How to cut a sediment core for ²¹⁰Pb geochronology

Abstract

²¹⁰Pb geochronology is described mathematically, but how to cut a sediment core is not explicit. Thick sectioning may reduce dating reliability; but on the contrary, thin sectioning is time-consuming. Considering the counting error of excess ²¹⁰Pb, a new method was proposed for the determination of meaningful thickness for sectioning. The authors applied this method to a core from Tokyo Bay. To increase the thickness with depth, the treatment is helpful in reducing the sample number for measurement and improving the dating accuracy. In addition, the averaging effect involved in sectioning was discussed, and it was confirmed that the averaging effect in the new method on ²¹⁰Pb geochronology may generally be neglected.

Appendix

Solution of Eq. 5

Equation 5 can be transformed into

$$ A_0 \exp \left( {\frac{{ - \lambda }} {s}\sum\limits_{i = 1}^{n - 1} {h_i } } \right) - \sqrt {A_0 } \exp \left[ {\frac{{ - \lambda }} {s}\left( {\sum\limits_{i = 1}^{n - 1} {h_i + \frac{1} {2}h_n } } \right)} \right] - A_0 \exp \left[ {\frac{{ - \lambda }} {s}\left( {\sum\limits_{i = 1}^{n - 1} {h_i + \frac{1} {2}h_n } } \right)} \right] = 0. $$

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Let ${{\sqrt {A_0 } } \mathord{\left/ {\vphantom {{\sqrt {A_0 } } {A_0 }}} \right. \kern-\nulldelimiterspace} {A_0 }} = \varepsilon _1$, Eq. 15 can be written as

$$ \exp \left( {\frac{{ - \lambda }} {{2s}}h_n } \right) + \varepsilon _1 \exp \left( {\frac{\lambda } {{2s}}\sum\limits_{i = 1}^{n - 1} {h_i } } \right)\exp \left( {\frac{{ - \lambda }} {{4s}}h_n } \right) - 1 = 0. $$

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Let $\exp \left( { - \frac{{\lambda h_n }} {{4s}}} \right) = y,$ the equation can be rewritten as

$$ y^2 + \varepsilon _1 \exp \left( {\frac{\lambda } {{2s}}\sum\limits_{i = 1}^{n - 1} {h_i } } \right)y - 1 = 0. $$

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The solution as shown in Eq. 6 is therefore obtained.

The transformation procedure for D

For $A_x = A_0 \exp \left( { - \frac{{\lambda x}} {s}} \right),\quad n \geqslant 1,h > 0,{\enspace} x_n = \left( {n - \frac{1} {2}} \right)h,$

$$ \begin{aligned} D & = \frac{1} {h}\int\limits_{(n - 1)h}^{nh} {A_0 \exp \left( { - \frac{\lambda } {s}x} \right)} {\text{d}}x - A_n \\ & = \frac{{A_0 }} {h}\int\limits_{\left( {n - 1} \right)h}^{nh} {\exp \left( { - \frac{\lambda } {s}x} \right)\,{\text{d}}x} - A_0 \exp \left[ { - \frac{\lambda } {s}\left( {n - \frac{1} {2}h} \right)} \right] \\ & = \frac{{A_0 }} {h}\left[ { - \frac{s} {\lambda }\exp \left( { - \frac{\lambda } {s}x} \right)} \right]_{(n - 1)h}^{nh} - A_0 \exp \left[ { - \frac{\lambda } {s}\left( {n - \frac{1} {2}h} \right)} \right] \\ & = - \frac{{A_0 s}} {{h\lambda }}\left( {\exp \left( { - \frac{\lambda } {s}nh} \right) - \exp \left[ { - \frac{\lambda } {s}(n - 1)h} \right]} \right) - A_0 \exp \left[ { - \frac{\lambda } {s}\left( {n - \frac{1} {2}h} \right)} \right] \\ & = - \frac{{A_0 s}} {{h\lambda }}\exp \left( { - \frac{\lambda } {s}nh} \right) + \frac{{A_0 s}} {{h\lambda }}\exp \left( { - \frac{\lambda } {s}nh} \right)\exp \left( {\frac{\lambda } {s}h} \right) - A_0 \exp \left( { - \frac{\lambda } {s}nh} \right)\exp \left( {\frac{\lambda } {{2s}}h} \right) \\ & = A_0 \exp \left( { - \frac{\lambda } {s}nh} \right)\left( {\frac{s} {{\lambda h}}\exp \left( {\frac{\lambda } {s}h} \right) - \frac{s} {{\lambda h}} - \exp \left( {\frac{\lambda } {{2s}}h} \right)} \right). \\ \end{aligned} $$

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