helium chronometry of apatite and titanite using nd yag laser heating we developed a technique for extracting 4he from apatite and titanite using laser heating that permits measurement of u and th and hence u th he ages on a single sample aliquot unlike direct laser heating quantitative he extraction and u th retention can be achieved when samples are indirectly laser heated to temperatures below their melting point resulting he ages are consistent with those obtained from conventional furnace heating of larger samples the major advantages of this technique include faster analysis and far lower more reproducible he blanks that permit single crystal analyses single crystal dating opens a variety of new avenues for u th he chronometry

kimberlite u th he dating links surface erosion with lithospheric heating thinning and metasomatism in the southern african plateau documenting the surface response to processes at depth is a central challenge in continental tectonics kimberlites bear an unusual record of both surface and deep processes through their downrafted crustal clasts pipe erosion levels and mantle xenolith suites here we combine new apatite u th he ahe data from four south africa kimberlites ranging in emplacement age from ca 143 ma to ca 74 ma with a wealth of other geologic information from the pipes to resolve the timing patterns and causes of erosion across 200 km of the southern african plateau comparison of the ahe dates with the kimberlite eruption ages constrains a strong regional unroofing pulse ca 117 90 ma which was most intense from ca 100 to 90 ma this unroofing phase is synchronous with warming thinning and metasomatism of the mantle lithosphere recorded by xenocrysts and peridotite xenoliths within these same kimberlites these thermochemical changes could be related to still deeper dynamic mantle processes our ahe data appear to directly document the erosional response to surface uplift induced by these various events the ahe results also detect as much as 1 5 km of cenozoic unroofing that is spatially centered on a proposed mid tertiary paleotributary to the orange river suggesting drainage patterns as the cause our results not only tightly refine the surface history of the interior of the southern african plateau but show that kimberlite ahe dating can tie cratonic erosion phases with contemporaneous thermochemical changes in the mantle and thereby point to causative links

processes controlling vertical coupling and decoupling between the upper and lower crust of orogens results from fiordland new zealand the pre cenozoic configuration of western new zealand allows determination of the effects of magmatism and a changing lower crustal rheology on the evolution of a cretaceous orogen from upper to lower crustal levels 10 50 km beginning at 126 ma a composite batholith dominated by diorite was emplaced into the lower crust during emplacement deformation was partitioned into zones weakened by magma and heat leading to the development of two layer parallel shear zones at the upper and lower contacts of the batholith transient vertical decoupling of the crust above and below the batholith occurred from 126 ma until 120 ma as magma was emplaced into and moved through a weak thick lower crust by 116 ma however much of the batholith had crystallized and the lowermost crust had cooled from 750 c t 850 c to t 650 700 c cooling was aided by the juxtaposition of pre existing crust against hot new crust and by the efficient extraction of partial melts out of the lower crust cooling together with dehydration of the lower crust and mafic compositions led to the development of a strong dry lower crustal root by 116 ma a strong lower crust resulted in high degrees of vertical coupling between the upper and lower crust during contraction from 116 to 105 ma even as magma continued to be emplaced into the mid upper crust a narrow focused orogenic style in the upper crust at this time reflected a highly viscous lower crust through which compressional stresses were transferred vertically the results imply that changes in plate boundary dynamics rather than the thermal weakening of thick lower crust during convergence controlled the onset of regional extension at 108 105 ma

linking hematite u th he dating with the microtextural record of seismicity in the wasatch fault damage zone utah usa techniques directly dating fault slip are few limiting the ability to interpret the rock record of seismicity hematite is commonly found in fault zones amenable to u th he dating and slip surface hematite may be reset by shear heating events and or recrystallization glossy hematite coated fault surfaces in the wasatch fault footwall damage zone utah usa exhibit evidence of hematite cataclasis and preserve pliocene hematite u th he dates apatite u th he and fission track data from the host gneiss indicate footwall unroofing through 2 km by ca 4 5 ma internally reproducible but disparate hematite u th he dates 4 5 ma and younger from isolated locations on a single fault surface do not reflect ambient cooling we hypothesize that these dates and associated iridescence and annealed crystal texture document rapid cooling from friction generated heat during small seismic slip events between ca 4 5 and 2 5 ma thus hematite u th he dating offers the potential to decipher thermal anomalies in the rock record associated with slip on 10 5 10 6 yr time scales

long term tectonothermal history of laramide basement from zircon he age eu correlations abstract the long term 1 ga thermal histories of cratons are enigmatic with geologic data providing only limited snapshots of their evolution we use zircon u th he zircon he thermochronology and age composition correlations to understand the proterozoic phanerozoic thermal history of archean wyoming province rocks exposed in the northern laramide ranges of western north america zircon he ages from the wind river range 54 dates and bighorn mountains 32 dates show negative correlations with effective uranium eu a proxy for radiation damage zircon dates from the bighorns are between 960 ma low eu and 20 ma high eu whereas samples from the wind rivers are between 582 ma low eu and 33 ma high eu we applied forward modeling using the zircon radiation damage and annealing model zrdaam to understand this highly variable dataset a long term t t path that is consistent with the available geologic constraints successfully reproduced age eu correlations the best fit to the wind rivers data involves two phases of rapid cooling at 1800 1600 ma and 900 700 ma followed by slower cooling until 525 ma during the phanerozoic these samples were heated to maximum temperatures between 160 and 125 c prior to laramide cooling to 50 c between 60 and 40 ma data from the bighorn mountains were successfully reproduced with a similar thermal history involving cooler phanerozoic temperatures of 115 c and earlier laramide cooling between 85 and 60 ma our results indicate that age eu correlations in zircon he datasets can be applied to extract long term thermal histories that extend beyond the most recent cooling event in addition our results constrain the timing magnitude and rates of cooling experienced by archean wyoming province rocks between recognized deformation events including the 1 ga period represented by the regionally extensive great unconformity

4he behavior in calcite filling viewed by u th he dating 4he diffusion and crystallographic studies fault filling calcite crystals sampled from the eocene oligocene gondrecourt graben paris basin eastern france have been studied in order to test the potentiality of calcite u th he dating based on recognized he retention behavior in crystal lattice at surface temperature copeland et al 2007 the samples have been selected because of their relatively old eocene to oligocene precipitation age and cold thermal history 40 c since precipitation they were sorted into three main tectonic and morphological groups in order of precipitation including 1 micro fracture calcites 2 breccia and associated geodic calcites and 3 vein and associated geodic calcites u th he dating of 63 calcite fragments yields ages dispersed from 0 2 0 02 to 35 8 2 7 ma as well as two older dates of 117 10 and 205 28 ma 1 these he ages correlate with grain chemistry such as sr ree concentrations or la yb n ratios likely reflecting parent fluid evolution only the oldest he ages which correspond to the most recently precipitated crystals have preserved the total 4he budget since precipitation to better understand both the age dispersion and why calcites precipitated earlier show younger ages 4he diffusion experiments have been conducted on 10 gondrecourt calcite fragments from 3 samples with he ages of 0 2 6 ma additionally a crystallographic investigation by x ray diffraction xrd performed on similar samples reveals that crystal structure evolves with increasing temperature beginning with micro cracks and cleavage opening these xrd results shed light on the u th he data indicating that in fault filling calcite he retention is controlled by multiple diffusion domains mdd with various sizes and therefore evolves through time with strong consequences on u th he age we thus interpret the gondrecourt calcite u th he age scatter as a consequence of the production of defects due to successive calcite crystallization phases associated with the deformation history

patterns of late cenozoic exhumation deduced from apatite and zircon u he ages from fiordland new zealand new apatite and zircon u th he ages from the fiordland region of new zealand s south island expand on earlier results and provide new constraints on patterns of late cenozoic exhumation and cooling across this region zircon u th he cooling ages in combination with increased density of apatite ages show that in addition to a gradual northward decrease in cooling ages that was seen during an earlier phase of this study there is also a trend toward younger cooling ages to the east distinct breaks in cooling age patterns on southwestern fiordland appear to be correlated to the location of previously mapped faults the northward decrease in ages may reflect asynchronous cooling related to migration in the locus of exhumation driven by subduction initiation or it may reflect synchronous regional exhumation that exposed different structural levels across fiordland or some combination of these effects in either case differential exhumation accommodated by major and minor faults that dissect fiordland basement rocks apparently played an important role in producing the resulting age patterns

magnetite u th he dating and its application to the geochronology of intermediate to mafic volcanic rocks abstract we present a new approach to dating intermediate to mafic volcanic rocks using magnetite u th he geochronology magnetite is common in volcanic rocks that typically do not contain easily datable minerals such as sanidine or zircon analytical procedures for producing magnetite u th he ages have been developed including mineral separation sample air abrasion to correct for ejection effects he extraction measurement sample dissolution and anion exchange column chemistry procedures dated magnetite crystals were non skeletal euhedral to subhedral and 100 300 m in size to test the reliability of this new geochronometer four basaltic to andesitic samples lacking sanidine or zircon were dated by both magnetite u th he and whole rock 40 ar 39 ar methods for two samples the ages from the different geochronometers are in excellent agreement 40 ar 39 ar age spectrum affected by 39 ar recoil yielded a well defined magnetite u th he age that is consistent with 40 ar 39 ar age data from similar nearby volcanic rocks the final sample however exhibited a near 40 discrepancy between the two methods despite yielding reproducible magnetite u th he ages in all cases the multi aliquot magnetite u th he ages n 7 exhibit 3 11 2 variation about the mean age indicating that reproducibility for magnetite u th he ages is comparable to that of apatite and zircon u th he analyses in order to assess the he retentivity we conducted a single magnetite helium diffusion experiment yielding a well behaved arrhenius relationship and a closure temperature of 250 c d t d t 10 c myr magnetite s high he retentivity coupled with u th he age reproducibility demonstrates good potential for magnetite u th he dating as an alternative volcanic geochronometer particularly in cases where samples yield inconclusive or uninterpretable 40 ar 39 ar ages

α emitting mineral inclusions in apatite their effect on u th he ages and how to reduce it u th rich mineral inclusions in apatite are often held responsible for erroneously old u th he ages because they produce parentless he three aspects associated with this problem are discussed here first simple dimensional considerations indicate that for small mineral inclusions the parentless helium problem might not be as serious as generally thought for example a mineral inclusion that is 10 the length width and height of its host apatite needs to be a thousand times more concentrated in u and th to produce an equal amount of he therefore single isolated inclusions smaller than a few m are unlikely to contribute significant helium for larger or more abundant inclusions the parentless helium problem can be solved by dissolution of the apatite and its inclusions in hot hf second besides creating parentless helium inclusions also complicate ejection corrections mathematical exploration of this latter problem for spherical geometries reveals that for randomly distributed inclusions the probability distribution of single grain ages is predicted to have a sharp mode at the mean age with tails towards younger and older ages multiple grain measurements will yield accurate and precise age estimates if 10 or more randomly distributed emitting mineral inclusions are present in a sample third thermal modeling indicates that mineral inclusions have a non trivial but minor 5 c effect on the closure temperature these predictions were tested on apatites from rapidly cooled migmatites of naxos greece which contain abundant u rich zircon inclusions thirty seven samples were subjected to two kinds of treatment the pooled age i e the synthetic multi grain age computed from a number of single grain analyses of four inclusion free samples 13 apatites prepared in hno3 is 10 9 ma close to apatite and zircon fission track ages from the same rock u th he ages of 14 inclusion bearing samples dissolved in hno3 range between 9 and 45 ma with a pooled age of 22 6 ma the ages of 19 hf treated samples range between 5 and 16 ma with 10 of 14 single grain samples between 9 and 13 ma and a pooled age of 10 9 ma these observations agree with the theoretical predictions and support the addition of hf treated apatite u th he dating to the thermochronological toolbox

low temperature anisotropic diffusion of helium in zircon implications for zircon u th he thermochronometry abstract in the last decade the zircon u th he zhe thermochronometer has been applied to a variety of geologic problems although bulk diffusion coefficients for he in zircon are available from laboratory step heating experiments little is known about the diffusion mechanism s and their dependence on the crystallographic structure of zircon here we investigate the diffusion of he in perfectly crystalline zircon using atomistic simulation methods that provide insights into the structural pathways of he migration in zircon empirical force fields and quantum mechanical calculations reveal that the energy barriers for he diffusion are strongly dependent on structure the most favorable pathway for he diffusion is the 0 0 1 direction through the open channels parallel to the c axis e 001 13 4 kj mol 1 activation energy for tracer diffusion of a he atom along 0 0 1 in contrast energy barriers are higher in other directions where narrower channels for he diffusion are identified such as 1 0 0 1 0 1 and 1 1 0 e of 44 8 101 7 and 421 3 kj mol 1 respectively molecular dynamics simulations are in agreement with these results and provide additional insight in the diffusion mechanisms along different crystallographic directions as well as the temperature dependence below the closure temperature of he in zircon t c 180 c reiners p w spell t l nicolescu s and zanetti k a 2004 zircon u th he thermochronometry he diffusion and comparisons with ar 40 ar 39 dating geochim cosmochim acta 68 1857 1887 diffusion is anisotropic as he moves preferentially along the 0 0 1 direction and calculated tracer diffusivities along the two most favorable directions differ by approximately five orders of magnitude d 001 d 100 10 5 at t 25 c above this temperature he atoms start to hop between adjacent 0 0 1 channels along 1 0 0 and 0 1 0 directions perpendicular to the c axis the diffusion along 1 0 0 and 0 1 0 is thermally activated such that at higher temperatures he diffusion in zircon becomes nearly isotropic d 001 d 100 10 at t 580 c these results suggest that the anisotropic nature of he diffusion at temperatures near the closure temperature should be considered in future diffusivity experiments furthermore care should be taken when making geologic interpretations e g exhumation rates timing of cooling etc from this thermochronometer until the effects of anisotropic diffusion on bulk ages and closure temperature estimates are better quantified

is apatite u th zonation information necessary for accurate interpretation of apatite u th he thermochronometry data abstract new u th and ce concentration maps were acquired by la icpms for 70 apatites from 18 cratonic basement samples from the canadian shield to characterize the nature and variability in apatite u th zonation all apatites are zoned in effective u concentration eu with 80 exhibiting modest zoning that varies from a factor of 1 2 to 2 4 zonation patterns include those with a general eu decrease from core to rim 25 a core to rim eu increase 35 and patchy or irregularly shaped eu distributions 40 most samples consist of individual apatites with markedly different and opposing eu profiles cathodoluminescence cl images were obtained for 258 apatites in 25 samples comparison of eu and cl patterns reveals no consistent relationships suggesting that cl is not a reliable proxy for apatite eu zonation we explore the implications of eu zonation for the interpretation of apatite u th he ahe data by comparing the age predictions for representative and endmember apatite eu profiles with those for unzoned apatites we focus on thermal histories that should magnify eu zonation effects including slow monotonic cooling extended he partial retention zone heprz residence and protracted reheating and cooling application of incorrect ejection correction f t factors different he concentration gradients and variable intracrystalline retentivity due to heterogeneous radiation damage may cause the ahe dates for zoned and unzoned apatites to differ in our dataset the magnitude of the f t effect is by simulating zoned apatite suites from representative and endmember samples we find that eu zoning adds modest age dispersion to most samples the commonly opposing eu profiles for individual apatites in most samples have different effects on the predicted age and partially cancel each other out to reduce the predicted age difference at the sample level higher predicted age differences between zoned and unzoned samples generally correlate with greater sample scatter the age deviations generally fall within the 15 uncertainty range that we commonly apply for sample ahe dates except in unusual circumstances it appears unlikely that the unzoned apatite assumption will lead to misinterpretation of ahe datasets thus precluding the need to routinely acquire apatite eu zonation information as part of ahe dating studies

tempo of burial and exhumation within the deep roots of a magmatic arc fiordland new zealand the u pb thermochronology of titanite apatite and rutile from a crustal profile through a mesozoic magmatic arc in fiordland new zealand is used to constrain the timing and duration of significant vertical movements during arc construction and evolution titanite data from deep crustal 12 13 kbar basement and cover rocks of central fiordland imply that contractional arc thickening 25 km occurred by 111 1 113 4 ma within a few million years of a major phase of mid to deep crustal magmatism this finding suggests that this cycle of magmatism arc thickening and high grade metamorphism occurred in 6 2 m y in contrast to rapid burial significant unroofing of the central fiordland gran ulites was more protracted requiring an additional 40 45 m y these new data are con sistent with continued residence of the granulites in thickened arc crust for 15 20 m y with subsequent major unroofing recorded by rutile cooling to 450 8c by ca 70 ma such temporal constraints are essential for comprehensive models of the growth modifi cation and unroofing of magmatic arc systems

u th ne and multidomain u th he systematics of a hydrothermal hematite from eastern grand canyon a hydrothermal hematite from the redwall limestone in eastern grand canyon arizona usa was analyzed for u th he and u th ne systematics the dense polycrystalline aggregate has u and th concentrations of 16 ppm each neon produced by particle capture on 18 o is abundant in the sample it constitutes 90 of total 21 ne and indicates a u th ne age of 217 5 ma this value provides a lower bound for the formation age of the sample but detailed interpretation is not possible given the current absence of neon diffusion data for hematite the 130 2 ma u th he age is much younger than the ne age and therefore must be a cooling age highly reproducible step heating he release experiments reveal the existence of a spectrum of diffusion domain sizes spanning about 3 orders of magnitude in radius corresponding to he closure temperatures ranging from 0 to 150 c these domains may correspond to crystallites in the specimen which are observed to range from smaller than a few nm to a few m 4he 3he age spectra acquired from the same experiments define a distinctive sigmoidal shape with step ages increasing monotonically from 0 to 210 ma this pattern clearly records the presence of the multiple domains as the hematite acquired its radiogenic helium coupling of the 4he 3he spectra and the diffusion data allow modeling of permissible time temperature paths experienced by the hematite these paths can be compared with independent thermal history reconstructions for eastern grand canyon characterized by 170 m yr of sedimentary burial followed by regional unroofing and canyon incision beginning at 80 ma the unroofing portion of the path inferred from the hematite is in good agreement with apatite u th he and fission track data the results permit rapid cooling in late cretaceous early tertiary time suggest a subsequent period of slow cooling through temperatures of 50 70 c with accelerated cooling ensuing after 20 ma in contrast the hematite data demand temperatures in some part s of the mesozoic that are at least 100 c hotter than expected based on burial depth estimates this may indicate that the sample experienced a reheating event associated with the passage of fluids similar to those observed at nearby mineralized breccia pipes overall these data provide motivation for further work to understand he and ne behavior in hematite and their potential use in thermochronometry

multistage exhumation and juxtaposition of lower continental crust in the western canadian shield linking high resolution u pb and 40ar 39ar thermochronometry with pressure temperature deformation paths 1 new u pb titanite apatite rutile and 40ar 39ar hornblende muscovite biotite data are linked with existing pressure temperature deformation paths to impose detailed temporal constraints on the juxtaposition and unroofing of domains in an extensive 20 000 km2 region of exhumed lower crust in the east lake athabasca region of the western canadian shield in the 200 m y between circa 1 9 ga high pressure granulite facies metamorphism and circa 1 7 ga unconformable deposition of athabasca basin sediments on the exhumed rocks our analysis reveals at least three distinct phases of unroofing at different rates specifically we distinguish 1 an early phase of extensional unroofing in the chipman domain at rates of 1 5 2 0 km m y from 1 0 gpa to 0 8 gpa associated with mafic magmatism and metamorphism 2 an episode of regional contractional uplift and associated unroofing along the legs lake shear zone at circa 1850 ma from 0 7 0 8 gpa to 0 4 0 5 gpa and 3 a final period of extensional unroofing at rates of 0 2 0 3 km m y from 0 4 to 0 5 gpa that culminated in transport of current exposures to near surface conditions the cooling patterns and retrograde assemblages are consistent with pauses between exhumational phases the apparent convergence of disparate higher temperature histories in several deep crustal domains at circa 1 89 1 88 ga implies their juxtaposition at 0 7 0 8 gpa conditions regional east directed thrusting of the deep crustal domains as a coherent unit at 1 85 ga juxtaposed the granulites with middle crustal hearne domain rocks at 0 4 0 5 gpa conditions the detailed exhumational history allows correlation with changing regional tectonic regimes associated with the amalgamation of laurentia the temporal and spatial heterogeneity of exhumation patterns in the east lake athabasca region may be a common feature of the unroofing histories of lower crustal rocks

influence of radiation damage on titanite he diffusion kinetics titanite is a common u th bearing accessory mineral that previous study suggested has a u th he closure temperature tc of 210 175 c although it has been recognized that radiation damage influences the he retentivity of apatite and zircon this effect has not been documented in titanite we acquired 51 the analyses from 11 archean basement samples across the kaapvaal craton south africa the the dates range from 20 to 1200 ma and are negatively correlated with their 10 970 ppm span of effective uranium concentration eu raman spectra acquired for 4 of these samples display peak broadening with increasing eu 4he diffusion experiments on the same 4 samples suggest tc values that vary by 175 c estimated alpha dose versus tc for our results and published the diffusion data document tc values of 150 210 c at alpha doses 50 1016 g with a sharp decrease in he retentivity above this threshold this damage diffusivity pattern is similar to zircon but the damage level at which the reduction in retentivity occurs appears to be lower for titanite because titanite typically has substantially lower eu than zircon for the same protracted thermal history the dates are likely to record higher temperature portions of the history than zircon u th he dates the results demonstrate that the dating can access a much larger temperature range and potentially be used to decipher more detailed time temperature paths than previously thought

effect of alpha damage annealing on apatite u th he thermochronology recent data on he diffusion challenge the temperature sensitivity of apatite u th he thermochronology the damage induced by recoil of u and th decay series during emission of particles recoil damage has been proposed to modify he diffusion properties through time however we propose that annealing of these irradiation defects may be an important phenomenon and may be significant in case of slowly cooled or reheated basement rocks to test this hypothesis we developed a quantitative model including an explicit treatment of recoil damage annealing and their effect on he diffusion kinetics and calibrate it against literature data our model is based on two hypotheses 1 helium is in equilibrium between an apatite crystal and its defects and 2 alpha recoil damage annealing can be described analogously to fission track annealing this model has been embedded into a monte carlo simulation of helium production ejection diffusion and applied to data from the french massif central a complex slowly cooled terrain with burial reheating where the thermal history has been constrained by previous fission track ft data including ft length distributions u th he ages are close to the ft ages from the same samples and are generally reproducible among replicates but some samples present he age dispersion that is not correlated with crystal size our model reproduces the massif central data very well except for three samples where he ages are older than corresponding ft ages we show that annealing of irradiation damage has an important impact on retentivity of helium and that the he content he is only a rough approximation of the damage level in particular our results show that independence of he ages on crystal sizes in case of reheated samples is a clear indication of the higher he retentivity induced by recoil defects and that an explicit treatment of defect annealing is required for a correct interpretation of u th he ages in such a case more generally a correlation with the crystal size can bring information on the thermal path only if the age of defects well represented by the fission track age is available due to the dependence of the partial retention zone on damages conversely in case of rapid cooling or for samples having low u and th contents damage effects can be ignored without significant effects on he ages

the potential of crinoids as u th sm he thermochronometers abstract u th sm he dating of crinoid fossils from late paleozoic rocks from a variety of sites in north america suggest that this material can be used as a useful low t thermochronometer under certain conditions available data suggest that common he is present in crinoids and at the low production rates typical of calcite this common he will be sufficient to produce spurious ages in crinoid samples of any age heating of samples to temperatures in excess of the opening temperature will expel all he common and radiogenic with all subsequent he accumulation being exclusively radiogenic such samples have potential for being robust thermochronometers sensitive to temperatures in the range of 60 110 c without common he analysts can take advantage of the large size of crinoid columnals and produce reliable ages even when young miocene or with low production rates eu 0 1 ppm samples not subjected to post depositional heating above the opening temperature burial to depths of 2 to 4 km depending on geothermal gradient sedimentation rate and residence time at peak t will likely retain common he and be unsuitable for thermochronologic studies

influence of crystal size on apatite u th he thermochronology an example from the bighorn mountains wyoming near surface tectonic and geomorphic processes involve cooling of rocks through low temperatures 50 200 c because rates of helium diffusion in apatite titanite and zircon are sensitive to temperature variations in this range uranium thorium helium thermochronometry u th he dating is well suited to establishing the timing and rates of these processes in the geologic record however because fractional loss of he is controlled by crystal size such that larger crystals retain a larger fraction of radiogenic he u th he ages must vary not only with thermal history but also with crystal size here we present crystal size correlated he ages from co existing apatites from the bighorn mountains wyoming that range from 100 to 350 ma these correlations are a sensitive indicator of the rock s thermal history in a temperature range below the system s nominal closure temperature t c 70 c for apatite and are consistent with a thermal history involving residence in the upper 2 3 km of crust since the precambrian with maximum temperatures of 65 80 c just prior to laramide orogenic exhumation the influence of crystal size on he ages will be most apparent in rocks where temperatures have been in the range of partial he retention for long periods of time 30 70 c for 10 7 years in such cases accurate interpretation of u th he ages must incorporate the effect of crystal size and this method may provide insights to thermal histories of rocks in previously inaccessible low temperature ranges

apatite u th he thermochronometry using a radiation damage accumulation and annealing model helium diffusion from apatite is a sensitive function of the volume fraction of radiation damage to the crystal a quantity that varies over the lifetime of the apatite using recently published laboratory data we develop and investigate a new kinetic model the radiation damage accumulation and annealing model rdaam that adopts the effective fission track density as a proxy for accumulated radiation damage this proxy incorporates creation of crystal damage proportional to production from u and th decay and the elimination of that damage governed by the kinetics of fission track annealing the rdaam is a version of the helium trapping model hetm shuster d l flowers r m and farley k a 2006 the influence of natural radiation damage on helium diffusion kinetics in apatite earth planet sci lett 249 148 161 calibrated by helium diffusion data in natural and partially annealed apatites the chief limitation of the hetm now addressed by rdaam is its use of he concentration as the radiation damage proxy for circumstances in which radiation damage and he are not accumulated and lost proportionately from the crystal by incorporating the rdaam into the hefty computer program we explore its implications for apatite u th he thermochronometry we show how u th he dates predicted from the model are sensitive to both effective u concentration eu and details of the temperature history the rdaam predicts an effective he closure temperature of 62 c for a 28 ppm eu apatite of 60 m radius that experienced a 10 c ma monotonic cooling rate this is 8 c lower than the 70 c effective closure temperature predicted using commonly assumed durango diffusion kinetics use of the rdaam is most important for accurate interpretation of u th he data for apatite suites that experienced moderate to slow monotonic cooling 1 0 1 c ma prolonged residence in the helium partial retention zone or a duration at temperatures appropriate for radiation damage accumulation followed by reheating and partial helium loss under common circumstances the rdaam predicts u th he dates that are older sometimes much older than corresponding fission track dates nonlinear positive correlations between apatite u th he date and eu in apatites subjected to the same temperature history are a diagnostic signature of the rdaam for many but not all thermal histories observed date eu correlations in four different localities can be explained with the rdaam using geologically reasonable thermal histories consistent with independent fission track datasets the existence of date eu correlations not only supports a radiation damage based kinetic model but can significantly limit the range of acceptable time temperature paths that account for the data in contrast these datasets are inexplicable using the durango diffusion model the rdaam helps reconcile enigmatic data in which apatite u th he dates are older than expected using the durango model when compared with thermal histories based on apatite fission track data or other geological constraints it also has the potential to explain at least some cases in which u th he dates are actually older than the corresponding fission track dates

low long term erosion rates and extreme continental stability documented by ancient u th he dates zircon and apatite crystals from the western canadian shield yield u th he dates that are the oldest yet reported for terrestrial rocks zircon dates from 1 73 to 1 58 ga are consistent with independent geological and thermochronological constraints and indicate that the rocks were at temperatures 180 c and crustal depths 7 10 km since ca 1 7 ga apatite dates from 0 95 to 0 55 ga suggest residence of rocks at temperatures 40 50 c and crustal depths 1 5 2 km for 1 0 0 6 b y when interpreted using conventionally accepted apatite he diffusion kinetics and considering the proposed effect of radiation damage on apatite he retentivity our analysis implies long term integrated unroofing rates of 2 5 m yr since ca 1 7 ga these rates are significantly lower than the long term rates suggested by previous thermochronological data sets in continental interiors but are within the range of short term erosional estimates the results are consistent with the extreme stability of this region since the proterozoic

helium diffusion in rutile and titanite and consideration of the origin and implications of diffusional anisotropy abstract diffusion of helium has been characterized in natural sphene titanite and synthetic rutile polished slabs of sphene and rutile were implanted with 100 kev 3he at a dose of 5 1015 3he cm2 and annealed in 1 atm furnaces 3he distributions following experiments were measured with nuclear reaction analysis using the reaction 3he d p 4he for diffusion in rutile we obtain the following arrhenius relations d a 2 26 1 0 1 0 exp 126 11 kj mol 1 rt m 2 sec 1 d c 1 75 1 0 8 exp 12 0 7 kj mol 1 rt m 2 sec 1 although activation energies for diffusion parallel to the c and a axes are comparable there is marked diffusional anisotropy with diffusion parallel to the c axis about 2 orders of magnitude faster than transport parallel to the a axis these diffusivities bracket the values determined for he diffusion in rutile in bulk release experiments stockli et al 2005 stockli et al 2007 wolfe 2009 although the role of anisotropy could not be directly evaluated in those measurements in titanite the following arrhenius relation was obtained over the temperature range 252 550 c for diffusion parallel to the a axis d 2 14 1 0 6 exp 148 8 kj mol 1 rt m 2 sec 1 in contrast to rutile and zircon cherniak et al 2009 titanite shows little evidence of anisotropy as diffusivities parallel to the a and c axes are similar and diffusivities for titanites from two different localities are similar he diffusion coefficients obtained in this study are similar to those measured through bulk release of he by step heating shuster et al 2003 over the investigated temperature range diffusion of he in titanite is similar to that of he diffusion in rutile parallel to the c axis but much faster than diffusion parallel to the a axis since the diffusion of he in rutile exhibits such pronounced anisotropy we model diffusional loss of he with a recently developed finite element code watson et al 2010 created to simulate diffusion in cylindrical geometry with differing radial and axial diffusion coefficients we present example applications to evaluate helium losses from rutile grains as a function of grain size and length to diameter ratios in efforts to better understand the occurrence of pronounced anisotropy for he diffusion in some crystals e g rutile zircon but not in others e g apatite titanite we consider the density and distribution of interstitial apertures in the crystal structure that might permit he migration these determinations an extension of the concept of ionic porosity are consistent with observations of relative diffusivities and the existence or absence of significant anisotropy for the minerals rutile zircon apatite and titanite

natural age dispersion arising from the analysis of broken crystals part i theoretical basis and implications for the apatite u th he thermochronometer abstract over the last decade major progress has been made in developing both the theoretical and practical aspects of apatite u th he thermochronometry and it is now standard practice and generally seen as best practice to analyse single grain aliquots these individual prismatic crystals are often broken and are fragments of larger crystals that have broken during mineral separation along the weak basal cleavage in apatite this is clearly indicated by the common occurrence of only 1 or no clear crystal terminations present on separated apatite grains and evidence of freshly broken ends when grains are viewed using a scanning electron microscope this matters because if the 4 he distribution within the whole grain is not homogeneous because of partial loss due to thermal diffusion for example then the fragments will all yield ages different from each other and from the whole grain age here we use a numerical model with a finite cylinder geometry to approximate 4 he ingrowth and thermal diffusion within hexagonal prismatic apatite crystals this is used to quantify the amount and patterns of inherent natural age dispersion that arises from analysing broken crystals a series of systematic numerical experiments were conducted to explore and quantify the pattern and behaviour of this source of dispersion using a set of 5 simple thermal histories that represent a range of plausible geological scenarios in addition some more complex numerical experiments were run to investigate the pattern and behaviour of grain dispersion seen in several real data sets the results indicate that natural dispersion of a set of single fragment ages defined as the range divided by the mean arising from fragmentation alone varies from c 7 even for rapid c 10 c ma monotonic cooling to over 50 for protracted complex histories that cause significant diffusional loss of 4 he the magnitude of dispersion arising from fragmentation scales with the grain cylindrical radius and is of a similar magnitude to dispersion expected from differences in absolute grain size alone spherical equivalent radii of 40 150 m this source of dispersion is significant compared with typical analytical uncertainties on individual grain analyses c 6 and standard deviations on multiple grain analyses from a single sample c 10 20 where there is a significant difference in the u and th concentration of individual grains eu the effect of radiation damage accumulation on 4 he diffusivity assessed using the rdaam model of flowers et al 2009 is the primary cause of dispersion for samples that have experienced a protracted thermal history and can cause dispersion in excess of 100 for realistic ranges of eu concentration i e 5 100 ppm expected natural dispersion arising from the combined effects of reasonable variations in grain size radii 40 125 m eu concentration 5 150 ppm and fragmentation would typically exceed 100 for complex thermal histories in addition to adding a significant component of natural dispersion to analyses the effect of fragmentation also acts to decouple and corrupt expected correlations between grain ages and absolute grain size and to a lesser extent between grain age and effective uranium concentration eu considering fragmentation explicitly as a source of dispersion and analysing how the different sources of natural dispersion all interact with each other provides a quantitative framework for understanding patterns of dispersion that otherwise appear chaotic an important outcome of these numerical experiments is that they demonstrate that the pattern of age dispersion arising from fragmentation mimics the pattern of 4 he distribution within the whole grains thus providing an important source of information about the thermal history of the sample we suggest that if the primary focus of a study is to extract the thermal history information from u th he analyses then sampling and analytical strategies should aim to maximise the natural dispersion of grain ages not minimise it and should aim to analyse circa 20 30 grains from each sample the key observations and conclusions drawn here are directly applicable to other thermochronometers such as the apatite rutile and titanite u pb systems where the diffusion domain is approximated by the physical grain size

diffusion of helium in zircon and apatite abstract diffusion of helium has been characterized in natural zircon and apatite polished slabs of zircon and apatite oriented either normal or parallel to c were implanted with 100 kev 3he at a dose of 5 1015 3 he cm2 diffusion experiments on implanted zircon and apatite were run in pt capsules in 1 atm furnaces 3he distributions following experiments were measured with nuclear reaction analysis using the reaction 3he d p 4he for diffusion in zircon we obtain the following arrhenius relations d c 2 3 10 7 exp 146 11 kj mol 1 r t m 2 s 1 d c 1 7 10 5 exp 148 17 kj mol 1 r t m 2 s 1 although activation energies for diffusion normal and parallel to c are comparable there is marked diffusional anisotropy with diffusion parallel to c nearly 2 orders of magnitude faster than transport normal to c these diffusivities bracket the range of values determined for he diffusion in zircon in bulk release experiments although the role of anisotropy could not be directly evaluated in those measurements in apatite the following arrhenius relation was obtained over the temperature range of 148 449 c for diffusion normal to c d 2 10 10 6 exp 117 6 kj mol 1 r t m 2 s 1 in contrast to zircon apatite shows little evidence of anisotropy he diffusivities obtained in this study fall about an order of magnitude lower than diffusivities measured through bulk release of he through step heating and within an order of magnitude of determinations where ion implantation was used to introduce helium and he distributions measured with elastic recoil detection since the diffusion of he in zircon exhibits such pronounced anisotropy helium diffusional loss and closure cannot be modeled with simple spherical geometries and the assumption of isotropic diffusion a finite element code cylmod has recently been created to simulate diffusion in cylindrical geometry with differing radial and axial diffusion coefficients we present some applications of the code in evaluating helium lost from zircon grains as a function of grain size and length to diameter ratios and consider the effects of shape anisotropy where diffusion is isotropic as in the case of apatite but shapes of crystal grains or fragments may depart significantly from spherical geometry

dating kimberlite emplacement with zircon and perovskite u th he geochronology kimberlites provide rich information about the composition and evolution of cratonic lithosphere accurate geochronology of these eruptions is key for discerning spatiotemporal trends in lithospheric evolution but kimberlites can sometimes be difficult to date with available methods we explored whether u th he dating of zircon and perovskite can serve as reliable techniques for determining kimberlite emplacement ages we obtained zircon and or perovskite u th he zhe phe dates from sixteen southern african kimberlites most samples with abundant zircon yielded reproducible zhe dates 15 dispersion that are in good agreement with published eruption ages the majority of dated zircon were xenocrystic zircons with reproducible dates were fully reset during eruption or resided at temperatures above the zhe closure temperature prior to entrainment in the kimberlite magma not dating hazy and radiation damaged grains can help avoid anomalous results for more shallowly sourced zircons that underwent incomplete damage annealing and or partial he loss during the eruptive process all seven kimberlites dated with phe yielded reproducible 15 dispersion and reasonable results we conducted two preliminary perovskite 4he diffusion experiments which suggest a phe closure temperature of 300 c perovskite in kimberlites is unlikely to be xenocrystic and its relatively high temperature sensitivity suggests that phe dates will typically record emplacement rather than post emplacement processes zhe and phe geochronology can effectively date kimberlite emplacement and provide useful complements to existing techniques this article is protected by copyright all rights reserved

dating paleochannel iron ore by u th he analysis of supergene goethite hamersley province australia u th he dating of late stage authigenic goethite combined with corrections for diffusive loss of he 4 by the he 4 he 3 methodology reveals strong correlation between a sample s age and its depth in ferruginized channel sediments from the yandicoogina deposit western australia corrected ages ranging from ca 18 ma near the surface to ca 5 ma at the bottom of the profile indicate that ferruginization of the aggraded channels becomes progressively younger with depth this trend is consistent with goethite precipitation at the groundwater atmosphere interface during water table drawdown driven by the aridification of western australia during the neogene the results demonstrate that the u th he system is ideal for dating goethite if diffusive loss corrections are applied the approach is suitable for dating weathering reactions on earth and should also be suitable for dating fe oxyhydroxides in the martian regolith

helium diffusion and u th he thermochronometry of titanite we have explored the diffusivity characteristics of radiogenic he in titanite sphene and have developed analytical techniques for u th he dating of this mineral results of incremental he outgassing experiments performed on titanites from a variety of geological environments suggest a thermally activated volume diffusion mechanism with an activation energy of 44 6 3 4 2 kcal mol and a frequency factor of 60 cm 2 s diffusivity is highly linearly correlated with the inverse square of the grain size indicating that the he diffusion domain in titanite is the crystal itself for typical titanite grains of 200 to 800 m minimum dimension the he closure temperature is in the range 191 to 218 c for a cooling rate of 10 c myr there is no indication in the titanites we studied that radiation damage plays a major role in he diffusion u th he ages of titanites from quickly cooled rocks yield ages with 5 8 1 reproducibility that are concordant with known ages and u th he ages of titanites from slowly cooled rocks are consistent with independently established cooling paths supporting the closure temperature estimates these experiments suggest that titanite u th he ages may be useful for constraining cooling histories at temperatures near the lower limit of those accessed by feldspar 40 ar 39 ar dating but higher than apatite fission track or u th he dating

the influence of natural radiation damage on helium diffusion kinetics in apatite abstract stepwise degassing diffusion experiments on 39 different apatite samples using radiogenic 4he and proton induced 3he reveal a range in closure temperature tc from 50 to 115 c for a cooling rate of 10 c myr there is no correlation between helium diffusion and apatite chemistry including f cl ratio but the closure temperature is positively correlated with the radiogenic 4he concentration 4he in each sample we argue that 4he is a proxy for a sample s natural exposure to actinide radioactivity below the closure temperature and that helium diffusion in apatite is impeded by radiation induced damage to the apatite structure the kinetics must therefore be an evolving function of time measured diffusivities thus reflect a snapshot in time and cannot alone be applied to the thermochronometric interpretation of a given sample the effect of radiation damage on helium diffusion appears to far exceed other known controls on helium diffusivity including grain size our diffusion data are well described by a previously proposed quantitative model that consists of two arrhenius relations one for volume diffusion through undamaged mineral structure and one for release of helium from radiation damage traps the unknown parameters in this trapping model were determined from the diffusion experiments and allow us to develop a tentative mathematical function that relates diffusivity to temperature and 4he by inserting this function into a 4he production diffusion model we have explored how these results affect the interpretation of apatite u th he thermochronometry the model predicts that the effective 4he closure temperature of apatite will vary with cooling rate and effective u concentration eu and may differ from the commonly assumed tc of 70 c by up to 15 c the 4he partial retention zone will look similar to previous expectations but its depth will depend on accumulation time and on eu most notably samples subjected to reheating after accumulation of substantial radiation damage will be more retentive than previously expected these predictions are consistent with recent observations of unexpected apatite u th he ages in some settings

apatite u th sm he thermochronology of rapidly cooled samples the effect of he implantation apatite u th sm he ahe thermochronology is increasingly used for reconstructing geodynamic processes of the upper crust and the surface results of ahe thermochronology however are often in conflict with apatite fission track aft thermochronology yielding an inverted age relationship with ahe dates older than aft dates of the same samples this effect is mainly explained by radiation damage of apatite either impeding he diffusion or causing non thermal annealing of fission tracks so far systematic age inversions have only been described for old and slowly cooled terranes whereas for young and rapidly cooled samples too old ahe dates are usually explained by the presence of undetected u and or th rich micro inclusions we report apatite u th sm he results for rapidly cooled volcanogenic samples deposited in a deep ocean environment with a relatively simple post depositional thermal history robust age constraints are provided independently through sample biostratigraphy all studied apatites have low u contents 5 ppm on average while aft dates are largely in agreement with deposition ages most ahe dates are too old for leg 43 where deposition age of sampled sediment is 26 5 29 5 ma alpha corrected average ahe dates are up to 45 ma indicating overestimations of ahe dates up to 50 this is explained by he implantation from surrounding host u th rich sedimentary components and it is shown that ahe dates can be corrected by mechanically abrading the outer part of grains we recommend that particularly for low u th apatites the possibility of he implantation should be carefully checked before considering the degree to which the alpha ejection correction should be applied

radiometric dating and temperature history of banded iron formation associated hematite gogebic iron range michigan usa newly developed techniques were used to characterize the age origin and temperature history of coarse hematite contained in high grade iron ore three botryoidal specimens from two different mines in the gogebic iron range near ironwood michigan usa yielded u th 21 ne ages concordant with plateau 4he 3he ages these data reveal that while two of the specimens grew at ca 772 41 ma the third formed 300 m y later 453 14 ma 4he 3he polydomain thermochronometry indicates hematite formation at 80 c for the older specimens and 60 c for the younger fe mobilization and recrystallization in open cavities at 3 5 km subsurface has occurred episodically or possibly continuously over hundreds of millions of years despite a quiescent tectonic setting throughout this interval while the analyzed specimens are distinct from typical fe ore their ages along with structural relationships constrain the onset of iron enrichment to between 1060 and 770 ma cooling of 0 5 c m y ensued between ca 800 and ca 550 ma declining to 0 1 c m y for the last 550 m y these data suggest an erosion rate of 6 m m y and 5 km of total erosion of the superior upland province through the phanerozoic and limit unroofing associated with pleistocene glaciation to 1 km

interpretation of u th he single grain ages from slowly cooled crustal terranes a case study from the transantarctic mountains of southern victoria land low temperature thermochronologic techniques e g apatite fission track aft thermochronology and u th he dating constrain near surface t t paths and are often applied to uplift denudation and landscape evolution studies samples collected in vertical profiles from granitic walls on either side of the ferrar glacier southern victoria land antarctica were analyzed using aft thermochronology and apatite u th he dating to further constrain the lowest temperature thermal history of this portion of the transantarctic mountains aft central ages vary systematically with elevation and together with track length information define a multi stage cooling denudation history in the cretaceous and early tertiary apatite u th he single grain age variation with elevation is not as systematic with considerable intra sample age variation although many complicating factors e g u and th rich micro inclusions fluid inclusions variation in crystal size particle ejection correction zonation and particle ejection correction implantation of he into a crystal or impediment of he diffusion out of a crystal and 147sm derived particles may contribute to age dispersion we found that variation in single grain ages correlated with cooling rate samples that cooled relatively quickly have less variation in single grain ages whereas samples that cooled relatively slowly 3 c m y or resided within an u th he partial retention zone heprz prior to more rapid cooling have a comparatively greater variation in ages decay of u and th via particle emission creates a 4he concentration profile dependent upon the initial parent u th within a crystal variation of single grain ages for samples with non homogeneous u th distributions will be enhanced with long residence time in the partial retention zone i e slow cooling because of the relative importance of loss via volume diffusion and loss via particle ejection with respect to the u th zonation and the grain boundary correction of ages for particle ejection ft correction factor typically assumes uniform u and th distribution within the crystal and when applied to a population of crystals with different u and th distributions will enhance the variation in ages most complicating factors listed above for apatite u th he ages result in ages that are too old we propose that if considerable variation in u th he single grain ages exists that a weighted mean age is determined once outlier single crystal ages are excluded using the criterion of chauvenet or a similar approach we suggest that the true age or most representative age for that age population lies between the minimum u th he age and the weighted mean age we apply this approach coupled with composite age profiles to better constrain the t t history of the profiles along the ferrar glacier significant intra sample variation in single crystal apatite u th he ages and other minerals dated by the u th he method should be expected especially when the cooling rate is slow the variation of u th he single crystal ages is therefore another parameter that can be used to constrain low temperature thermal histories

effects of u th rich grain boundary phases on apatite helium ages abstract single grain apatite u th sm he ages he ages from non detrital samples sometimes show larger dispersions than can be explained by known sources of age dispersion such as grain size radiation damage parent zonation fracturing or intracrystalline inclusions we present observations and model age bias effects of an additional source of apatite he age dispersion observed in some cases u th rich grain boundary phases gbps precipitated on apatite crystal surfaces apatite grains from several samples with dispersed apatite he ages are coated or partially coated by reddish orange gbps rich in fe u and th these gbps are heterogeneous in thickness grain coverage and composition and have effective u concentrations eu up to 1000 ppm these phases may have large effects on the bulk eu and 4he compositions of apatite grains and can produce significant he age biases the direction and magnitude of this age bias is primarily a function of four factors 1 the host grain s size 2 the gbps thickness and eu relative to the host grain 3 the timing of gbp formation relative to the cooling age of the host grain and 4 whether the gbps are preserved and analyzed with the grain during he dating some of the most severe effects occur when gbps formed before or near the time of the apatite cooling age but were lost just prior to analysis for example during mineral separation in this case gbps of commonly observed thicknesses 1 10 m with eu 2 10 times that of the host apatite grain implant enough 4he into typical sized apatite grains to produce hundreds of percent positive age biases in contrast when u th rich gbps are preserved and analyzed with apatite grains the resulting he ages can be negatively biased though the heterogeneity variable preservation and ambiguous formation ages of gbps preclude a general quantitative and practical solution to this problem our model demonstrates that observed age dispersions in some samples are consistent with the effects of u th rich grain boundary phases and illustrates the conditions under which and by how much they will bias he ages

natural age dispersion arising from the analysis of broken crystals part ii practical application to apatite u th he thermochronometry abstract we describe a new numerical inversion approach to deriving thermal history information from a range of naturally dispersed single grain apatite u th he ages the approach explicitly exploits the information about the shape of the 4he diffusion profile within individual grains that is inherent in the pattern of dispersion that arises from the common and routine practice of analysing broken crystals additional dispersion arising from differences in grain size and in u and th concentration of grains and the resultant changes to helium diffusivity caused by differential accumulation and annealing of radiation damage is explicitly included in this approach we calculate the ingrowth and loss due to both thermal diffusion and the effects of ejection of helium over time using a finite cylinder geometry broken grains are treated explicitly as fragments of an initially larger crystal the initial grain lengths l0 can be treated as unknown parameters to be estimated although this is computationally demanding a practical solution to the problem of solving for the unknown initial grain lengths is to simply apply a constant and sufficiently long l0 value to each fragment we found that a good value for l0 was given by the maximum fragment length plus two times the maximum radius of a given set of fragments currently whole crystals and fragments with one termination are taken into account a set of numerical experiments using synthetic fragment ages generated for increasingly complex thermal histories and including realistic amounts of random noise 5 15 are presented and show that useful thermal history information can be extracted from datasets showing very large dispersion these include experiments where dispersion arises only from fragmentation of a single grain length 400 m and radius 75 m c 6 50 dispersion including the effects of grain size variation for spherical equivalent grain radii between 74 and 122 m c 10 70 dispersion and the combined effects of fragmentation grain size and radiation damage for eu between 5 and 150 ppm c 10 107 dispersion additionally we show that if the spherical equivalent radius of a broken grain is used as a measure of the effective diffusion domain for thermal history inversions then this will likely lead to erroneous thermal histories being obtained in many cases the viability of the new technique is demonstrated for a real data set of 25 single grain u th he apatite ages obtained for a gabbro sample from the bk 1 bierkraal borehole drilled through the bushveld complex in south africa the inversion produces a well constrained thermal history consistent with both the u th he data and available fission track analysis data the advantage of the new approach is that it can explicitly accommodate all the details of conventional schemes such as the effects of temporally variable diffusivity zonation of u and th and arbitrary grain size variations and it works equally effectively for whole or broken crystals and for the most common situation where a mixture of both are analysed for the routine application of the apatite u th he thermochronometry technique with samples where whole apatite grains are rare our experiments indicate that 15 20 single grain analyses are typically required to characterise the age dispersion pattern of a sample the experiments also suggest that picking very short crystal fragments as well as long fragments or even deliberately breaking long crystals to maximise the age dispersion in some cases would ensure the best constraints on the thermal history models the inversion strategy described in this paper is likely also directly applicable to other thermochronometers such as the apatite rutile and titanite u pb systems where the diffusion domain is approximated by the physical grain size