### Age Determination

### Methods – Age

### 1. Purpose

The analytical procedures described here aim to determine the post mortal interval (PMI) of elephants for the purpose of the protection of elephant species. Its development and validation has been described in the final report to the Federal Agency for Nature Conservation (SCHUPFNER 2012). The analytical procedures applied are validated to age representative elephant ivory samples of masses ranging from about 7 g to about 12 g. For this purpose, the values of the specific activity of ^{14}C/C and ^{90}Sr/Ca and ^{228}Th/^{232}Th are determined and compared to calibration curves of the values of PMI as a function of the analyses results.

### 2. Representative ivory samples

According to Brunnermeier et al. (2012), the sample must be taken from the stump of the elephant tusk to be representative for the purpose of age determination applying the analyses methods to determine the isotope profile of ^{14}C/C and ^{90}Sr/Ca and ^{228}Th/^{232}Th.

### 3. Analysis procedures to determine the isotopic profile of ^{14}C/C and ^{90}Sr/Ca and ^{228}Th/^{232}Th

#### 3.1 Sample Preparation

When determining the concentration of ^{14}C in the ivory sample, direct absorption and liquid scintillation counting (LSC) can be applied. During this process, the carbon has to be released as carbon dioxide by incineration of ivory. Finally, the carbon is adsorbed to 20 ml through the application of a scintillation cocktail, which serves to detect the concentration of ^{14}C. The scintillation cocktail named OXYSOLVETM is suited very well for this purpose.

If there is a need of analysing ^{90}Sr and/or ^{228}Th/^{232}Th, additional to the determination of ^{14}C/C, the residue of the carbon release should be ashed for about 16 hours in a muffle furnace at a temperature of about 650 °C. After this, the ivory ash has to be moistening with a few mL of nitric acid (65%) and it has to be heated at a laboratory sand-bath at level 4 for a few hours to dryness. Ashing and moistening should be repeated until a snow-white ash remains which is free of carbon and of nitrate as far as possible. The mean value of the ratio of ash to wet weight of ivory has been found to be about (54 ± 5)% (see table 1). At this point, the ivory ash is prepared for combined radiochemical purification and the enrichment with ^{90}Sr and thorium.

#### 3.2 Determination of ^{14}C/C applying direct absorption and liquid scintillation counting

##### 3.2.1 Radiochemical purification and concentration of carbon isotopes (^{12}C, ^{13}C, ^{14}C)

The carbon isotopes (^{12}C, ^{13}C, ^{14}C) which are released by incineration must be transformed into carbon dioxide which is purified from all radionuclides, stable elements or chemical compounds potentially interfering the absorption and the following nuclear detection process. This procedure combining several precipitation steps is described in detail by Brunnermeier et al. (2011) and Brunnermeier (2012).

##### 3.2.2 Detection of ^{14}C applying low-level LSC

^{14}C and all stable carbon isotopes like ^{12}C and ^{13}C which have been released and purified from the ivory sample are absorbed in 20 mL of OXYSOLVETM which is in a PE-LSC-Vial of 20 mL volume. This LSC-cocktail ^{14}CO_{2} mixture produces flashes which can be detected sensitively by a low-level scintillation counter of type “Quantulus 1220”, produced by LKB Wallac. This scintillation counter combines passive and active shielding to realize both at once; a high counting efficiency of about 73% and low-background counting rates of about 3.1 cpm.

The ^{13}C is corrected. The calibration data, ^{13}C correction and the validation are shown at Brunnermeier (2012). With this analysis, the quality criteria are realized at a counting time of about 2000 minutes. These are shown in table 1 according to Schupfner (2012).

#### 3.3 Combined radiochemical purification of ^{90}Sr/Ca and ^{228}Th/^{232}Th applying Ion-Exchange Chromatography at elevated temperature.

The determination of ^{90}Sr/Ca and ^{228}Th/^{232}Th in elephant ivory can result in a more unambiguous dating of ivory under several conditions which are shown in section 4.2.

The combined radiochemical purification procedure is able to purify, enrich and prepare both for low-level nuclear detection, ^{90}Sr and the thorium isotopes ^{228}Th, ^{230}Th and ^{232}Th. Its basic principle is the application of ion-exchange chromatography at elevated temperature. This method has been firstly developed to purify ^{90}Sr from about 5 to 6 g of ivory ash, and secondly to purify the thorium containing part of the elution solution by applying well suited reversed-phase extraction chromatography. The ^{90}Sr fraction is prepared to be detected with a low-level gas-counter. Therefore, it is precipitated as strontium sulphate at a paper filter of 55 mm diameter. The purified thorium is prepared as a very thin layer by applying electroplating for alpha-spectrometry with silicon semiconductor detectors. These methods are described in detail by Schmied (2012) in the case of ^{90}Sr and Brunnermeier (2012) in the case of the thorium isotopes.

#### 3.4 Nuclear radiation detection methods

##### 3.4.1 Detection of ^{90}Sr applying a low-level-gas counter

^{90}Sr (t_{1/2} _{1}= (28.90 ± 0.03) years (NNDC 2016)) is a long living beta emitting fission product which decays to the short-living ^{90}Y (t_{1/2} 1= (64.053 ± 0.020) hours (NNDC 2016) to the stable isotope ^{90}Zr.

To detect the high energetic beta-radiations of ^{90}Sr (endpoint energy = (546.0 ± 1.4) keV Browne (1997)) and ^{90}Y (endpoint energy = (2280.1 ± 1.6) keV (Browne 1997) a low-level-gas-flow-counting device, for example a 10-small bowl-low-level-beta counter of type LB 770, produced by Berthold, is suited very well. In order to begin with the counting, ^{90}Y, which has been removed from the sample during the radiochemical purification, is build up according to the lowest level of radioactivity.

During this process, ^{90}Y is about to reach the secular radioactive equilibrium with ^{90}Sr. Therefore, the activities of ^{90}Y and ^{90}Sr are nearly equal during the following counting period. This results in the lowest possible limit of detection with a high counting efficiency of about 66% and low-background counting rates of about 0.5 cpm. At a counting time of about 10000 minutes with this analysis the quality criteria are realized which are shown in table 1 according to Schupfner (2012).

^{1} t_{1/2}: half life time according to the National Nuclear Data Center Brookhaven National Laboratory, based on ENSDF and on the Nuclear Wallet Cards (NNDC 2016)

##### 3.4.2 Detection of ^{228}Th, ^{230}Th and ^{232}Th applying alpha-spectrometry with Si-detectors

These thorium isotopes are all alpha-emitters from the naturally occurring element thorium. Their half-life times range from (1.9116 ± 0.0016) years (NNDC 2016) for ^{228}Th, (75400 ± 300) years (NNDC 2016) for ^{230}Th and (1.40 ± 0.01) · 1010 years (NNDC 2016) for ^{232}Th. ^{230}Th serves as a nuclide indicating a thorium contamination of the ivory sample. A well known amount of about 0.01 to 0.05 Bq of ^{229}Th with a half-life time of (7932 ± 28) years (NNDC 2016) is added to the sample ash, controlling the chemical losses of thorium during the radiochemical purification process of applying isotope dissolution analyses. All these radionuclide are identified and quantified with a single counting period by applying alpha-spectrometry. Therefore, Si- semiconductors and detectors with high energy resolution are used. The values of the FWHM^{2} range from about 30 to 50 keV, which also depends on the quality of sample preparation. The alpha-energies are specific for the thorium isotopes and they increase as follows:(4012.3 ± 1.4) keV (Abasaleem 2014) for ^{232}Th, (4687.0 ± 1.5) keV^{3} (Akovali 1996) for ^{230}Th, (4845.3 ± 1.2) keV^{3} (Jain et. al 2009) for ^{229}Th and (5423.15 ± 0.22) keV^{3} (Jain et. al 2009) for ^{282}Th. This results in a very low limit of detection with a high counting efficiency of about 30 to 40% and low-background counting rates of about 0.0001 to 0.01 cpm depending on detector and energy. At a counting time of about 12000 minutes with this analysis the quality criteria are realized which are shown in table 1 according to Schupfner (2012).

#### 3.5 Quality criteria of the determination of the isotope profile of ^{14}C/C and ^{90}Sr/Ca and ^{228}Th/^{232}Th

With the analyses methods shown in sections 3.1 to 3.4 the quality criteria are realized as shown in table 1 according to Schupfner (2012, 2016).

**Table 1 a**: Characteristic values of sample and analytical parameters according to Schupfner (2012, 2016).

^{2} FWHM: Full Width at Half Maximum

^{3} only the energies of the alpha transitions with the highs transition probability is shown here

#### 3.6 Other determination methods

There are a number of other determination methods which are able to determine the isotope profile of ^{14}C/C and ^{90}Sr/Ca and ^{228}Th/^{232}Th as well as the methods described here or even better. In the following section the calculation of the PMI^{4} does not assume the application of a certain analytical method as described here. It is only assumed that the determination parameters are similar or even more precise than the values shown in table 1b.

**Table 1b**: Characteristic values of detection parameters compared with the reference data according to Schupfner (2012, 2016).

^{4} PMI: Post Mortal Interval: PMI is the period of time from the time of death of the animal whose ivory is analysed to the time at which the detection started

### 4. Calculation of the PMI^{5} from data of the isotopic profile of ^{14}C/C and ^{90}Sr/Ca and ^{228}Th/^{232}Th

According to the calibration curves which have been received to analyse independent dated ivory samples the specific activities of ^{14}C/C and ^{90}Sr/Ca as well as the activity ratios of ^{228}Th/^{232}Th the PMI of an elephant can be calculated under certain conditions. The calibration curves are shown in several studies like Schupfner (2012, 2016) which are based on the data published by Brunnermeier et al. (2011, 2012) and by Brunnermeier (2012) and Schmied (2012). All studies found that the values of ^{14}C/C can enable a sufficient unambiguous determination of the PMI, if the analyses results are within a suited range. In the following section, the interpretation and calculation of the PMI as results of the ^{14}C/C determination are shown.

#### 4.1 Calculation of the time of death T_{D}(^{14}C/C)

The results of ^{14}C/C in units of pMC^{6} are found to be within a range from about 97 pMC to about 109 pMC. The mean value of the relative reference value of the uncertainty u_{rel} (^{14}C/C) is about (5 ± 3)% which has been observed during the validation process analysing 69 samples (Schupfner 2012). This is a typical range applying the method of low-level LSC after direct ^{14}CO_{2} absorption which has been described in section 3.1 and 3.2. Data taken from recently completed studies by Schupfner (2016) and Auerhammer (2015) indicate that the mean value of ^{14}C/C of ivory samples from 2010 to 2015 is (103 ± 6) pMC. Before the nuclear weapon fallout caused a significant increase of ^{14}C/C in global terrestrial food chains, i.e. before 1956. The value of (98 ± 2) pMC has been found representative based on data of Hua and Barbetti (2004) who analysed wood samples. The value of (97.8 ± 2.0) pMC is regarded as representative as stated in Brunnermeier (2012) who analysed independently dated ivory. Other references like Eisenbud (1987), the UNSCEAR report of 1993 and Lieser (1980) present similar values of ^{14}C/C before 1956. On the other hand, the deviation of their values from the mean value is about ± 10%. Within the scope of uncertainties of the analyses no significant difference between wood and ivory can be observed before 1956.

It has been shown in a recently completed study Schupfner (2016) that nearly 50% of the samples with a PMI < 5 years are below the upper limit of ^{14}C/C, which is representative for wood and ivory before 1954.

The calibration curve of ^{14}C/C which shows the dates of death T_{D} versus ^{14}C/C has been actualised for 1965 ≤ T_{D} ≤ 2015. The question, whether the elephant lived at the south or at the north hemisphere do not cause a significant difference at all T_{D} < 1964 and T_{D} > 1967.

T_{D} is defined as T_{D} =: T_{A} - PMI

where

T_{A} : Date of analysis

PMI: Post mortal interval

The unit of pMC T_{D}(^{14}C/C) is calculated by equation 1 within an interval of 1965 < T_{D} < 2015, determining ^{14}C/C. According to the calibration curves based on data of Brunnermeier (2012), Schupfner (2016), Auerhammer (2015) and Hua and Barbetti (2004), the polynomials of degree 4 or 5 are suggested to determine T_{D} within an interval of 1954 < T_{D} < 2015:

T_{D} (^{14}C/C) = a_{5} · (^{14}C/C)^{5} + a_{4} · (^{14}C/C)^{4} + a_{3} · (^{14}C/C)^{3} + a_{2} · (^{14}C/C)^{2} + a_{2} · (^{14}C/C)^{2} + a_{1} · ^{14}C/C + a_{0}

Eq. 1

The coefficients of the polynomial functions a_{0} to a_{5} are given in the following table 2.

**Table 2**: Coefficients of the polynomial functions a_{0} to a_{5} determining T_{D} (^{14}C/C) within an interval of T_{D} ranging from about 1954 to 2015.

^{5} PMI: Post Mortal Interval: PMI is the period of time from the time of death of the animal whose ivory is analysed to the time at which the detection started

^{6} pMC: Percentage Modern Carbon. Unit of specific activity of ^{14}C/C. Definition see Brunnermeier 2012

According to the date of Hua and Barbetti (2004), the distinction between north and south hemisphere is necessary only in a period ranging from about 1963 to 1965. In 1963, the maximum value of ^{14}C/C in wood was reached at about 199 pMC at the north hemisphere whereas in 1965 the maximum value of ^{14}C/C with about 168 pMC was observed at the south hemisphere (see Hua and Barbetti (2004)). Therefore, only the different values of the coefficients a_{0} to a_{5} from about 1956 to 1963 are suggested. If it is known, it should also be also taken into consideration whether an analysed ivory sample origins from an elephant living on the southern or the northern hemisphere. Above 1965, a distinction between north and south hemisphere is no longer suggested to be necessary.

#### 4.2 Conditions justifying the determination of ^{90}Sr/Ca and ^{228}Th/^{232}Th

##### 4.2.1 Suggestion to determine ^{90}Sr/Ca

According to data taken from several studies, it is recommended to determine ^{90}Sr/Ca in elephant ivory additional to ^{14}C/C, if the specific activity is equal or lower than 109 pMC. Therefore, it is not possible to distinguish between a time of death before 1958 and those after 2010 with a necessary confidence level. If the value of the specific activity of ^{90}Sr/Ca is significantly above the lower limit of detection of about 0.002 Bq ^{90}Sr/g Ca, the time of death is after 2010 with a confidence level of 95%. If ^{90}Sr/Ca is below 0.002 Bq ^{90}Sr/g Ca, the time of death is before 1956 with a confidence level of 95%.

##### 4.2.2 Suggestion to determine ^{228}Th/^{232}Th

Based on the data taken from several studies, it is suggested to determine ^{228}Th/^{232}Th in elephant ivory additional to ^{14}C/C, if the specific activity is in the range of 109 pMC < ^{14}C/C < 147 pMC. Moreover, the time of death could be above 1.2.1976 and an unambiguous dating is not possible no matter how precise the result is. In this case, the knowledge of the value of ^{90}Sr/Ca does not help to solve the problem because of the similar shapes displayed by the bomb curves of ^{90}Sr/Ca and ^{14}C/C. In contrast to the bomb curves, the thorium calibration curve of PMI (^{228}Th/^{232}Th) (Brunnermeier 2012, Singh 2015) is a continual decreasing function between about 4 years < PMI < about 60 years. Therefore, it is suggested to determine ^{228}Th/^{232}Th, if ^{14}C/C is in the range mentioned above.

#### 4.3 Calculation of the time of death T_{D}(^{90}Sr/Ca)

The trend of ^{90}Sr/Ca of T_{D} > 1965 can be fitted by two different curves. The curves of significantly lower values fit a portion of about 27% of the available data quite well. About 73% of the ^{90}Sr/Ca results with T_{D} > 1965 are fitted much better by the higher curve.

All the analysed ivory included in a recently completed study (Schupfner 2016) is sampled from animals living at the north hemisphere. The (UNSCEAR report of 1993) presents the assessment of the population weighed deposition density of the most important radionuclides from the atmospheric nuclear test explosions on the north and south hemisphere. They state that the deposition of ^{90}Sr was a factor of about 4, which is higher at the north hemisphere than at the southern hemisphere (Eisenbud 1987). Perhaps, this fact declares these relatively high values. The maximum of ^{90}Sr/Ca in bone tissues shown by Eisenbud (2015) and in ivory shown by Schmied (2012) are both from 1965. However, their values differ by a factor of about 25. In ivory, much higher maximum values of ^{90}Sr/Ca are observed than in human bone tissue. Before 1960, there is no difference detectable between human bones tissue and ivory. After 1960 until about 1982 the ratio of ^{90}Sr/Ca in ivory compared to human bone tissue decreases from about 25 to 13 compared with the ivory calibration curve (high). With reference to the calibration curves of Schupfner (2012, 2016), on the basis of the data of Schmied (2012), logarithmic functions are suggested to determine T_{D} within an interval of 1956 < T_{D} < 2015:

T_{D}(^{90}Sr/Ca) = a_{0} · ln (^{90}Sr/Ca) + t_{1}

Eq. 2

The coefficients are given in the following table 3.

**Table 3**: Coefficients of the calibration function to determine T_{D} (^{90}Sr/Ca) within an interval of 1956 < T_{D} < 2015.

The values of ^{90}Sr/Ca of the recently analysed samples are significant above the detection limit in ivory of low PMI. In contrast to ^{14}C/C, the following conclusions may be drawn:

If ^{90}Sr/Ca is found above the LLD ≤ 0.002 Bq ^{90}Sr/g Ca in elephant ivory, a PMI corresponding to a date of death T_{D} before 1956 is excluded.

For T_{D} > 1965 two calibration curves describe the behaviour of T_{D}(^{90}Sr/Ca) fairly well. The lower curve calculated with equation 2 (a_{0} = -7.661 and t_{1} = 1964.6) describes the values of ^{90}Sr/Ca which are about as factor of 4 to 5 below the upper curve calculated with equation 2 with the a_{0} = -11.495 and t_{1} = 1973.2. A reasonable interpretation of the observed difference is the geographic difference of the origin of the ivory samples. It can be assumed that the ivory samples with a low value of ^{90}Sr/Ca are taken from elephants living mainly at the south hemisphere, whereas the high values of ^{90}Sr/Ca are related to elephants living mainly at the north hemisphere. Nevertheless, the analysis of ^{90}Sr/Ca with sufficient low LLD of about 0.002 Bq ^{90}Sr/g Ca can prevent the impending blindness of ^{14}C/C age determination up to about 2045, if the trend of the green calibration curve of figure 2 would continue.

If ^{90}Sr/Ca is found below the LLD ≤ 0.002 Bq ^{90}Sr/g Ca and ^{14}C/C is about 100 pMC in elephant ivory, a PMI corresponding to a date of death T_{D} before 1956 is not excluded, but it cannot be stated that the death had occurred before 1956. Accordingly, it can happen that the elephant lived at the south hemisphere and its value of ^{90}Sr/Ca in ivory is low, according to Schmied (2012). In order to enable a correct calculation of the T_{D} the determination of ^{228}Th/^{232}Th is suggested in such cases.

#### 4.4 Calculation of the time of death T_{D}(^{228}Th/^{232}Th)

Due to the fact that thorium isotopes are naturally occurring radionuclides, the analytical blank has to be corrected carefully. In the following section 4.4.1 it is suggested how this can be done.

##### 4.4.1 Test for thorium contamination applying ^{230}Th as indicator radionuclide

Based on the data of several studies Schupfner (2012, 2016), it can be stated that there is a linear relationship between the activity concentration of a(^{230}Th) and a(^{232}Th) related to ash weight. Expected values of ^{232}Th a_{exp}(^{232}Th) can be calculated from a(^{230}Th) according to the following equation:

a_{exp}(^{232}Th) = c · a(^{230}Th)

Eq. 3

where c is a constant with c ± u(c) = (0.55 ± 0.34)

It is assumed that a_{exp}(^{232}Th) = 0, if a(^{230}Th) = 0. This is an assumption, because on the one hand it is not possible to avoid an analytical blank, on the other hand it cannot be excluded that a sample does not contain any thorium. Before calculating the most probable ratio activity concentration of ^{228}Th/^{232}Th, the following contamination test are performed:

- a) If a(
^{232}Th) < 1.1·10-4 Bq^{232}Th/g aw and - b) If a(
^{230}Th) < 2.1·10-4 Bq^{230}Th/g aw and - c) a
_{exp}(^{232}Th) is expected to be within a range given by u(c) in equation 3.

Depending on the result of this contamination test the following conclusions are drawn.

^{7} activity concentration a is calculated as activity in Bq per sample ash weight

**No Blank Correction**

If no increase of ^{232}Th is observed, i.e. the above statements a) and b) and c) are all true, then the activity ratio of ^{228}Th/^{232}Th is calculated according to the following equation in 4a. This will be derived from the application of the isotope dilution analysis of energy resolved alpha-spectrometry with silicon barrier surface detectors (see section 3.4.2).

The activity concentration of ^{232}Th is given by:

a(^{232}Th) = A_{tr} · Y_{tr} · N(^{232}Th)Y_{232Th} · N(tr) · m_{a}

Eq. 4a

where

tr: radiochemical yield tracer, here ^{229}Th; added to the dissolved sample before beginning with the radiochemical purification procedure.

0.01 Bq < A_{tr} < 0.1 Bq: added amount of activity of the radiochemical yield tracer

Y_{tr}, Y_{232Th}: Emission probability of the alpha-transition for ^{229}Th and ^{232}Th respectively

Y_{tr} = 0.9840 (Bq·s) - 1;(Jain et al. 2009) Y_{232Th} = 1.0000 (Bq·s)-1 (Abasaleem 2014)

N(tr), N(^{232}Th): Number of background corrected counts within the region of interest of the alpha-peak of ^{229}Th and ^{232}Th respectively:

N(x) = N ́(x) – N_{0}(x)

where

- N ́(x): Total number of counts within the alpha-peak of interest
- N
_{0}(x): expected number of background counts within the alpha-peak of interest - x: alpha-peak of interest for
^{229}Th and^{232}Th respectively - m
_{a}: mass of the sample ash being analysed

In analogy to ^{232}Th the activity concentration of ^{228}Th is given by:

a(^{228}Th) = A_{tr} · Y_{tr} ·N(^{228}Th)Y_{228Th} · N(tr) · m_{a}

Eq. 4b

where

Y_{228Th} = 0.9994 (Bq·s) - 1 (Singh 2015): Emission probability of the alpha-transition for ^{228}Th

N(^{228}Th) = N ́(^{228}Th) – N_{0}(^{228}Th): Number of background corrected counts within the region of interest of the alpha-peak of ^{228}Th.

As shown above a contamination with ^{232}Th is excluded. Therefore, the calculation of the ratio of the activity concentration of ^{228}Th/^{232}Th can be simplified in the following way:

^{228}Th/^{232}Th =: a(^{228}Th)a(^{232}Th) = Y_{tr} · N(^{228}Th) Y_{228Th} · N(tr) · m_{a} Y_{tr} · N(^{232}Th)Y_{232Th} · N(tr) · m_{a}

Eq. 5a

This can be simplified to:

^{228}Th/^{232}Th = Y_{232Th} · N(^{228}Th)Y_{228Th} · N(^{232}Th)

Eq. 5b

This equation results in the most probable value if:

a_{LLD}(^{232}Th) < a(^{232}Th) < a_{mean}(^{232}Th) +1.69·SD(^{232}Th)

The ratio ^{228}Th/^{232}Th being calculated according to equation 5b tends to result in slightly too high values than too low values.

If a_{LLD}(^{232}Th) ≥ a(^{232}Th) the magnitude of N(^{232}Th) in equation 5b must be replaced by the number of counts N_{LLD}(^{232}Th), corresponding to the lower limit of detection LLD

N_{LLD}(^{232}Th) = 3 · [(1 + t_{L,s}t_{L0}) · N_{0tL,s}(^{232}Th)]^{1/2}

where t_{L,s}, t_{L0} are the counting time t_{L} (L abbr. for life time) of alpha spectrometry of the sample and the background counting time t_{L0}, respectively. Then:

^{228}Th/^{232}Th ≥ Y_{232Th} · N(^{228}Th)Y_{228Th} · N_{LLD}(^{232}Th)

Eq. 5c

The ratio ^{228}Th/^{232}Th being calculated according to equation 5c tends to result in rather too low than too high values.

**Blank Correction**

If a significant increase of ^{232}Th is observed according to the criteria in a), b) or c) above, then the activity ratio of ^{228}Th/^{232}Th is calculated according to the following equation taking into consideration a blank correction.

^{228}Th/^{232}Th = a(^{228}Th) - a_{blank}(^{228}Th)a(^{232}Th) - a_{blank}(^{232}Th)

Eq. 6a

Analysing ivory combs made before 1945, the ratio of the analytical blank a_{blank}(^{228}Th) and a_{blank}(^{232}Th) have been found (Schupfner 2012, Brunnermeier 2012) to be

a_{blank}(^{228}Th)a_{blank}(^{232}Th) = 1.19 ± 0.30

Therefore, equation 6a can be simplified to:

^{228}Th/^{232}Th = a(^{228}Th) - 1.19 · a_{blank}(^{232}Th)a(^{232}Th) - a_{blank}(^{232}Th)

Eq. 6b

The ratio ^{228}Th/^{232}Th being calculated according to eq. 6c tends to give slightly too low values than too high ones.

##### 4.4.2 Equations calculating the time of death T_{D}(^{228}Th/^{232}Th)

Applying the actualized calibration curves received on the basis of the data taken from Brunnermeier 2012 and Schupfner 2016, the PMI can be calculated by applying the following equations 7a and b:

The value of PMI in the case of PMI > 4 years is calculated with:

PMI = T_{D} - T_{0} = c_{1}·ln(^{228}Th/^{232}Th – 1) + c_{0}

Eq. 7a

where

c_{1}= - 8.771 years

c_{0} = 36.123.

To calculate the value of PMI in the case of PMI ≤ 4 years is calculated with:

PMI = T_{D} - T_{0} = c_{2}·ln(^{228}Th/^{232}Th – 1)^{χ}

Eq. 7b

where

c_{2} = 5.866·10^{-6} years

χ = 3.828

The uncertainty of age determination is about ± 7 years, for values of PMI below 50 years. If the ratios of ^{228}Th/^{232}Th are about 1.2 or below, the uncertainty of age determination increases. In this cases it has to be stated that the PMI is > 50 years.