω 𝑖 ) 2 + ( 𝑠 𝜁 𝐼𝐶𝑃 𝜁 𝐼𝐶𝑃 ) 2 ] 1 2 (16)
To assess if the pooled age equation is appropriate (i.e. if all the grains belong to the same population) we follow the formulation of the chi-squared homogeneity test of [START_REF] Galbraith | Statistics for LA-ICP-MS Fission Track Dating[END_REF]:

𝜒 𝑠𝑡𝑎𝑡 2 = ∑(𝑧 𝑖 2 𝜎 𝑖 2 ⁄ ) - ∑(𝑧 𝑖 𝜎 𝑖 2 ⁄ ) 2 ∑1 𝜎 𝑖 2 ⁄ ( 17
)
where zi = ln(ti) and i=set i /ti with set i defined as the standard error on the age that takes into account only the analytical uncertainties and ignores the uncertainties on the zeta factor and on the zeta fractionation factor [START_REF] Vermeesch | Statistics for LA-ICP-MS based fission track dating[END_REF]. The p-value can then be calculated; a p-value > 0.1 implies no evidence against the null hypothesis that the grains are a single population, between 0.05 and 0.1 implies very weak evidence of multiple populations being present; between 0.01 and 0.05 moderately strong evidence against the null hypothesis and < 0.01 strong evidence that multiple populations are present. If the p-value calculated for the sample is too low to use the pooled age, a central age [START_REF] Galbraith | Statistical models for mixed fission track ages[END_REF]) can be calculated using for example QTQt [START_REF] Gallagher | Transdimensional inverse thermal history modeling for quantitative thermochronology[END_REF] or IsoplotR [START_REF] Vermeesch | IsoplotR : A free and open toolbox for geochronology[END_REF]. The latter software package can also be used to deconvolute different potential age populations.

It should be noted that the chi-squared test is applied to a pool of apatite singlegrain ages to assess whether these ages are consistent with being from a single grainage population. When it is suspected that a pool of apatite single-grain ages may be derived from more than one source (e.g. different detrital populations), it is recommended to use independent data (e.g. trace element analyses, Dpar values or multi-kinetic data) to sort analyses into different coherent populations, and then apply the chi-squared test. In old and slowly cooled terranes, a small difference in apatite chemistry in either crystalline bedrock or detrital samples can also lead to the 'failure'

Journal Pre-proof J o u r n a l P r e -p r o o f of the chi-square test. However this compositional heterogeneity can then be exploited using multi-kinetic modelling approaches on separate apatite populations to increase the thermal sensitivity of the apatite fission track technique to from c. 50 to 175 °C (e.g. [START_REF] Schneider | Application of Low Temperature Thermochrnology to Hydrocarbon Exploration[END_REF]Issler, 2019, McDannell et al., 2019). The LA-ICP-MS apatite fission track dating approach lends itself well to assessing trace element chemical variations in apatite as the technique permits acquisition of independent data (e.g. U-Pb age information, trace element or Cl abundances) during fission track analysis to sort the data into separate coherent populations.

Discussion

The various topics discussed below primarily cover issues related to apatite fission track dating using our LA-ICP-MS protocol; they are not intended to represent a comprehensive discussion on apatite fission track dating in general. These topics include sample preparation, absolute U concentration determinations by LA-ICP-MS and the issues of zero track grains and U zoning in LA-ICP-MS protocols, and are accompanied by tests of our methods.

Sample preparation and reusing the Durango zeta material

Typically, between 80 and 100 Durango shards are counted and analysed in the We strongly advocate the use of epoxy-only grain mounts for LA-ICP-MS fission track dating (cf [START_REF] Donelick | Apatite Fission-Track Analysis[END_REF]. Epoxy-only grain mounts have the advantage over epoxy-on-glass mounts in that all the grains are at approximately at the same exposure level following grinding and polishing (assuming the grains are roughly of similar size). This is clearly beneficial for repolishing and reusing Durango zeta mounts, but is also key to ensuring minimal loss of grains during ablation of unknowns. If all the grains are at approximately at the same exposure level, a mount can be polished down ensuring a large fraction of the apatite grains is not polished down too far yet still reveals internal surfaces with 4π geometry. We recommend removing approximately one-third of the grain which ensures a significant proportion of the grain is still firmly embedded in the epoxy resin. Subsequent HNO3 etching to reveal fission tracks also etches (and thus loosens) the margins of the grain in the epoxy mount, and grains where over half the grain has been ground away are much more likely to be lost during ablation. For a representative sample of plutonic apatites (typical grain sizes of 75 to 200 µm) we estimate that the grain loss rate for a well-prepared epoxy-only grain mount is <10%, whereas in an epoxy-on-glass mount up to half the grains could be lost during ablation. other laboratories using the EDM. Additionally, U-Pb age data were simultaneously acquired for apatite grains from samples S1 and S2 and are in agreement with previously determined crystallisation ages for both samples [START_REF] Ketcham | Reproducibility of thermal history reconstruction from apatite fission-track and (U-Th)/He data[END_REF],

Journal

showing that simultaneous acquisition of U-Pb data (as well as a large suite of trace elements) is not prejudicial to the accuracy or the precision of the LA-ICP-MS fission track age.

Interestingly, [START_REF] Ketcham | Reproducibility of thermal history reconstruction from apatite fission-track and (U-Th)/He data[END_REF] suggest that the LA-ICP-MS analyses from different laboratories typically show greater dispersion in single grain ages than EDM analyses. Although this could be interpreted as LA-ICP-MS fission track dating causing over-dispersion, it is also possible EDM induces a bias towards under-dispersion.

Usually more grains are analysed with LA-ICP-MS protocols (ca. 30-40 grains) than with EDM (ca. 20), which increases the likelihood of producing over-dispersed grain data (McDannell et al., 2019). One potential additional advantage of the LA-ICP-MS technique is that there is no possibility (unlike EDM) of subconsciously tracking if grains belong to one population using the Ns/Ni ratio, which would result in artificially low chisquare test values [START_REF] Donelick | Apatite Fission-Track Analysis[END_REF].

Apatite grains from Mt Dromedary were also dated by the EDM method (Fig. 5A). Mt Dromedary apatite has the advantage for an EDM vs LA-ICP-MS comparison in that it is similar to typical natural samples in terms of defects and U zoning, yet its age is well constrained from multiple studies. We also dated a sample from [START_REF] Cogné | Postbreakup tectonics in southeast Brazil from thermochronological data and combined inverse-forward thermal history modelling[END_REF] by both EDM and LA-ICP-MS on the same grains (Fig. 4D and5B). This sample (Br5) was chosen because it has a relatively low track spontaneous track density (1.32 x 10 5 tracks/cm 2 ) and therefore potential U zonation would be difficult to detect (see section 5.5 for discussion on zonation). Detailed single grain data are provided in the Supplementary Material (Table S6). For both samples, all EDM and Journal Pre-proof J o u r n a l P r e -p r o o f LA-ICP-MS single grain ages are comparable at the 2level, as are the resultant pooled ages (Figs 5C and 5D). Therefore, our LA-ICP-MS protocol is able to reproduce ages with those obtained by the EDM method and with similar precision, even for samples with relatively low track densities.

Zero-track grains

For young and/or low U apatite samples, grains with no spontaneous tracks are sometimes encountered; when Ns=0 then the error on the fission track age cannot be determined (equation 4). There is no simple way to deal with zero-track count data with the LA-ICP-MS method. EDM data are based on the ratios of two independent Poisson variables (Ns and Ni) which greatly simplifies all subsequent statistical analyses (including calculations of the uncertainty on zero-track grains); in contrast LA-ICP-MS fission track data are based on mixed ratios of Poisson and (log)normally distributed variables that are not easy to model statistically.

One possibility is to recast the age equation in an EDM form as shown by Vermeesch (2017, section 5 therein). The age equation in EDM form converts the 238 U/ 43 Ca ratio to a Ni equivalent (see Vermeesch, 2017 for further details). A non-zero age is then obtained by adding 0.5 to both Ns and the Ni equivalent value, and the error on the fission track age can then be calculated. An alternative approach involves delimiting a 95% confidence interval that goes from 0 Ma (the age of the zero-track grain) to an age calculated for Ns=3, because in the Poisson distribution an observation of zero tracks denotes a 95% probability that the underlying mean is between zero and three. Both approaches yield similar results and the latter method can be easily implemented in the spreadsheet provided in the Supplementary Material. However, we favour the approach of [START_REF] Vermeesch | Statistics for LA-ICP-MS based fission track dating[END_REF], as it is also enables zero-track grains to

Journal Pre-proof J o u r n a l P r e -p r o o f also be included in a chi-squared homogeneity test. If a zero-track grain is present in a dataset then the single-grain age ti given by equation 3 is 0 Ma and the chi-squared homogeneity test (equation 17) cannot be employed using this grain to calculate the  2 value. Recasting the age equation into an EDM form allows the use of equation 17 with a zero-track grain; we then follow the transformation proposed in equations 6.38 and 6.39 of [START_REF] Vermeesch | Statistics for fission-track thermochronology[END_REF] for low-track density grains prior to applying the chisquared homogeneity test.

U zoning in apatite

One of the major strengths of the EDM is that data are collected from identical areas on individual grains and their mirror-images in the muscovite external detector, and therefore within-grain heterogeneity in uranium concentration can be accommodated by the EDM. The distribution of induced tracks in the external detector can therefore be considered as a reliable proxy map for the uranium distribution in its mirror-image apatite grain. This induced track "map" also records depth-integrated variations in uranium concentration, as induced fission (similar to spontaneous fission)

generates tracks in the detector that are produced by uranium up to half a fission-track length below the apatite grain surface. This is a major advantage of our method as it facilitates checking if the background and drift corrections applied during the secondary LA-ICP-MS session were fully appropriate.

U zoning in unknowns is potentially more problematic but can be accounted for.

U zoning with depth is easily dealt with by generating depth-weighted U/Ca ratios during the initial data reduction stage. Comparison of the depth-weighted U/Ca ratio with the U/Ca ratio from the same laser ablation spot analysis is a useful monitor for U zoning. Our depth-weighted U/Ca function in the Iolite DRS (section 3.1) extends down to 8 µm (half a fission track length) and accounts for U closer to the mount surface contributing more to the spontaneous track count. A graph of depth-weighted U/Ca vs U/Ca measured for the total pit depth is shown in Figure 6. These data should plot on a line with a slope very close to unity which is what is observed (Fig. 6); deviation from this line is accounted for by downhole U zonation. Importantly the sample in Figure 6 (Sample K1 from , 2015;[START_REF] Petrus | A new approach to laser-ablation inductively-coupled-plasma mass-spectrometry (LA-ICP-MS) using the flexible map interrogation tool[END_REF][START_REF] Ubide | High-resolution LA-ICP-MS trace element mapping of igneous minerals: in search of magma histories[END_REF].

Moreover, extracting data (elemental abundances and/or isotopic ratios) from grain maps from a specific region of interest (a polygon) that is identical to the counted area is now possible using the flexible 'Monocle' map interrogation tool for Iolite [START_REF] Petrus | A new approach to laser-ablation inductively-coupled-plasma mass-spectrometry (LA-ICP-MS) using the flexible map interrogation tool[END_REF]. Grain mapping by LA-ICP-MS can thus be employed as an alternative to laser ablation spot analysis in the case of grains with low spontaneous track densities.

In both approaches the time it takes to undertake the analysis and the subsequent data reduction is longer than conventional single-spot ablations and thus these approaches are best suited only for samples with low spontaneous track densities. from the study of [START_REF] Turab | Tectonics drives rapid exhumation of the western Himalayan syntaxis: Evidence from low Journal Pre-proof J o u r n a l P r e -p r o o f temperature thermochronometry of the Neelum River region, Pakistan[END_REF].

Journal Pre-proof

  r n a l P r e -p r o o f version (Trace Elements FTD) of the Trace Elements DRS[START_REF] Woodhead | Isotopic and elemental imaging of geological materials by laser ablation Inductively Coupled Plasma mass spectrometry[END_REF], and which is provided in the supplementary material. Although a user can easily use their own data reduction protocols (e.g. other software packages or an in-house spreadsheet), Iolite can comfortably handle long (> 6h) LA-ICP-MS sessions with additional functionality such as the generation of depth-weighted U/Ca values as described below. NIST 612 is used as primary reference material to correct for LA-ICP-MS session drift. LA-ICP-MS systems drift slowly during extended analytical runs due to deposition on the cones, gradual expulsion of atmospheric gases from the laser cell and changes in the electrical properties of the interface and ion lens system which can result in lower signal intensities with time. Normalization relative to a stoichiometric internal standard isotope in the reference material (typically 43 Ca or 44 Ca in apatite) removes much of the effects of session drift; nevertheless minor residual drift in Canormalised elemental abundances may still remain and necessitates drift correction.In this study 'semi-quantitative' standardisation is employed (a term employed within Iolite to denote baseline correction followed by normalization of unknowns relative to the drift-corrected reference material). Semi-quantitative normalization allows for optimal drift correction as separate spline fits can be applied to both the 238 U and 43 Ca channels. The modifications in the Trace Elements FTD DRS produce both (1) a U/Ca channel produced by dividing each time slice of the final U ppm and Ca ppm channels to correct for variations in ablation volume (i.e. a time-resolved U/Ca ratio) and (2) a depth-weighted U/Ca value for fission track dating. The depth weighting follows Chew

  ), and may result from an unsuitable spline fit being applied to the NIST 612 reference materials during the initial data reduction of the secondary LA-ICP-MS session. If the comparison of the Durango shards from a secondary LA-ICP-MS session with their corresponding values Journal Pre-proof J o u r n a l P r e -p r o o f from the primary LA-ICP-MS session appears appropriate (typically within a ± 5% range; Fig. 2) then the session-specific zeta fractionation factor (s) is applied to all the unknown grains analysed during the secondary LA-ICP-MS session. If a one shard

  primary LA-ICP-MS session for reuse later in secondary LA-ICP-MS sessions with unknowns. Between 15 -25 laser ablation spots can fit on each shard, and a suite of 80 -100 Durango shards would allow approximately 7000 to 10000 separate singlegrain fission track ages to be determined (assuming 4 Durango zeta shard analyses for every 20 single-grain unknowns). Once the Durango zeta material is used up, we have successfully employed the following strategy to reuse the mount further. The three reference points on the sample mount (the central portion of three SEM target Journal Pre-proof J o u r n a l P r e -p r o o f grids) are ablated by the LA system using a small laser ablation spot (c. 25 µm) down to a depth of c. 50 µm. The mount is then gently repolished down by a depth of c. 20 µm to remove the ablation pits of previous analyses and can then be reused. Given the evidence for the U-homogeneity over the counted area on the Durango shards (cf Chew et al., 2016), significant variations in U content down to a depth of c. 20 µm are considered negligible.

  Figure3A, the U/Ca ratio is plotted against the U content (in ppm) for 42 Durango zeta

Firstly, we consider

  the issue of potential U zoning in the Durango shards employed in the zeta calculation. LA-ICP-MS mapping of our Durango crystal and replicate spot analyses of our primary zeta grains demonstrates that U zoning in our Durango crystal is minor and on a significantly larger scale than that of an individual shard.In subsequent secondary LA-ICP-MS sessions with unknowns, the 238 U/ 43 Ca values of the Durango shards may be offset by a session-specific zeta fractionation Journal Pre-proof J o u r n a l P r e -p r o o f factor s (which is not related to variations in U content in a given shard). This zeta fractionation factor can then be applied to the 238 U/ 43 Ca values of a given Durango shard in a secondary LA-ICP-MS session, so it can be ratioed against its corresponding value in the primary LA-ICP-MS session. This ratio for each shard (Ri, equation 14) in a secondary LA-ICP-MS session should cluster around unity; a systematic deviation from unity during the secondary LA-ICP-MS session indicates an inappropriate drift correction while a single outlier might indicate U zoning in a given Durango shard which can then be discarded and not used in subsequent secondary LA-ICP-MS sessions.

( 2017 )

 2017 suggests undertaking multiple laser ablation spots analyses on a single crystal and calculating the U content by assuming a log-normal distribution of U in the grain. An alternative is to produce a 238 U/ 43 Ca map of the entire grain using the LA-ICPMS Journal Pre-proof J o u r n a l P r e -p r o o f technique that captures intra-grain U-variation and thus closely mimics the role of the muscovite external detector employed in the EDM. Rapid characterisation of elemental zonation by two-dimensional mapping of the grain surface is now possible by employing LA-ICP-MS systems with fast cell washout (e.g the aerosol rapid induction system or ARIS, van Malderen et al.

Figure 1 :

 1 Figure Captions

Figure 2 :

 2 Figure 2: Plot of the individual Ri ratios for each analysed Durango shard (primary LA-

Figure 3 :

 3 Figure 3: Plots illustrating the difficulty in accurate U concentration determinations by

Figure 4 :

 4 Figure 4: Results of fission track dating by LA-ICP-MS (all errors are at 2 level. All

Figure 5 :

 5 Figure 5: Comparison of fission track dating by EDM and LA-ICP-MS on the same

Figure 6 :

 6 Figure 6: Plot of the downhole-weighted U/Ca ratio vs the U/Ca ratio for the whole

Figure 1 Figure 2 Figure 3 Figure 4 Figure 5

 12345 Figure1

Figure 6

 6 Figure6

  ratio by up to 20% between these two approaches, even though it makes typically very little difference (< 1%) to the pooled age of the sample.U zoning on the etched surface of the grain mount (i.e. the counting area) can be more difficult to assess. In the EDM (in samples with appreciable U) the corresponding induced track image can act as a guide for U zoning; such information is not available in the LA-ICP-MS fission track technique. For samples with high spontaneous track densities (i.e. with old ages and/or significant U contents) then the spontaneous track density can be used as an indication for U zoning. In such cases the counted area should be sited on a zone with a homogenous spontaneous density, or alternatively the counted area should be the same size and in the same position as the laser ablation spot. The uncertainty on the single grain age is mainly dependant on the number of tracks counted; reducing the counted area to the size of the laser ablation spot should not significantly penalise the single grain age accuracy in samples with high spontaneous track densities. The simplest solution is to count on an area that mimics the exact size and location of the laser spot. However, this solution has the disadvantage of increasing the age uncertainty as the number of counted tracks will be low, while the choice of counted area (and thus spot location) in samples with very few tracks may be inadvertently affected by operator bias. To circumvent this, two different approaches have been developed that allow for counting of the whole grain surface. Vermeesch

	Journal Pre-proof
	In samples with low track density (i.e. young or low U samples) the problem of U zoning is potentially greater, although it is still possible to date such samples by LA-J o u r n a l P r e -p r o o f ICP-MS.
	Turab et al., 2017) is a young sample (3.7 Ma) with a low
	spontaneous track count (2.83 × 10 4 tracks/cm 2 ) which could not be used as a guide
	to potential U zonation. In samples with U zoning, individual grains may differ in U/Ca

J o u r n a l P r e -p r o o f grains from the same DUR-DCa crystal analysed by [START_REF] Chew | LA,Q)-ICPMS trace-element analyses of Durango and McClure Mountain apatite and implications for making natural LA-ICPMS mineral standards[END_REF] for three separate analytical sessions. The apatite U content is again determined by using NIST 612 glass as an external reference material and using internal standardisation. During the two runs of a primary zeta session acquired during the same day, the U ppm vs U/Ca ratios are virtually identical. The third run was performed on the same shards several months later during an unknown session. Individual analyses from the latter session clearly lie on a different U ppm vs U/Ca linear array (Fig. 3A), with the mean U ppm value being offset by about -3.8% and U/Ca by about -7.8% from the average of the first two zeta runs (Fig. 3B). Figure 3C shows the deviation of several session averages of U ppm and U/Ca from the mean of the first two zeta runs. There is a correlation between the session mean U ppm and U/Ca values (Inset Fig. 3D); with changes in the U/Ca fractionation from session to session inducing changes in the mean U ppm value when normalised to NIST612 standard glass, with a > 15% range observed between the maximum and minimum U ppm values (Fig. 3C). For this reason, we do not recommend reporting U ppm values for LA-ICP-MS fission track data as they are not accurate, and dependent on variations in U/Ca fractionation between sessions and also different instrumental setups / protocols. While the U/Ca fractionation between sessions and illustrated in Fig. 3C is easily accommodated by our protocol, the offsets in absolute U concentration measurements would be impossible to detect if an absolute dating approach (using NIST glass with no apatite reference material) was employed. The extent of these systematic offsets when using NIST612 as a primary reference material is such that until a Uhomogenous fluoroapatite primary reference material is found or produced, routine LA-ICP-MS absolute fission track dating is not possible.

Journal Pre-proof J o u r n a l P r e -p r o o f

Tests of the method

The -based approach to LA-ICP-MS apatite fission track dating has been tested by dating three different age standards, Durango (using a crystal different that is different to that used for zeta calibration and treated as an unknown; reference age 31. 440.18 Ma, McDowell et al., 2005), Fish Canyon tuff ( 28 Radial plots (drawn using IsoplotR, [START_REF] Vermeesch | IsoplotR : A free and open toolbox for geochronology[END_REF] for these samples are shown on Fig. 4 and the detailed data are provided in the Supplementary Material. These tables also show all the parameters that are employed in the calculation and that should be presented in publications that use FT data generated using our protocol (i.e Ns, area, 238 U/ 43 Ca with error and primary zeta factor with error). The three ages determined using our protocols are indistinguishable from the accepted ages at 2 level, with Durango giving a pooled age of 31.7  2.1 Ma, Fish Canyon tuff a pooled age of 27.9  3.2 Ma and Mount Dromedary a pooled age of 99.1  6.7 Ma. These results show that our protocol is able to reproduce well the age of standards, with a precision similar to what is usually obtained with the EDM method.

In 2014 the TCD laboratory participated in an international inter-laboratory exercise to test the reproducibility of thermochronological data [START_REF] Ketcham | Reproducibility of thermal history reconstruction from apatite fission-track and (U-Th)/He data[END_REF].

The test consisted of 'blind' dating of two samples, and was undertaken by two of the authors of this study (NC and DC) using the protocol described here. Results for NC are reported on Fig. 4 and the detailed data are provided in the Supplementary Material. The results of [START_REF] Ketcham | Reproducibility of thermal history reconstruction from apatite fission-track and (U-Th)/He data[END_REF] shows that for both analysts the ages calculated with our protocol are in excellent agreement with the mean of the 15 ages and 13 ages reported from the different labs for samples S1 and S2 respectively. The associated uncertainties are in the same range as the uncertainties determined by J o u r n a l P r e -p r o o f between sessions which in turn propagate through to systematic variations in the accuracy of absolute fission track age determinations, while the spontaneous fission decay constant is not well constrained. -based approaches do not employ the spontaneous fission decay constant while absolute U concentration determinations are not required -it is only assumed that the relative differences in 238 U/ 43 Ca ratios between grains (unknowns and Durango zeta grains) are constant in a secondary LA-ICP-MS session with unknowns. These -based approaches have often employed a ICP factor that had to be calculated for every LA-ICP-MS session with unknowns, which results in either a ICP factor with low precision or time-consuming counting to produce a precise ICP factor for each session.

The method developed here overcomes these problems by measuring a single, high-precision ICP in a primary LA-ICP-MS session that can be reused for subsequent secondary LA-ICP-MS sessions. It involves revisiting the same grains (shards) of Durango that were used for the primary LA-ICP-MS session and calculating a sessionspecific zeta fractionation factor to account for 238 U/ 43 Ca fractionation differences resulting from variations in LA-ICP-MS tuning. This approach also allows the user to verify that the drift correction based on NIST 612 is appropriate. The method has been tested and validated against apatite fission track standards of known age and by participating in an international survey that employed 'blind' dating of unknown samples. The method has also been tested by undertaking LA-ICP-MS and EDM fission track dating on the same grains on one apatite fission track standard (Mt Dromedary) and one low track density sample (Br5 from [START_REF] Cogné | Postbreakup tectonics in southeast Brazil from thermochronological data and combined inverse-forward thermal history modelling[END_REF]. The LA-ICP-MS method developed here for fission track dating of apatite can be easily modified for other U-bearing minerals such as zircon or titanite. For titanite 43 Ca can also be used as an internal elemental standard, while for zircon either 29 Si or a Zr isotope can be employed. The main issue is to find a sufficiently large titanite or zircon reference material with no small-scale U zoning (i.e. U is homogenous over the scale of a shard or crystal), such that they can be employed in an analogous fashion to Durango is in our LA-ICP-MS apatite protocol.