On the use of 60°C “cooling ages” obtained using projected fission-track lengths in apatite

Chemical Geology (Isotope Geoscience Section), 111 (1994) 263-267

263

Elsevier Science B.V., Amsterdam [PDI

On the use of 60 °C "cooling ages" obtained using projected fission-track lengths in apatite David A. Coylea and Roger PowelP aDepartment of Geology, La Trobe University, Bundoora, Vic. 3083, Australia bDepartment of Geology, University of Melbourne, Parkville, Vic. 3052, Australia (Received November 15, 1992; revised and accepted June 30, 1993 )

ABSTRACT The estimation of fission-track ages using counts of projected track lengths, in particular, 60 °C cooling ages, is based on determining the proportion of tracks greater than a particular length. We show that errors on 60 ° C cooling ages are significantly greater than have been previously estimated; for example, a minimum error on such ages as they are conventionally determined, is + 22% at the 95% confidence level. In addition, the equation used to determine projected track length ages, involving a constant relationship between track shortening, conventional age reduction, and the proportion of projected track lengths greater than 10/tm, is shown not to apply generally. It is recommended that the practise of estimating 60 °C cooling ages using projected track length distributions be substantially re-evaluated.

I. Introduction

The application of analysis in apatite has recently made the transition from a simple, inexpensive method of dating the time at which a rock has cooled through ~ 110°C (e.g., G.A. Wagner, 1968; Gleadow and Lovering, 1978; Gleadow and Brooks, 1979 ), to a very sophisticated technique for determining the complete thermal history of a rock below ~ 130 ° C (Duddy et al., 1988; Green, 1989; Green et al., 1989 ). Instrumental in this has been the observation that the lengths of fission tracks are very sensitive to annealing when heated in the geologic environment to temperatures greater than 60 ° C. The reconstruction of low-temperature thermal histories has relied upon the distribution of horizontal confined track lengths .1 in apatite, whereas the projected track length dis*1The length of fission tracks whose complete etched length is observed, as measured on the surface of the crystal.

tribution .2 early found not to reflect subtle differences in thermal history as revealed by horizontal confined track length distributions (Gleadow et al., 1986; Hurford et al., 1989). However, it has been proposed that the distributions of projected lengths in apatite can record the time at which those crystals cooled through 60 ° C (G.A. Wagner, 1988 ). The estimation of this age, tf, is based upon counting the proportion of projected track lengths in a sample of spontaneous tracks that are > 10/~m, (Cs), with respect to this same proportion observed in a sample of induced tracks (Ci): tf=tm(Cs/Ci)

(1)

where tm is the conventional measured age [G.A. Wagner et at., 1989a, equation (4)]. This is believed to be valid because it has been observed that in the Urach III drill-hole in *2The length of the observed part of etched fission tracks which intersect the surface of a crystal, as measured on the surface of the crystal.

0009-2541/94/$07.00 © 1994 Elsevier Science B.V. All fights reserved. SSD10009-2541 (93 ) EO 179-W

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Germany, few horizontal confined tracks > 10 #In exist at a present-day temperature of 60 ° C (M. Wagner, 1985). However, this observation is not typical of all drill-holes which intersect the partial annealing zone (Gleadow et al., 1983; Kill and Gleadow, 1989; K a m p and Green, 1990). The uncertainty associated with tf was originally assumed to be a fixed 15% (G.A. Wagner, 1988), but subsequent publications dispense with estimates of error altogether (Wagner et al., 1989b; Hejl and Wagner, 1990). Given that the value of tf is based upon measured data, and that the distribution of natural lengths in apatite is well known (Laslett et al., 1982 ) it is possible to determine the spectrum of true errors on all potential values of tf. More importantly, it is possible to determine the n u m b e r o f tracks which would need to be measured in order to achieve reasonable uncertainties on tr. The two major questions with respect to tf are: ( 1 ) Is the 60 °C cooling age equation valid? (2) What is the uncertainty on a 60 ° C cooling age?

2. The validity of the 60°C cooling age equation Underlying the tf approach is Eq. 1, which involves the assumption that, as the m e a n length of fission tracks decreases with annealing, the relationship between track shortening and Cs is linear. If it is not, it is not possible to relate the measured age to trin this simple way. To show that the relationship is not linear, the change of Cs with annealing was calculated using the probability distribution outlined in the Appendix. Fig. 1 illustrates this, using published values o f the m e a n and standard deviation o f normalised horizontal confined track lengths from annealing experiments [data for Durango apatite from table 1 o f Green (1988) ]. Curve A is for fixed track lengths, whereas curve B is for Gaussian-distributed track lengths. Fig. 1 shows that for most an-

D.A. CELLO AND R. POLL

1 0.8

o2

~

.

~

~

0.6 .

°

0.4 0.2

0.2

0.4 0.6 tf/tm

0.8

Fig. 1. The relationship between C $ / C i and tf/tm. The diagonal line corresponds to Eq. 1, the equation normally used to calculated tf. The calculated curves are for, A, a fixed track lengthprior to annealingof 15.91 (after Green, 1988 ), and for, B, a mean track length prior to annealing of 15.91 with a standard deviation of + 0.09 (after Green, 1988). The positions of the curves were calculated with C~/Ci obtained from Eq. A-3, with Ci set to the value of C~for tf/tm= 1, SOthat CJC~= 1 in the absence of annealing. The value of tr/tm is taken to be directly related to the track length reduction due to annealing, normalised to the mean track length prior to annealing. Thus, with no annealing, tf/tm = 1, whereas for length reduction due to annealing such that 95% of the track lengths are < 10 #m, tf/ t~ is taken to be zero. The length reduction corresponding to lf/lm=O is 100[ 1- 10/(15.91 + 1.645tr) ], giving 37.1% for curve A (with a=0), and 37.7% for curve B (with a=0.09). (The constant, 1.645, is the 95% point on a standard Gaussian distribution.) nealing conditions, Eq. 1 underestimates tf; whereas for very small tf, it is overestimated. Clearly Eq. 1 is not sufficient for determining tf; moreover, a more appropriate equation would depend on normally unavailable inform a t i o n concerning the track length distribution prior to annealing.

3. The uncertainty on a 600C cooling age Acknowledging that Eq. 1 is only approximately correct, it is possible to calculate the uncertainty on a 60°C cooling age from the uncertainties on Cs and Ci. For c measured

265

60 °C "COOLING AGES" OBTAINED USING PROJECTED FISSION-TRACK LENGTHS

projected track lengths, the n u m b e r of lengths greater than a particular length, c ' = cp, is binomially distributed, where p is the proportion of lengths greater than a particular length. Because a large n u m b e r o f observations are involved, c' approximates a Gaussian distribution, with mean, cp, and standard deviation, [cp(1 - p ) ] 1/2. Therefore p is also approximately Gaussian, with a mean, p, and a standard deviation, [ p ( 1 - p ) / c ] 1/2. As Cs is the proportion of spontaneous track lengths greater than 10/~m, in this case p is C~, and the standard deviation on C~ is [ C~( 1 - C~)/c ] 1/2, and the 95% confidence interval on Cs is: 20-c, = 2 [C,( 1 -C~)lc] 1/2

(2)

Similarly, the 95% confidence interval on Ci is: 20-Ci = 2 [Ci(1 - C i ) / c ] 1/2

(3)

Projected track length ages are most commonly d e t e r m i n e d when the population m e t h o d is used to measure the fission-track age tin. Because the population m e t h o d produces a pair o f samples from the apatite separate, one with spontaneous tracks and one with induced tracks (Gleadow, 1981 ), the same n u m b e r o f projected track lengths are measured in both samples to determine the ratio C,/Ci which is used to calculate tr. Thus it is reasonable to determine the uncertainty on the ratio CJC~ for a particular value of c. Observing that, for tm considered to be relatively well known: 0.2tf ~ t m 2

0-2 Cs/Ci

n u m b e r o f projected track lengths measured, c, via ac, and aci, and the degree of annealing represented by Cs/Ci. Fig. 2 shows how the relative error varies as Cs decreases from C~= Ci, to C~= 0.1 Ci. It can be seen that the error is as small as ___7% at the 95% confidence level but only when c = 10,000 tracks are measured (for Cs = Ci). W h e n c = 1000, this error is ~ _+22%. For values o f C~ less than C~, and c = 1000, the relative errors increase sharply. The final relative error on tf is not simply equal to the relative error on CJCi. The measured fission-track age, lm also has an uncertainty which must be incorporated into the final error on tf. This serves to increase the errors proportionally, such that the relative error on tf is ~ _+90% when tm has a relative error of _ 10%, and Cs/Ci= 1 ( c = 5 0 0 ) . Clearly such large uncertainties seriously limit the usefulness of projected track lengths in determining 60 °C cooling ages.

4. Discussion and conclusions It would be helpful if there existed some independent m e t h o d of confirming the validity 60 c-

50

-O

g 4o N 3o

and therefore, the relative error on tr is: ~ 2o 0-tf - - 0-Cs/Ci

tf

(4)

Cs/ Ci

At the 95% confidence level this is to a first approximation:

20-tf=2Fl( a - C s + l - C i ' l l 1/z tf

Lc\

C,

Ci J J

(5)

This equation allows us to investigate the dependence o f the relative error on 60°C cooling ages, calculated with Eq. 1, in terms of the

0

2000

4000

6000

8000

10000

Number of tracks measured (each of C s and Ci)

Fig. 2. The percent error at 95% confidence on the ratio CJCi using lo= 15.91 and a standard deviation on l/lo of +0.09 (after Green, 1988). Lines at 10% error and 1000 tracks measured are included for reference.

D.A.CELLOANDR.POLL

266

of tf estimates. Unfortunately, it is impossible to determine whether or not tf does in fact measure the time at which a sample has cooled through 60 ° C, because there is no independent thermo-chronometer to which trvalues can be compared. All that can be done is to examine existing data from conventional apatite fission-track analysis to determine if the results are plausible. More seriously, tf can never have an impossible value because, from the form of Eq. I, it is bounded by the measured fission-track age, tin, and a zero age. Because results will always look plausible, given that very little is generally known about the time of cooling through 60 ° C, it is tempting to believe that they are indeed correct. However, even a correct result with a large relative error is of no value in thermochronology. Even if the problem of the form of the age equation is overcome, in order to determine 60 °C cooling ages with errors small enough to make the age useful, the number of tracks counted will need to be increased by an order of magnitude. This will probably never be possible with most sedimentary rocks, particularly for samples from drill-holes that are used in basin analysis. Even for granitic rocks, where it is feasible to collect appropriately large volumes of material, the time involved in counting such large numbers of tracks will be prohibitively long unless the process can be automated.

Appendix - - The probability distribution of projected track lengths The parameter, Cs, the proportion of (spontaneous) fission tracks of projected track length greater than 10/~m, is directly related to the value of the cumulative distribution function of the proj ected track length for 10 ttm. The geometrical relationship for the projected track length, x, is: 1

x=~z sin 0 - y tan 0

(A-1)

in which the position of the centre of the original track with respect to the plane of observation, is y, the actual track length is z, and the angle of the track from the vertical to the plane of observation is 0. In Eq. A-l, it is convenient to consider x, y and z to be normalized to the mean of the actual track lengths. The cumulative distribution function of x, F(x), may be determined from the joint probability distribution of y, 0 and z, f(y,O,z), using the cumulative distribution approach (CDA, e.g., Mood et al., 1974). Taking y, 0 and z to be independent, then f(y,O,z) =f(y)f(z)f(O). The probability distribution o f y will be taken to be uniform, so f(y)= 1. The probability distribution of z will be taken to be Gaussian with mean 1 and standard deviation a. Taking the orientation of tracks to be uniformly distributed, then, by the CDA:

f( O)=dF( O)/dO=sin O Thus: 1

)]

(z--l) 2

f(y,O,z)=sinO[~exp(-20.2

(A-2)

Applying the CDA to get the cumulative distribution function of x, F(x), then:

Acknowledgements x

D.A.C., who was barely supported by a postgraduate research scholarship from La Trobe University, thanks Andy Gleadow for bringing projected track lengths to his attention, and also Geoff Laslett, Paul Green, and Giinther Wagner for discussions on the subject of projected track lengths. We thank Peter Kamp, Giinther Wagner, and particularly Rex Galbraith and Peter Deines, for helpful reviews.

Note

that

if a ~ 0 ,

then

this

simplifies to give

f(x)=dF(x)/dx=2(1-x), which is the distribution found by Dakowski (1978). Eq. A-3 was used in the calculation of Fig. 1, with the help of the software package Mathematica© (Wolfram, 1991 ) in performing the integration numerically (in the absence of an analytical solution), noting that Cs is F(x) for x = I0 (i.e. 10 #m) divided by the mean of the actual track lengths.

60°C "COOLING AGES" OBTAINED USING PROJECTED FISSION-TRACK LENGTHS

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