The mathematical basis for the US-ESR dating method

Quaternary Geochronology 30 (2015) 1e8

Contents lists available at ScienceDirect

Quaternary Geochronology journal homepage: www.elsevier.com/locate/quageo

Research paper

The mathematical basis for the US-ESR dating method
res d, Qingfeng Shao a, *, John Chadam b, Rainer Grün c, Christophe Falgue e d Jean-Michel Dolo , Jean-Jacques Bahain a

College of Geography Science, Nanjing Normal University, 1, Wenyuan Road, 210023 Nanjing, China Department of Mathematics, University of Pittsburgh, Pittsburgh, PA 15260, USA c Research School of Earth Sciences, The Australian National University, ACT 0200 Canberra, Australia d  Departement de Pr ehistoire, Mus eum National d'Histoire Naturelle, UMR7194, 1, Rue Ren e Panhard, 75013 Paris, France e CEA, I2BM, 91401 Orsay Cedex, France b

a r t i c l e i n f o

a b s t r a c t

Article history: Received 20 January 2015 Received in revised form 15 July 2015 Accepted 28 July 2015 Available online 1 August 2015

Over the past two decades, the combined electron spin resonance (ESR) and U-series dating method has been widely applied to date fossil teeth for archaeological studies through the use of the US-ESR model. The obtained age, compatible with both the ESR and U-series data determined in all dental tissues and the burial environment, is more flexible and reliable than the parametric uptake ages (by early, linear or recent uptake models), for which selection was often based on the expected age of the site. In this paper, the mathematical basis of the US-ESR model is described in detail, from the U-uptake description to the calculation of accumulation dose in the sample and the US-ESR age determination. An example is used to illustrate the calculation of the US-ESR age, associated dose rates and U-uptake parameters. While the description in this paper is specific to US-ESR model and more largely combined ESR/U-series dating of fossil teeth, we expect that some of the principles can be used in other applications of U-series dating. © 2015 Elsevier B.V. All rights reserved.

Keywords: ESR/U-series dating method US-ESR model U-uptake Dose rate evolution Fossil teeth

1. Introduction Grün et al. (1988) proposed a U-uptake model for ESR dating of fossil teeth based on a smooth diffusion function: U8(t) ¼ U8m(t/ T)pþ1, where U8(t) is the 238U concentration at a time t in a dental tissue, U8m the measured, present day 238U concentration, T the apparent age of the tooth and p the U-uptake parameter (1  p < ∞). In this model, ESR dose rate evaluations are derived from measured U-series disequilibria between 238U, 234U and 230Th in all dental tissues (enamel, dentine and cement) contributing to the irradiation of the tooth enamel. This dating approach is often called combined ESR/U-series dating and yields US-ESR age estimates. Over the past two decades, the combined ESR/U-series dating approach has become a commonly used geochronological tool and has produced hundreds of age estimates for archaeological sites. In this paper, we describe the mathematical basis of this dating model. While the description in this paper is specific to combined ESR/U-

* Corresponding author. E-mail addresses: [email protected] (Q. Shao), [email protected] (J. Chadam),
res), jean-michel. [email protected] (R. Grün), [email protected] (C. Falgue [email protected] (J.-M. Dolo), [email protected] (J.-J. Bahain). http://dx.doi.org/10.1016/j.quageo.2015.07.002 1871-1014/© 2015 Elsevier B.V. All rights reserved.

series dating of fossil teeth, we expect that some of the principles can be used in other applications of U-series dating. 2. Mathematical model The US model is based on four assumptions (Grün et al., 1988): 1) there are no U-series nuclides in the dental tissues before burial (i.e., U(t) ¼ 0 for t ¼ 0); 2) the 234U/238U activity ratio (r0) in the burial environment is constant over time; 3) there is no Th uptake; 4) there is no net loss of any of the isotopes at any time. For the equations that follow, it is convenient to define the following parameters: p: U-uptake parameter, T: age of the sample, U8m: measured, present day 238U concentration in dental tissues, U8, U4, U0: concentrations of 238U, 234U and 230Th, l8, l4, l0: decay constants of 238U, 234U and 230Th (e.g., Cheng et al., 2013),

2

Q. Shao et al. / Quaternary Geochronology 30 (2015) 1e8

R48, R08: 234U/238U and 230Th/238U activity ratios, r0: 234U/238U activity ratio in the burial environment, S8, S4: concentrations of 238U and 234U diffusing into the dental tissues per unit time. With this model, the dynamics of the three U-series nuclides of interest, 238U, 234U and 230Th, in the dental tissues are described as follows:

(see Appendix 1) yields:

U8 ðtÞ ¼ U8m ðt=TÞpþ1

(12)

3 2 Z1 l8 U8 ðtÞ 4 U4 ðtÞ ¼ r0  l4 tðr0  1Þ spþ1 el4 tð1sÞ ds5 l4

dU8 ðtÞ ¼ S8 ðtÞ  l8 U8 ðtÞ dt

(1)

2 Z1 l8 U8 ðtÞ 4 tðr0 l0  l4 Þ spþ1 el0 tð1sÞ ds U0 ðtÞ ¼ l0  l4

dU4 ðtÞ ¼ S4 ðtÞ  l4 U4 ðtÞ þ l8 U8 ðtÞ dt

(2)

Z1

0

 l4 tðr0  1Þ

e

ds5:

(14)

(3)

Each of these differential equations is of the form:

dy ¼ Ay þ BðtÞ dt

3. Relationship between p and T

(4)

and has a unique solution:

yðtÞ ¼ yðt0 ÞeAðtt0 Þ þ

s

3

pþ1 l4 tð1sÞ

0

dU0 ðtÞ ¼ l4 U4 ðtÞ  l0 U0 ðtÞ: dt

Zt

(13)

0

eAðttÞ BðtÞdt:

(5)

Using Eqs. (12)e(14), the 234U/238U and 230Th/238U activity ratios at the putative age, T, of the sample can be expressed as:

R48 ðTÞ ¼

l4 U4 ðTÞ ¼ r0  l4 Tðr0  1Þ l8 U8 ðTÞ

Z1

spþ1 el4 Tð1sÞ ds

(15)

0

t0

Using the initial condition of U8(t0) ¼ U4(t0) ¼ U0(t0) ¼ 0 (with t0 ¼ 0), the concentrations of 238U, 234U and 230 Th at a given time are:

Zt

S8 ðtÞel8 ðttÞ dt

U8 ðtÞ ¼

l0 U0 ðTÞ l8 U8 ðTÞ 2 Z1 l0 4 ¼ Tðr0 l0  l4 Þ spþ1 el0 Tð1sÞ ds l0  l4

R08 ðTÞ ¼

0

(6)

Z1

0

 l4 Tðr0  1Þ

Zt U4 ðtÞ ¼

s

e

ds5:

(16)

0

½l8 U8 ðtÞ þ S4 ðtÞel4 ðttÞ dt

(7)

l4 U4 ðtÞel0 ðttÞ dt:

(8)

0

Zt U0 ðtÞ ¼ 0 238

If U concentration diffusing into a dental tissue is expressed as S8(t) ¼ ctp, it follows from Eq. (6):

Zt U8 ðtÞ ¼ c

tp el8 ðttÞ dtyc

0

Zt 0

S8 ðtÞ ¼ ðp þ 1Þ

Inserting the measured values of the activity ratios for R48(T) and R08(T) and solving Eqs. (15) and (16) for r0 (see Appendix 2) results in:

Z 1 R48  l4 T spþ1 el4 Tð1sÞ ds 0 r0 ¼ Z 1 spþ1 el4 Tð1sÞ ds 1  l4 T

(17)

0

and

ct pþ1 tp dt ¼ pþ1

R48  r0 ¼

U8m p t : T pþ1

(10)

tp

:

(11)

Using (9)e(11) in the solutions (6)e(8) and introducing s ¼ t/t



Z 1 R08  l4 T spþ1 el0 Tð1sÞ ds 0 : Z 1 spþ1 el0 Tð1sÞ ds 1  l0 T

l0 l4 l0

(18)

0

Eliminating r0 by combination of Eqs. (17) and (18) gives the peT relationship in the form of (see Appendix 2):

a4 yðp; cxÞ  ½a4  ð1  cÞa0 yðp; xÞ þ ð1  cÞ ¼ 0

(19)

where x ¼ l0T, c ¼ l4/l0, a4 ¼ R48 e 1 and a0 ¼ R08 e 1 with

2

Combining this with Assumption 2 results in:

l8 S8 ðtÞ l ðp þ 1ÞU8m ¼ r0 8 l4 l4 T pþ1



(9)

using the fact that exp(l8(tt)) z 1, because l8(tt) < l8T z 1.551  105 is much less than l4T z 0.282 or l0T z 0.917 for the ages T  105 years. The measured value of 238U at T, U8m ¼ U8(T) ¼ cT(pþ1)/(p þ 1), implies that c ¼ (p þ 1)U8m/T(pþ1), resulting in:

S4 ðtÞ ¼ r0

3

pþ1 l4 Tð1sÞ

yðp; xÞ ¼ 41  x

31

Z1 s

pþ1 xð1sÞ

e

ds5

:

(20)

0

The above equations provide the relationship between p and T

Q. Shao et al. / Quaternary Geochronology 30 (2015) 1e8

3

derived for experimentally measured values of the activity ratios.

where r0 is obtained from Eq. (17) or (18).

4. Dose accumulation

4.2. External dose from dentine, Dden(T, p)

The US-ESR procedure for determining the age relies on using the measured value of the total dose obtained by ESR analysis on tooth enamel to provide a second equation in p and T. In the following section, we will use the following notations:

The dentine dose is generated by b-particles from the U incorporated into the dentine and its daughter isotopes. Enamel sample preparation usually involves the removal of the outer 50 mm, which eliminates the volume that received irradiation by external a-particles. The g-dose from dentine is also considered to be negligible because g-rays have long effective ranges (>20 cm) compared with enamel thickness (0.1e0.2 cm). However, for larger teeth (e.g., elephants or mammoths) the gamma dose rate generated within the tooth may contribute significantly to the enamel dose rate (Nathan and Grün, 2003). The integrated dentine b-dose in the enamel from 0 to T can be expressed as:

a8b, a0b: beta attenuation factors for 238Ue 234U and 230The 206Pb (Marsh, 1999), b8b, b0b: beta self absorption factors for b-dose rates 238Ue234U and 230The 206Pb (Marsh, 1999), d8a, d4a, d0a: alpha dose rates for 238Ue234U, 234Ue 230Th and 230 rin et al., 2011), The 206Pb (Gue 8 rin db, d0b: beta dose rates for 238Ue234U and 230The 206Pb (Gue et al., 2011), faw, fbw: a, b water attenuation factors (Grün, 1994), k: alpha efficiency (Grün and Katzenberger-Apel, 1994), C8, C4, C0: total dose rates for 238Ue234U, 234Ue 230Th and 230The 206 rin et al., Pb, the latter can be adjusted for 222Rn loss (Gue 2011), rin et al., 2011), CU, CTh, Ck: total dose rates from U, Th and K (Gue Dena, Dden, Dcem, Denv: total dose contributed by enamel, dentine, cement and environment (mainly sediments and cosmic rays) from the radioactive isotopes in enamel, dentine, cement and the environment. De: dose value measured by ESR analysis in enamel, De ¼ Dena þ Dden þ Dcem þ Denv.

4.1. Internal dose, Dena(T, p) The internal dose, Dena, is generated by a and b-particles emitted from the U incorporated into enamel and its daughter isotopes. The accumulated internal dose from the time the tooth was formed (t ¼ 0) to today (t ¼ T) can be expressed by:

 ZT  l l C8 U8 ðtÞ þ 4 C4 U4 ðtÞ þ 0 C0 U0 ðtÞ dt: Dena ðT; pÞ ¼ l8 l8

ZT  Dden ðT; pÞ ¼

 l0 0 C0 U0 ðtÞ dt: l8

C80 U8 ðtÞ þ

0

Note that the decay from 234U to 230Th does not involve bemissions. Following the same approach, C80 , the total dose rate for 238 Ue234U in dentine and cement, is given by: 0

C8 ¼

1 fwb

fb8 a8b d8b :

(25)

C00 , the total dose rate for 230Th to 206Pb in dentine and cement, is in the similar form. Introducing U8(t) and U0(t) from Eqs. (12) and (14) into the previous equation and following the same type of calculations as in Appendix 3, yields:

C80 TU8m l0 þ C0 U T 2 pþ2 l0  l4 0 8m 2 Z1 Z1 4ðr0 l0  l4 Þ upþ2 spþ1 el0 uTð1sÞ duds

Dden ðT; pÞ ¼

0

0

 l4 ðr0  1Þ

U8m TðC8 þ C4 r0 Þ  C4 U8m l4 T 2 ðr0  1Þ pþ2 Z1 Z1 l0 upþ2 spþ1 el4 uTð1sÞ duds þ C U T2 l0  l4 0 8m

e

duds5:

0

4.3. External dose from cement, Dcem(T, p) The expressions of external dose from dentine, Eqs. (24)e(26), are also applicable to cement, if it covers one side of the enamel surface. Hence, the Dcem(T, p) is:

2 0 C8 TU8m 0 l0 24 þ Dcem ðT; pÞ ¼ C U T ðr0 l0 pþ2 l0  l4 0 8m Z1 Z1

0

4ðr0 l0  l4 Þ

s

(22)

The dose rates of C4 and C0 are in the similar form. Inserting Eqs. (12)e(14) into (21) and introducing u ¼ t/T allows the calculation of Dena(T, p) as follows (see details of calculation in Appendix 3):

0

pþ2 pþ1 l4 uTð1sÞ

(26)

k 1 C8 ¼ a d8a þ b b8b d8b : fw fw

2

u 0

The corresponding dose rate of C8 is:

Dena ðT; pÞ ¼

3

Z1 Z1

(21)

0

(24)

Z1

Z1

0

0

u

pþ2 pþ1 l0 uTð1sÞ

Z1

Z1

0

0

 l4 ðr0  1Þ

s

e

 l4 Þ duds

0

u

s

e

0

3

Z1 Z1

3 pþ2 pþ1 l4 uTð1sÞ

upþ2 spþ1 el0 uTð1sÞ duds

 l4 ðr0  1Þ

duds5

u 0

pþ2 pþ1 l4 uTð1sÞ

s

e

duds5:

0

(27) (23)

4

Q. Shao et al. / Quaternary Geochronology 30 (2015) 1e8

4.4. External dose from environment, Denv(T)

given by Eq. (20),

The dose from the environment into the enamel is defined here as the sum of b (if the enamel is not shielded by cement) and gdoses generated by U, Th and K from burial environment (mainly sediments) and the dose generated by cosmic rays, over a time from 0 to T:

0:37yðp; 0:308xÞ  0:294yðp; xÞ þ 0:692 ¼ 0:

Denv ðTÞ ¼ DU ðTÞ þ DTh ðTÞ þ DK ðTÞ þ Dcos ðTÞ:

(28)

If the U and Th decay chains are in secular equilibrium state, the calculations are straightforward. Note that the external gamma dose rate is usually measured in situ rather than derived from the analysis of a sediment sample. The calculations of the beta and gamma dose rates involve beta attenuation factors and corrections for water contents (see above). The cosmic rays have to be corrected for sediment shielding (Prescott and Hutton, 1994). Finally, the total dose is given by:

De ðT; pÞ ¼ Dena ðT; pÞ þ Dden ðT; pÞ þ Dcem ðT; pÞ þ Denv ðTÞ:

(29)

(31)

For each p ̶ 1 one can numerically solve Eq. (20) for x (see Appendix 4) and plotting the results one obtains the peT relationship (Fig. 1A). In spite of the complicated functional form of Eq. (20), its graph is very nearly a straight line (see Appendix 4). With the calculated peT relationship, the data of Table 1 can be inserted into Eqs. (21)e(28) to calculate the dose components, Dena(T, p), Dden(T, p), Dcem(T, p), Denv(T) (Fig. 1B). The g-dose from the external environment was assumed to be constant over time. The sum of each component yields the paleodose, Di (Fig. 1C). Finally, comparing the measured De value (1196 ± 60 Gy) with the calculated Di, the age estimate (726 þ 87/84 ka) can be determined (Fig. 1C). Then, one can derive the time averaged internal dose rate (444 ± 147 mGy/a), dentine b-dose rate (222 ± 66 mGy/a), cement b-dose rate (149 ± 43 mGy/a) and total dose rate (1647 ± 177 mGy/a) (Fig. 1B). Using the age estimate and the calculated peT relationships, p-values (0.50 ± 0.33, 0.32 ± 0.28 and 0.33 ± 0.27) for enamel, dentine and cement, respectively, are readily determined (Fig. 1A).

5. US-ESR age determination 7. Conclusions The estimated age, Te, is then obtained from the measured data R48, R08 and De with the following procedure: Step 1 e Take an increasing sequence of values of pi < piþ1 and use the measured values of R48 and R08 to calculate the corresponding increasing sequence of values of xi(¼l0Ti) using Eq. (19), in order to establish the peT relationship for each dental tissue. Step 2 e Use these values from step 1 in Eq. (17) or (18) for r0 and in Eq. (29) to calculate the dose components (Dena(T, p), Ddent(T, p), Dcem(T, p)). The single and double integrals that appear can be numerically calculated using commercial software packages (e.g., Matlab©, see Appendix 4). This provides the model's estimate Di of the total dose for the uptake parameter, pi. Continue this procedure to identify values pi < piþ1 which provide sufficiently tight bounds of the form:

This paper presents the mathematical basis of the US-ESR dating model. This model has been successfully applied at numerous sites and provides a major advance over the parametric early, linear or recent uptake ages, whose selection was often based on the expected age of the site. With laser ablation we are now able to measure the spatial distribution of U concentrations and U-series isotopes. Initial results indicate that U-uptake is a fast process (Grün et al., 2014), i.e., the CSUS-ESR model may be more appropriate for such samples (Grün, 2000). It may seem timely to develop a model that accounts for measured spatial distributions of U-series isotopes. Until then we strongly advise using the measured ESR and U-series data for the calculation of both US-ESR and CSUS-ESR age estimates. The range provided by these age estimates encompasses all continuous U-uptake histories, and is thus independent of the

Table 1 The data used as an example for US-ESR age calculation.

Di < De < Diþ1 :

(30)

Step 3 e The final step is to use the method of bisection to find the value of pe, and the corresponding xe(¼l0Te) for which D(pe, Te) ¼ De. In applications of dating fossil teeth, the US-ESR age calculation can be performed either on the DATA program (Grün, 2009) or the USESR program (Shao et al., 2014). 6. Example of US-ESR age calculation To illustrate the age determination, the data of Table 1 (also used in Shao et al., 2014) were used to calculate a US-ESR age estimate and the associated parameters. The first step is to calculate the peT relationship for each dental tissue. For the enamel, taking the measured activity ratios, R48 ¼ 1.37 and R04 ¼ 0.81, and using the decay rates of l4 ¼ 2.82  106 a1 and l0 ¼ 9.17  106 a1, one has c ¼ l4/ l0 ¼ 0.308, a4 ¼ 0.37, a0 ¼ 0.1097. Thus Eq. (19) becomes, for y(p, x)

Parameter

Material

Value

Error

De (Gy) U (ppm) 234 U/238U 230 Th/234U 210 Pb/230Th Water (wgt%) U (ppm) 234 U/238U 230 Th/234U 210 Pb/230Th Water (wgt%) U (ppm) 234 U/238U 230 Th/234U 210 Pb/230Th Water (wgt%) Total thickness (mm) Removed side 1 (mm) Removed side 2 (mm) U (ppm) Th (ppm) K (%) Water (wgt%) Cosmic dose rate (mGy/a) In situ dose rate (mGy/a)

Enamel Enamel Enamel Enamel Enamel Enamel Dentine Dentine Dentine Dentine Dentine Cement Cement Cement Cement Cement Enamel Enamel Enamel Sediment Sediment Sediment Sediment

1196 2.85 1.37 0.81 0.65 0 68.00 1.36 0.83 0.35 7 48.44 1.37 0.83 0.29 7 1389 127 126 3.0 7.0 0.98 13 0 832

60 0.23 0.03 0.02 0.10 0 4.08 0.03 0.03 0.08 1 2.91 0.03 0.03 0.08 2 167 20 20 0.2 0.4 0.03 3 0 58

Q. Shao et al. / Quaternary Geochronology 30 (2015) 1e8

5

Fig. 1. An example illustrating the calculations of the US-ESR age, associated dose rates and U-uptake parameters.

Inserting U8(t) ¼ U8(t) (t/t)pþ1 into Eq. (7), and from Eq. (11) follows:

specifics of U-uptake assumptions. Acknowledgments We gratefully thank Prof. Henry P. Schwarcz, McMaster University, for his thoughtful advice in setting up this research. The authors thank N. McLean and an anonymous reviewer, who provided insightful comments on this paper. Q.S. thanks the Erasmus Mundus program and the MNHN for providing scholarship for this study. Appendix 1. Solving for the concentrations With the approximations made in Section 2, Eq. (12) is obtained:

U8 ðtÞ ¼ U8m ðt=TÞpþ1 :

(A1)

S4 ðtÞ ¼

. l8 r0 U ðtÞðp þ 1Þtp t pþ1 l4 8

(A2)

And Eq. (13) can be expressed as:

l U4 ðtÞ ¼ 8 U8 ðtÞ l4

Zt h

. i l4 ðt=tÞpþ1 þ r0 ðp þ 1Þtp t pþ1 el4 ðttÞ dt:

0

(A3) Using s ¼ t/t yields:

6

Q. Shao et al. / Quaternary Geochronology 30 (2015) 1e8

U4 ðtÞ ¼

l8 U ðtÞ l4 8

Zt

h

i ðl4 tÞspþ1 þ r0 ðp þ 1Þsp el4 tð1sÞ ds:

(A4)

2 Z1 l8 U8 ðtÞ 4 l tð1  r0 Þ spþ1 el4 tð1sÞ ds  ðl4 U0 ðtÞ ¼ l0  l4 4 0

0

Z1  r0 l0 Þt

An integration by parts allows the second integral to be written as:

Z1

s

pþ1 l0 tð1sÞ

e

3 ds5

(A11)

0

ðp þ 1Þsp el4 tð1sÞ ds ¼ 1  l4 t

0

Z1

which is the expression in Eq. (14).

spþ1 el4 tð1sÞ ds

(A5)

0

and Eq. (A4) becomes:

Appendix 2. Derivation of the peT relationship

2 3 Z1 l8 pþ1 l4 tð1sÞ 5 4 U4 ðtÞ ¼ U8 ðtÞ r0 þ l4 tð1  r0 Þ s e ds l4

For notational convenience we write:

(A6)

Z1 I0 ¼ I0 ðp; TÞ ¼ l0 T

0

spþ1 el0 Tð1sÞ ds

(A12)

0

which is Eq. (13) of Section 2. For U0(t), an integration by parts in Eq. (8) gives:

and similarly for I4. The Eq. (15) can be written as:

2 3 Zt l4 4 0 l0 ðttÞ U4 ðtÞ  U4 ðtÞe U0 ðtÞ ¼ dt5 l0

R48 ðTÞ ¼ r0  ðr0  1ÞI4

(A13)

yielding:

0

2 3 Zt l4 4 l0 ðttÞ ¼ U4 ðtÞ  ðS4 ðtÞ þ l8 U8 ðtÞ  l4 U4 ðtÞÞe dt5 l0

r0 ¼ ðR48 ðTÞ  I4 Þ=ð1  I4 Þ: Also, Eq. (16) is:

0

   l  ðr0  1ÞI4  4  r0 I0 l0     l0 l4 ¼ R48 ðTÞ  r0   r0 I0 l0  l4 l0

(A7)

R08 ðTÞ ¼ using Eq. (2) in Eq. (A7). Thus

2 l0 U0 ðtÞ ¼ l4 4U4 ðtÞ 

Zt

3 ðS4 ðtÞ þ l8 U8 ðtÞÞel0 ðttÞ dt5

þ l4

l4 U4 ðtÞel0 ðttÞ dt:

(A8)

0

Eq. (8) shows that the last integral is U0(t), which results in:

2 ðl0  l4 ÞU0 ðtÞ ¼ l4 4U4 ðtÞ 

l0 l0  l4



(A15)

using (A13). Solving for r0 in (A15) gives:

0

Zt

(A14)

    l  l4 l R08 ðTÞ  4 I0 ð1  I0 Þ: r0 ¼ R48 ðTÞ  0 l0 l0

(A16)

Eqs. (A14) and (A16) then directly lead into the Eq. (19) in the main text.

3

Zt

l0 ðttÞ

ðS4 ðtÞ þ l8 U8 ðtÞÞe

dt5:

Appendix 3. Calculating the total dose

0

(A9)

Eq. (21) defines:

As above in Eq. (A3), introducing s ¼ t/t results in:

l8 U ðtÞ l4 8 ¼

Z1 h 0

ZT  Dena ðT; pÞ ¼

i

0

ðl4 tÞspþ1 þ r0 ðp þ 1Þsp el0 tð1sÞ ds 2

l8 U ðtÞ4r0 þ ðl4  r0 l0 Þt l4 8

Z1

C8 U8 ðtÞ þ

(A17)

Considering each of the terms separately and using the solutions in Eqs. (12)e(14) one has:

3 spþ1 el0 tð1sÞ ds5

 l4 l C4 U4 ðtÞ þ 0 C0 U0 ðtÞ dt: l8 l8

(A10)

0

with the same integration by parts as above. Substituting from (A6) for U4(t) yields:

ZT U8 ðtÞdt ¼

C8 0

C8 U8m T pþ1

ZT t pþ1 dt ¼ 0

C8 U8m T : pþ2

After carrying out the integral using Eqs. (12) and (13)

(A18)

Q. Shao et al. / Quaternary Geochronology 30 (2015) 1e8

l4 C l8 4

ZT U4 ðtÞdt ¼ r0 0

ZT 0

0 U8 ðtÞt @

C4 U8m T  l4 ðr0  1ÞC4 pþ2

Z1 0

xðpÞ ¼ xð1Þ þ bðp þ 1Þa :

C U T spþ1 el4 tð1sÞ dsAdt ¼ r0 4 8m pþ2 0

ZT t

pþ2 @

0

1

Z1 s

pþ1 l4 tð1sÞ

e

dsAdt:

0

(A19)

Similarly, using Eqs. (12) and (14) as above

(A21)

This can be accomplished by taking the natural log of Eq. (A19) in the form:

1

U l4  C4 8m ðr0  1Þ T pþ1

7

lnðxðpÞ  xð  1ÞÞ ¼ a lnðp þ 1Þ þ lnb

(A22)

and doing a linear fit of the data (x(pi) ¼ x(i) listed above) in (x(pi)  x(1)) vs pi þ 1 for i ¼ 2, …, n. The Matlab code for this is:

0 1 1 3 Z ZT 6 ðr l  l Þ U ðtÞt @ spþ1 el0 tð1sÞ dsAdt 7 4 8 7 6 0 0 ZT 7 6 l0 l0 7 6 0 0 C0 U0 ðtÞdt ¼ C0 6 0 1 7: T 1 7 Z Z l8 l0  l4 6 7 6 0 4 l4 ðr0  1Þ U8 ðtÞt @ spþ1 el4 tð1sÞ dsAdt 5 2

0

(A20)

0

Making the change of variables u ¼ t/T in the iterated integrals, adding and grouping terms results in the Eq. (23) for Dena(p, T). The other terms are obtained by following the same procedure. Appendix 4. Numerical schemes 4.1. Computing the peT relationship For a given p  1 the function in Eq. (19) can be solved numerically for x(¼l0T) using Matlab with the integral appearing in Eq. (20) calculated using the trapezoidal approximation. For R48 ¼ 1.37; R04 ¼ 0.81 and c ¼ l4/l0 ¼ 0.308 the Matlab code is:

Using the plot function in Matlab: plot (root/l0, p) results in the graph in Fig. 1A. While the graph appears almost linear, it is possible to obtain a functional relation of the form

4.2. Computing the total dose At each step of the procedure outlined in Section 5, a value of pi and the corresponding xi ¼(l0Ti) are given and one must calculate the corresponding total dose D(pi, Ti) for comparison with the measured total dose, De. This is a straightforward calculation in

8

Q. Shao et al. / Quaternary Geochronology 30 (2015) 1e8

Matlab with the single and double integrals that appear calculated numerically using the functions. The Matlab code for calculating the integrated dose components of, Dena (T, p), Dden (T, p), and Dcem (T, p) is:

References Cheng, H., Edwards, R.L., Shen, C.-C., Polyak, V.J., Asmerom, Y., Woodhead, J., €tl, C., Wang, X., Alexander, E.C., 2013. ImHellstrom, J., Wang, Y., Kong, X., Spo provements in 230Th dating, 230Th and 234U half-life values, and U-Th isotopic measurements by multi-collector inductively coupled plasma mass spectrometry. Earth Planet. Sci. Lett. 371e372, 82e91. Grün, R., 1994. A cautionary note: use of ‘water content’ and ‘depth for cosmic dose rate’ in the AGE and DATA programs. Anc. TL 12, 50e51. Grün, R., 2000. An alternative model for open system U-series/ESR age calculations: (closed system U-series)-ESR, CSUS-ESR. Anc. TL 18, 1e4. Grün, R., 2009. The DATA program for the calculation of ESR age estimates on tooth enamel. Quat. Geochronol. 4 (3), 231e232. Grün, R., Eggins, S., Kinsley, L., Mosely, H., Sambridge, M., 2014. Laser ablation Useries analysis of fossil bones and teeth. Palaeogeogr. Palaeoclimatol. Palaeoecol. 416, 150e167. Grün, R., Katzenberger-Apel, O., 1994. An alpha irradiator for ESR dating. Anc. TL 12,

35e38. Grün, R., Schwarcz, H.P., Chadam, J., 1988. ESR dating of tooth enamel: coupled correction for U-uptake and U-series disequilibrium. Nucl. Tracks Radiat. Meas. 14, 237e241. rin, G., Mercier, N., Adamiec, G., 2011. Dose-rate conversion factors: update. Anc. Gue TL 29, 5e8. Marsh, R.E., 1999. Beta-gradient Isochrons Using Electron Paramagnetic Resonance: Towards a New Dating Method in Archaeology (M.Sc. thesis). McMaster University, Hamilton. Nathan, R., Grün, R., 2003. Gamma dosing and shielding of a human tooth by a mandible and skull cap: Monte Carlo simulations and implications for the accuracy of ESR dating of tooth enamel. Anc. TL 21, 79e84. Prescott, J.R., Hutton, J.T., 1994. Cosmic ray contributions to dose rates for luminescence and ESR dating: large depths and long-term time variations. Radiat. Meas. 23, 497e500.
res, C., 2014. Monte Carlo simulation of USShao, Q., Bahain, J.-J., Dolo, J.-M., Falgue ESR age uncertainty. Quat. Geochronol. 22, 99e106.