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ESR dating of tooth enamel: Coupled correction for U-uptake and U-series disequilibrium
Nucl. Tracks Radiat. Meas., Vol. 14, Nos I/2, pp. 237-241, 1988 Int. J. Radial Appl. Instrum., Part D
0191-278X/88 S3.00+ 0.00 Pergamon Press pie
Printed in Great Britain
ESR DATING OF TOOTH ENAMEL: COUPLED CORRECTION FOR U-UPTAKE A N D U-SERIES DISEQUILIBRIUM RAINER GRON* and HENRY P. SCHWARCZ
Department of Geology, McMaster University, Hamilton, ON, L8S 4MI, Canada and JOHN CHADAM
Department of Mathematics and Statistics, McMaster University, Hamilton, ON, L8S 4K1, Canada
(Received 1 August 1987; in final form 4 November 1987) Abstract--ESR dates on tooth enamel require a knowledge of the history of uptake of uranium (U) by both the dentin and enamel. We present a method of determining the uptake history by simultaneously fitting a model uptake function to the U-series and accumulated dose data. The function is of the form U(t) = Uo(t/T) p+t; Uo is the observed U-238 content of the tooth; p is a parameter to be determined in the fitting procedure; T is the age of the sample, and U(t) is the U content at any time t.
1. INTRODUCTION FOSSIL teeth of large mammals are commonly found as a component of archaeological and Pleistocene paleontological sites, and constitute an important datable material from those sites. The method of dating tooth enamel using ESR has been recently reviewed by Griin et al. (1987). These authors noted that the history of uptake of U by the tooth is of great importance in estimating the age from the accumulated dose (AD). They supposed that two possible uptake models would bracket the range of possible U-uptake histories: early uptake (EU) by which the present-day U concentration, Uo, of the tooth was attained soon after burial in the site; and linear uptake (LU) in which per time unit an equal increment of U was taken up by the tooth (U(t) = Uo(t/T) where U(t) is the U-concentration at any time t in the past and T is the age of the tooth). It would, however, be preferable to have some independent criterion from which to infer the history of U-uptake, that is, the mathematical form of U(t). A possible approach to solving this problem is presented by the observation that uranium-series "dates" for bone and teeth are also dependent on the form of U(t) (Szabo, 1982). The age of a tooth can be inferred from its 23°Th/23'U and :34U/238U activity ratios if the history of U-uptake is known. Normally, only "closed system" dates are quoted for bones and teeth; these are equivalent to the EU model (e.g. Szabo, 1979, 1980; McKinney, 1984). If, however, U(t) has some other form (e.g. linear uptake), then a given pair of activity ratios will correspond to some
other (older) age; at the same time, a given A D will also correspond to a specific (older) age determined by the form of U(t). Therefore it might be possible to obtain a specific function U(t) such that a given A D and pair of activity ratios will yield concordant ESR and U-series dates. Some limits can be placed on the possible form of U(t) as follows. (1) U(0) = 0: teeth of living organisms are essentially U-free. (2) U(T) = Uo, the concentration observed today. (3) U(t) increases monotonically; Uo is the maximum value ever achieved (i.e. no U-leaching has occurred). (4) The EU and LU models are special cases of U(t). A possible family of functions which satisfies these criteria is: u ( t ) = Uo
.
(l)
Note that: for p = - 1, U(t) = Uo (EU) for p = 0, U(t) = Uo(t/T) (LU). Figure 1 shows the family of curves defined by these functions. The third criterion may appear to be unnecessarily restrictive: could U(t) not reach a true maximum (dU(t)/dt = 0) for some t < T? In this case, however, we should observe that 23°Th/234U > 1.05 (the maximum value that could be
*Present address: Subdept. of Quaternary Research, University of Cambridge, Free School Lane, Cambridge, CB2 3RS, U.K. 237
238
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FIG. 1. Curves for U uptake as defined by equation (1) for some selected values of p. obtained by U-series decay). We normally observe, instead, that 23°Th/234U < 1.00. While this family of functions may seem to be overly simplistic, it puts some general limits on the behavior of U uptake, and allows us to see the consequences to combined Useries/ESR dating of tooth enamel. Qualitatively, we could describe the curves in Fig. 1 as follows: (a) - 1 < p < 0 decellerating, early uptake; (b) p ~ 0 uniform (linear) uptake; (c) p > 0 late (sublinear) uptake• We shall call p the "uptake parameter" as it uniquely specifies the form of U ( t , p ) . 2. S O L V I N G F O R T H E U P T A K E
200
300
t[ka)
FIG. 2. Function of p vs t for measured 234U/23sU and 23°Th/234U activity ratios of sample 145 (see Table la).
lationship is almost linear. The m i n i m u m value of t is of course that for p = - 1, which gives the closed system U-series age for the given pair of activity ratios. Note that these data could also be used to model U-series ages of teeth and bones; this will be discussed in a later paper• The ESR age of a tooth can be obtained from its A D value by iteratively solving the integral equation: AD =
(4)
D ( p , t) dt 0
where D ( p , t) is the dose rate as a function of time, including the effects of approach to secular equilibrium, and U uptake. We can now rewrite this as:
PARAMETER
We shall define Us(p, t) to be the time-dependent content of 2~8U in the tooth (either the enamel or the dentin) as defined in (l). Then we can easily derive functions for the content of 234U (U4) and 23°Th (Tho), respectively: U4(p, t) = (28/24)Us(p, t) {r - 24(r -- 1) or(O/t)P+~exp(--24(t -- O))d0}
(2)
and Tho(p, t) = (2s/(20 - ).4))Us (p, t) {(r20 - )-4)
AD
=
f:
D(p, t) dt =
f:
[C8 U8 (p, t)
+C4U4(p,t)+CoTho(p,t)+C]dt
(5)
where p is implicitly given by the form of the functions Us, U4, and Th0. C is a constant dose rate due to cosmic rays and ambient radiation from U, Th, and K in the soil. Cs, (74, and Co are coefficients that take into account the effects of or-efficiency, water content, fl-attenuation, and the energies of the corresponding parts of the E3sU decay chain. For given values of p, Z3°Th/Z~U and 2~U/238U, a specific value of A D corresponds to a particular t-value. Figure 3
or(O /t )P+ ~ e x p ( - 2o(t - 0))dO -
).4(r
-
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or(O/t~,+ 1 e x p ( - 24(t - 0)) dO}
(3)
where 2s.4.0 = decay constants of 23sU, 234U, and 23°Th r = initial 2r4U/23sU activity ratio. We can then derive expressions for 23°Th/234U and 234U/23sU as functions o f p and t. For a particular pair of these ratios, then, we can numerically obtain a unique relation between p and t (Fig. 2). Note that over the entire range of 23°Th/234U values, this re-
0
I00
200
300
t[~ca]
FIG. 3. Plot of AD(fl, t) vs 1. Dotted lines give the estimation of the US-ESR age from the measured AD.
ESR D A T I N G O F TOOTH E N A M E L shows the relation between A D and t determined by a particular set of data (see Table 1). We can now obtain the model age of the tooth by finding the value of T that corresponds to the observed value of AD. We shall call such a model age a U-series corrected age, or simply, a US--ESR age. Note that it is necessary in this procedure to compute the relation between A D and t for both enamel and dentin that was in contact with it, as the dentin also contributes to the radiation dose of the enamel via a fl-ray dose (Griin et al., 1987). In general, the activity ratios and uptake parameter of the dentin will be different from that of the enamel and, hence, must be measured separately. 3. EXAMPLE: THE HOXNE SITE
We have made replicate analysis of two tooth samples from the site of Hoxne, England. The samples came from the collection of the Department of Zoology, Cambridge University, and were donated by J. Wymer and A. Lister. As no environmental dosimetry measurements had been made at the sites where the teeth had been obtained, the external ~ and fl doses were estimated from U, Th and K analyses of soil found attached to the teeth, and assumed to be representative of the site. The (EU) and (LU) ESR ages of the two samples are shown in Table 1, along with the "closed system" (EU) U-series ages for the enamel and dentin. Note that these U-series ages are much younger than even the (EU) ESR age. This already suggests that Uuptake must have occurred relatively late for both of these samples, and for both components of the tooth: enamel and dentin. This is further confirmed when we compute the US/ESR age. Figure 2 shows the relation between p and t for the observed values of 23°Th/234U and 234U/238U,of enamel and dentin respectively. Figure 3 shows a computed value of A D vs t for sample 145A1. The observed A D s of the enamel were then used, together with such graphs, to obtain the model ages as shown in Table 1. They are uniformly much higher than the LU ages. The corresponding values of p for the enamel and dentin are also shown in Table 1. Figure 2 shows the actual history of U-uptake in one of the teeth, for dentin and enamel. As expected, we obtain from the US/ESR model the result that most of the U in both the enamel and dentin was taken up very late in the burial history of the tooth. As a result, most of the accumulated dose of this tooth would have been the result of the ambient radiation (given by the constant term in equation (5)). Consequently, our estimate of the age of the tooth is critically dependent on our knowledge of the ambient radiation field. As noted earlier, we did not have detailed close rate information for this sample, the only one which we have so far treated with the US/ESR model. Therefore it is also difficult to give estimates of the age-uncertainty: the plot A D ( t , p ) is principally only
239
dependent on the accuracy of the measurement of the U-series isotopes in the teeth fractions and the accuracy of measurement of the external dose rate. However, in this particular case, the influence of the U-isotopes is negligible due to the high value o f p and the individual U S - E S R age error is here only dependent on A D and external dose. The uncertainty of AD-measurements in teeth is rather low (in the range of 5%); the uncertainty of the external close from the analytical side is in the range of less than 1% (due to the detection limits of INAA, XRF, etc.). From Table lb we can also see that, whereas subsamples of each of the two analysed teeth were internally very consistent, on the other hand there was a significant difference between the model ages of the two teeth. This difference might be entirely attributable to the difference in the external dose rate experienced by the two teeth. The average of the two model ages would, in such a case, be the best estimate of the true age of the sedimentary layer in which the teeth were found. This average age of 319 _+ 38 ka would place the Hoxne deposit in isotope stage 9. 4. DISCUSSION For most tooth samples which we have studied so far (Griin et al., 1988; Zymela et al., 1988; Schwarcz et al., 1988) the U content of the enamel and dentin has been sufficiently high that the internal radiation dose would be the dominant source of the ESR signal for all cases except very late uptake of uranium. We would therefore expect that this model, to the extent that it accurately depicts the history of U-uptake, would greatly improve the accuracy of the age estimates. We would expect that in most cases, p ~<0, that is, uranium uptake has begun soon after burial of the tooth; for such samples closed system U-series ages would agree fairly well with ESR ages for the tooth. This method does not, however, provide a universal solution for all problems of ESR dating of teeth. We can envisage various problems, which are as follows. (1) The particular family of functions which we have prescribed may not resemble very closely the actual U uptake history of a given tooth. Indeed, the correct U(t) function may vary considerably from one tooth to another, depending on the local environmental history: temperature; rate of bacterial decay; water content of the soil; U-content of the soil; etc. Nevertheless, it is likely that our proposed family of U(t) functions will adequately approximate most cases of gradual U uptake, with the exception of cases where significant amounts of U have been lost from the tooth at some time in the past. (2) In case the U-series age is higher than the E U - E S R age, the presented model cannot give a solution. (3) No attempt has been made to model the spatial distribution of U in the tooth components. It has
240
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ESR D A T I N G OF TOOTH ENAMEL been shown (Grtin et al., 1987) that U very inhomogeneously distributed in both enamel and dentin, in a manner that can be approximately modelled by the diffusion of U from the surrounding soil into the tooth, proceeding more rapidly in the dentin than in the enamel. The present method assumes that only the variation in the volume-averaged U concentration need be known to calculate these open system ages. But, clearly, a thin zone of high-U concentration near the outside of a dentin layer would produce a much smaller/~ dose in the adjacent enamel than would a uniformly distributed content of uranium of the same average concentration. In future studies we shall attempt to model the effect of non-uniform Udistribution in systems with planar and spherical symmetry. (4) The effect of Rn-loss cannot be considered until it is actually measured. This method along with all formulas will be described in a forthcoming paper (Chadam et al., 1988).
Acknowledgements--We acknowledge the assistance of N.
Cesar in carrying out the U-series analyses of the tooth samples. The tooth samples used in this study were donated by J. Wymer and A. Lister. This research was funded by grants from the Natural Science and Engineering Research Council to H. P. Schwarcz, J. Chadam and D. C. Ford.
241
REFERENCES Chadam J., Griin R, and Schwarcz H. P. (1988) A model of U-diffusion into teeth for U-series disequilibria and ESR dating. Geochim. cosmochim. Acta (submitted). Griin R. (1986). Beta attenuation in thin layers. Ancient TL 4, I-8. Griin R., Schwarcz H. P. and Zymela S. (1987) Electron spin resonance dating of tooth enamel. Can. J. Earth Sci. 24, 1022-1037. Griin R. and Schwarcz H. P. (1987) Some remarks on ESR dating of bones. Ancient TL 5, !-9. Hille P. (1979) An open system model for uranium series dating. Earth Planet. Sci. Lett. 42, 138-142. McKinney C. (1984) Uranium-series dating of bones and teeth; present status. Presentation held at Symposium on Archaeometry, 14-18 May 1984, Washington, D.C. (Abstracts, p. 95). Schwarcz H. P., Griin R., Latham A. G., Mania D. and Brunnacker K. (1988) New evidence for the age of the Bilzingsleben archaeological site. Archaeometry 30, 5-17. Szabo B. J. (1979) Dating fossil bone from Cornelia, Orange Free State, South Africa. J. Archaeol. Sci. 6, 201-203. Szabo B. J. (1980) Results and assessments of uraniumseries dating of vertebrate fossils from Quaternary Alluviums in Colorado. Arct. Alp. Res. 12, 95-100. Szabo B. J. (1982) Uranium-series disequilibrium data for tooth fragments from the fossil hominid site at Ternifine, Algeria. S . A . J . Sci. 78, 205. Zymela S., Schwarcz H. P., Griin R., Stalker A. McS. and Churcher C. S. (1988) ESR dating of Pleistocene fossil teeth from Alberta and Saskatchewan. Can. J. Earth Sci. (in press).
0191-278X/88 S3.00+ 0.00 Pergamon Press pie
Printed in Great Britain
ESR DATING OF TOOTH ENAMEL: COUPLED CORRECTION FOR U-UPTAKE A N D U-SERIES DISEQUILIBRIUM RAINER GRON* and HENRY P. SCHWARCZ
Department of Geology, McMaster University, Hamilton, ON, L8S 4MI, Canada and JOHN CHADAM
Department of Mathematics and Statistics, McMaster University, Hamilton, ON, L8S 4K1, Canada
(Received 1 August 1987; in final form 4 November 1987) Abstract--ESR dates on tooth enamel require a knowledge of the history of uptake of uranium (U) by both the dentin and enamel. We present a method of determining the uptake history by simultaneously fitting a model uptake function to the U-series and accumulated dose data. The function is of the form U(t) = Uo(t/T) p+t; Uo is the observed U-238 content of the tooth; p is a parameter to be determined in the fitting procedure; T is the age of the sample, and U(t) is the U content at any time t.
1. INTRODUCTION FOSSIL teeth of large mammals are commonly found as a component of archaeological and Pleistocene paleontological sites, and constitute an important datable material from those sites. The method of dating tooth enamel using ESR has been recently reviewed by Griin et al. (1987). These authors noted that the history of uptake of U by the tooth is of great importance in estimating the age from the accumulated dose (AD). They supposed that two possible uptake models would bracket the range of possible U-uptake histories: early uptake (EU) by which the present-day U concentration, Uo, of the tooth was attained soon after burial in the site; and linear uptake (LU) in which per time unit an equal increment of U was taken up by the tooth (U(t) = Uo(t/T) where U(t) is the U-concentration at any time t in the past and T is the age of the tooth). It would, however, be preferable to have some independent criterion from which to infer the history of U-uptake, that is, the mathematical form of U(t). A possible approach to solving this problem is presented by the observation that uranium-series "dates" for bone and teeth are also dependent on the form of U(t) (Szabo, 1982). The age of a tooth can be inferred from its 23°Th/23'U and :34U/238U activity ratios if the history of U-uptake is known. Normally, only "closed system" dates are quoted for bones and teeth; these are equivalent to the EU model (e.g. Szabo, 1979, 1980; McKinney, 1984). If, however, U(t) has some other form (e.g. linear uptake), then a given pair of activity ratios will correspond to some
other (older) age; at the same time, a given A D will also correspond to a specific (older) age determined by the form of U(t). Therefore it might be possible to obtain a specific function U(t) such that a given A D and pair of activity ratios will yield concordant ESR and U-series dates. Some limits can be placed on the possible form of U(t) as follows. (1) U(0) = 0: teeth of living organisms are essentially U-free. (2) U(T) = Uo, the concentration observed today. (3) U(t) increases monotonically; Uo is the maximum value ever achieved (i.e. no U-leaching has occurred). (4) The EU and LU models are special cases of U(t). A possible family of functions which satisfies these criteria is: u ( t ) = Uo
.
(l)
Note that: for p = - 1, U(t) = Uo (EU) for p = 0, U(t) = Uo(t/T) (LU). Figure 1 shows the family of curves defined by these functions. The third criterion may appear to be unnecessarily restrictive: could U(t) not reach a true maximum (dU(t)/dt = 0) for some t < T? In this case, however, we should observe that 23°Th/234U > 1.05 (the maximum value that could be
*Present address: Subdept. of Quaternary Research, University of Cambridge, Free School Lane, Cambridge, CB2 3RS, U.K. 237
238
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FIG. 1. Curves for U uptake as defined by equation (1) for some selected values of p. obtained by U-series decay). We normally observe, instead, that 23°Th/234U < 1.00. While this family of functions may seem to be overly simplistic, it puts some general limits on the behavior of U uptake, and allows us to see the consequences to combined Useries/ESR dating of tooth enamel. Qualitatively, we could describe the curves in Fig. 1 as follows: (a) - 1 < p < 0 decellerating, early uptake; (b) p ~ 0 uniform (linear) uptake; (c) p > 0 late (sublinear) uptake• We shall call p the "uptake parameter" as it uniquely specifies the form of U ( t , p ) . 2. S O L V I N G F O R T H E U P T A K E
200
300
t[ka)
FIG. 2. Function of p vs t for measured 234U/23sU and 23°Th/234U activity ratios of sample 145 (see Table la).
lationship is almost linear. The m i n i m u m value of t is of course that for p = - 1, which gives the closed system U-series age for the given pair of activity ratios. Note that these data could also be used to model U-series ages of teeth and bones; this will be discussed in a later paper• The ESR age of a tooth can be obtained from its A D value by iteratively solving the integral equation: AD =
(4)
D ( p , t) dt 0
where D ( p , t) is the dose rate as a function of time, including the effects of approach to secular equilibrium, and U uptake. We can now rewrite this as:
PARAMETER
We shall define Us(p, t) to be the time-dependent content of 2~8U in the tooth (either the enamel or the dentin) as defined in (l). Then we can easily derive functions for the content of 234U (U4) and 23°Th (Tho), respectively: U4(p, t) = (28/24)Us(p, t) {r - 24(r -- 1) or(O/t)P+~exp(--24(t -- O))d0}
(2)
and Tho(p, t) = (2s/(20 - ).4))Us (p, t) {(r20 - )-4)
AD
=
f:
D(p, t) dt =
f:
[C8 U8 (p, t)
+C4U4(p,t)+CoTho(p,t)+C]dt
(5)
where p is implicitly given by the form of the functions Us, U4, and Th0. C is a constant dose rate due to cosmic rays and ambient radiation from U, Th, and K in the soil. Cs, (74, and Co are coefficients that take into account the effects of or-efficiency, water content, fl-attenuation, and the energies of the corresponding parts of the E3sU decay chain. For given values of p, Z3°Th/Z~U and 2~U/238U, a specific value of A D corresponds to a particular t-value. Figure 3
or(O /t )P+ ~ e x p ( - 2o(t - 0))dO -
).4(r
-
1) 50O.
or(O/t~,+ 1 e x p ( - 24(t - 0)) dO}
(3)
where 2s.4.0 = decay constants of 23sU, 234U, and 23°Th r = initial 2r4U/23sU activity ratio. We can then derive expressions for 23°Th/234U and 234U/23sU as functions o f p and t. For a particular pair of these ratios, then, we can numerically obtain a unique relation between p and t (Fig. 2). Note that over the entire range of 23°Th/234U values, this re-
0
I00
200
300
t[~ca]
FIG. 3. Plot of AD(fl, t) vs 1. Dotted lines give the estimation of the US-ESR age from the measured AD.
ESR D A T I N G O F TOOTH E N A M E L shows the relation between A D and t determined by a particular set of data (see Table 1). We can now obtain the model age of the tooth by finding the value of T that corresponds to the observed value of AD. We shall call such a model age a U-series corrected age, or simply, a US--ESR age. Note that it is necessary in this procedure to compute the relation between A D and t for both enamel and dentin that was in contact with it, as the dentin also contributes to the radiation dose of the enamel via a fl-ray dose (Griin et al., 1987). In general, the activity ratios and uptake parameter of the dentin will be different from that of the enamel and, hence, must be measured separately. 3. EXAMPLE: THE HOXNE SITE
We have made replicate analysis of two tooth samples from the site of Hoxne, England. The samples came from the collection of the Department of Zoology, Cambridge University, and were donated by J. Wymer and A. Lister. As no environmental dosimetry measurements had been made at the sites where the teeth had been obtained, the external ~ and fl doses were estimated from U, Th and K analyses of soil found attached to the teeth, and assumed to be representative of the site. The (EU) and (LU) ESR ages of the two samples are shown in Table 1, along with the "closed system" (EU) U-series ages for the enamel and dentin. Note that these U-series ages are much younger than even the (EU) ESR age. This already suggests that Uuptake must have occurred relatively late for both of these samples, and for both components of the tooth: enamel and dentin. This is further confirmed when we compute the US/ESR age. Figure 2 shows the relation between p and t for the observed values of 23°Th/234U and 234U/238U,of enamel and dentin respectively. Figure 3 shows a computed value of A D vs t for sample 145A1. The observed A D s of the enamel were then used, together with such graphs, to obtain the model ages as shown in Table 1. They are uniformly much higher than the LU ages. The corresponding values of p for the enamel and dentin are also shown in Table 1. Figure 2 shows the actual history of U-uptake in one of the teeth, for dentin and enamel. As expected, we obtain from the US/ESR model the result that most of the U in both the enamel and dentin was taken up very late in the burial history of the tooth. As a result, most of the accumulated dose of this tooth would have been the result of the ambient radiation (given by the constant term in equation (5)). Consequently, our estimate of the age of the tooth is critically dependent on our knowledge of the ambient radiation field. As noted earlier, we did not have detailed close rate information for this sample, the only one which we have so far treated with the US/ESR model. Therefore it is also difficult to give estimates of the age-uncertainty: the plot A D ( t , p ) is principally only
239
dependent on the accuracy of the measurement of the U-series isotopes in the teeth fractions and the accuracy of measurement of the external dose rate. However, in this particular case, the influence of the U-isotopes is negligible due to the high value o f p and the individual U S - E S R age error is here only dependent on A D and external dose. The uncertainty of AD-measurements in teeth is rather low (in the range of 5%); the uncertainty of the external close from the analytical side is in the range of less than 1% (due to the detection limits of INAA, XRF, etc.). From Table lb we can also see that, whereas subsamples of each of the two analysed teeth were internally very consistent, on the other hand there was a significant difference between the model ages of the two teeth. This difference might be entirely attributable to the difference in the external dose rate experienced by the two teeth. The average of the two model ages would, in such a case, be the best estimate of the true age of the sedimentary layer in which the teeth were found. This average age of 319 _+ 38 ka would place the Hoxne deposit in isotope stage 9. 4. DISCUSSION For most tooth samples which we have studied so far (Griin et al., 1988; Zymela et al., 1988; Schwarcz et al., 1988) the U content of the enamel and dentin has been sufficiently high that the internal radiation dose would be the dominant source of the ESR signal for all cases except very late uptake of uranium. We would therefore expect that this model, to the extent that it accurately depicts the history of U-uptake, would greatly improve the accuracy of the age estimates. We would expect that in most cases, p ~<0, that is, uranium uptake has begun soon after burial of the tooth; for such samples closed system U-series ages would agree fairly well with ESR ages for the tooth. This method does not, however, provide a universal solution for all problems of ESR dating of teeth. We can envisage various problems, which are as follows. (1) The particular family of functions which we have prescribed may not resemble very closely the actual U uptake history of a given tooth. Indeed, the correct U(t) function may vary considerably from one tooth to another, depending on the local environmental history: temperature; rate of bacterial decay; water content of the soil; U-content of the soil; etc. Nevertheless, it is likely that our proposed family of U(t) functions will adequately approximate most cases of gradual U uptake, with the exception of cases where significant amounts of U have been lost from the tooth at some time in the past. (2) In case the U-series age is higher than the E U - E S R age, the presented model cannot give a solution. (3) No attempt has been made to model the spatial distribution of U in the tooth components. It has
240
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~
. ~
ESR D A T I N G OF TOOTH ENAMEL been shown (Grtin et al., 1987) that U very inhomogeneously distributed in both enamel and dentin, in a manner that can be approximately modelled by the diffusion of U from the surrounding soil into the tooth, proceeding more rapidly in the dentin than in the enamel. The present method assumes that only the variation in the volume-averaged U concentration need be known to calculate these open system ages. But, clearly, a thin zone of high-U concentration near the outside of a dentin layer would produce a much smaller/~ dose in the adjacent enamel than would a uniformly distributed content of uranium of the same average concentration. In future studies we shall attempt to model the effect of non-uniform Udistribution in systems with planar and spherical symmetry. (4) The effect of Rn-loss cannot be considered until it is actually measured. This method along with all formulas will be described in a forthcoming paper (Chadam et al., 1988).
Acknowledgements--We acknowledge the assistance of N.
Cesar in carrying out the U-series analyses of the tooth samples. The tooth samples used in this study were donated by J. Wymer and A. Lister. This research was funded by grants from the Natural Science and Engineering Research Council to H. P. Schwarcz, J. Chadam and D. C. Ford.
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