In this chapter, we consider the linear radiative transfer model for a passive medium with no intrinsic thermal contributions. The domain is a one-dimensional slab for which reflectivity and transmissivity are determined from the total in- and outgoing radiative fluxes across the boundaries. Further, we assume isotropic scattering in the medium and no emissivity and reflectivity on the boundaries but thermal radiation from the latter and an isotropic incoming radiation from an external source. Our developments are based on the discrete ordinate method in the angular variable and a modified version of the finite volume method in the spatial variable. The solution is obtained by an iterative method, and we present a necessary condition for its convergence based on norm operations.

The direct carbonate procedure for accelerator mass spectrometry radiocarbon (AMS 14C) dating of submilligram samples of biogenic carbonate without graphitization is becoming widely used in a variety of studies. We compare the results of 153 paired direct carbonate and standard graphite 14C determinations on single specimens of an assortment of biogenic carbonates. A reduced major axis regression shows a strong relationship between direct carbonate and graphite percent Modern Carbon (pMC) values (m = 0.996; 95% CI [0.991–1.001]). An analysis of differences and a 95% confidence interval on pMC values reveals that there is no significant difference between direct carbonate and graphite pMC values for 76% of analyzed specimens, although variation in direct carbonate pMC is underestimated. The difference between the two methods is typically within 2 pMC, with 61% of direct carbonate pMC measurements being higher than their paired graphite counterpart. Of the 36 specimens that did yield significant differences, all but three missed the 95% significance threshold by 1.2 pMC or less. These results show that direct carbonate 14C dating of biogenic carbonates is a cost-effective and efficient complement to standard graphite 14C dating.

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