BLANK MODEL

The “large-mass” blank manifests itself most clearly for low Fm samples (see Figure 1, Panel B, “radiocarbon-free” sample). The large-mass blank can typically be explained by a combination of one or more of the following: sample pretreatment, carbon contamination in the Fe catalyst used in the reduction step, adsorption of atmospheric CO₂ on to the graphite before it is put under the vacuum of the ion source, ion source memory, or background in the energy spectrum of the AMS ¹⁴C detector. The effect of the “large-mass” blank on the Fm of an unknown sample (Fm_unknown) can be expressed as:

(1) $$F{m_{measured}} = F{m_{unknown}}{\rm{}} + F{m_{large - blank}}\left( {1 - {{F{m_{unknown}}} \over {F{m_{standard}}}}} \right)$$

where Fm_measured is the measured AMS Fm, and Fm_large-blank and Fm_standard, are the Fm of the “large-mass” blank and primary standard respectively. Equation (1) is effectively the same as Equation (26) in Donahue et al. (Donahue Reference Donahue, Linick and Jull1990). Conceptually, Equation (1) can be explained as follows: a sample that has the same measured Fm as that of the standard must have an actual Fm equal to that of the standard (Fm_measured = Fm_standard ⇔ Fm_unknown = Fm_standard), while a sample that has the same measured Fm as that of the blank must have an actual Fm equal to that of the blank (Fm_measured = Fm_large-blank ⇔ Fm_unknown = 0).

A “mass-balance” blank reflects the addition of a constant mass contaminant to the sample during sample processing/handling. The modelling of this blank can take the form of a contaminant with a certain mass and Fm (Brown Reference Brown and Southon1997; Hanke Reference Hanke, Wacker, Haghipour, Schmidt, Eglinton and McIntyre2017) or as a combination of modern and ¹⁴C-dead carbon contaminant (Santos Reference Santos, Southon, Drenzek, Ziolkowski, Druffel, Xu, Trumbore, Eglinton and Hughen2010; Welte Reference Welte, Hendriks, Wacker, Haghipour, Eglinton, Gunther and Synal2018). Both approaches are functionally equivalent. In general, the mass-balance measured Fm (Fm_measured) of an unknown sample can be expressed as:

(2) $$F{m_{measured}} = {{F{m_{unknown}} \cdot {m_{unknown}} + F{m_{cont}} \cdot {m_{cont}}} \over {{m_{unknown}} + {m_{cont}}}}$$

where Fm_unknown and Fm_cont are the Fm of the unknown sample and the contaminant, and m_unknown and m_cont are the mass of the unknown and the contaminant. The form of the “mass balance” blank is represented by dashed lines in Figure 1.

Our procedure to derive a Fm_unknown that is corrected for both types of blank is as follows: First, Fm_measured values are large-blank corrected by substituting Fm_{large-blank corrected} for Fm_unknown into Equation (1):

(3) $$F{m_{large - blank{\rm{}}\ corrected}} = F{m_{standard}}{{F{m_{measured}} - F{m_{large - blank}}} \over {F{m_{standard}} - F{m_{large - blank}}}}$$

with a corresponding uncertainty of:

(4) $$\delta _{F{m_{large - blank{\rm{}}\ corrected}}}^2 = {\left( {{{F{m_{standard}}} \over {F{m_{standard}} - F{m_{large - blank}}}}} \right)^2}\delta _{F{m_{measured}}}^2{\rm{}} + {\left( {{{F{m_{standard}}F{m_{measured}} - F{m_{standard}}^2} \over {{{(F{m_{standard}} - F{m_{large - blank}})}^2}}}} \right)^2}\delta _{F{m_{large - blank}}}^2$$

where δ_Fmmeasured, and δ_{Fmlarge-blank} are the uncertainties in the measured and large-blank Fm.

Subsequently, Fm_{large-blankcorrected} is corrected for mass balance blank by substituting Fm_{large-blankcorrected} for Fm_measured in Equation (2):

(5) $$F{m_{unknown}} = {{F{m_{large - blank{\rm{}}\ corrected}}({m_{unknown}} + {m_{cont}}) - F{m_{cont}} \cdot {m_{cont}}} \over {{m_{unknown}}}}$$

with a corresponding uncertainty of:

(6) $$\delta _{F{m_{unknown}}}^2 = {\left( {{{{m_{unknown}} + {m_{cont}}} \over {{m_{unknown}}}}} \right)^2}\delta _{F{m_{large - blank{\rm{}}\ corrected}}}^2{\rm{}} + {\left( {{{{m_{cont}}} \over {{m_{unknown}}}}} \right)^2}\delta _{F{m_{cont}}}^2{\rm{}} + {\left( {{{F{m_{cont}}{m_{cont}} - F{m_{large - blank{\rm{}}\ corrected}}{m_{cont}}} \over {{m_{unknown}}^2}}} \right)^2}\delta _{{m_{unknown}}}^2{\rm{}} + {\left( {{{F{m_{larg - blank{\rm{}}\ corrected}} - F{m_{cont}}} \over {{m_{unknown}}}}} \right)^2}\delta _{{m_{cont}}}^2$$

where δ_{Fmlarge-blank corrected}, δ_Fmcont, δ_munknown, and δ_mcont are the uncertainties in the large-blank corrected Fm, contaminant Fm, sample mass, and contaminant mass, respectively.

To determine the values of Fm_large-blank, Fm_cont, and m_cont, we measure and blank correct secondary standards as if they were unknowns. These values are then used to define a total chi-squared statistic:

(7) $${\chi ^2} = {1 \over N}\sum _{j = 1}^N\left\{ {{1 \over n}\sum _{i = 1}^n{{{{(F{m_{unknow{n_{ji}}}} - F{m_c}_{onsensu{s_j}})}^2}} \over {{\sigma _u}{{_{nknow{n_{ji}}}}^2}}}} \right\}$$

where Fm_consensus if the consensus value of the secondary standard, N is the number of unique secondary standards, each having a different consensus value Fm_consensus, and n is the number of individual measurements of that unique secondary. The χ² statistic is initially calculated using starting values of Fm_large-blank, Fm_cont, and m_cont. Then, using the Solver program in Microsoft Excel (2018), we minimize the χ² statistic by allowing Fm_large--blank, Fm_cont, and m_cont to vary, subject to certain constraints (e.g., Fm_cont ≤ 1, m_cont > 0, etc.). In practice, Fm_large-blank, Fm_cont, and m_cont values are different for different samples process types (e.g., organic versus inorganic).

DATA

Figures 2 and 3 show a subset of inorganic and organic secondary standards processed on our small-sample graphitization-line over the past 5 years (2013–2018). Using the measured and consensus Fm values of these secondary standards, we determined Fm_large-blank, Fm_cont, and m_cont, and for inorganic and organic samples by minimizing χ² (Equation 7). Resulting Fm_large-blank, Fm_cont, and m_cont, values for inorganic and organic secondary standards are listed in Table 1. No long-term time dependency in the values was observed. The black lines in Figures 2 and 3 show expected Fm_measured results using Table 1 blank and mass balance values.

Table 1 Calculated mass balance and large mass blank values for samples processed on NOSAMS’s “small-sample” graphitization line.

Figure 2 A subset of inorganic secondary standards processed on NOSAMS’s “small-sample” line. The black lines are expected results using the blank and mass-balance values listed in Table 1. The gray lines show model sensitivity to the contaminant Fm (dark gray, ±0.41) and mass (light gray, ±0.42 μg).

Figure 3 A subset of organic secondary standards processed on NOSAMS’s “small-sample” line. The black lines are expected results using the blank and mass balance values listed in Table 1. The gray lines show model sensitivity to the contaminant Fm (dark gray, ±0.18) and mass (light gray, ±0.84 μg).

Uncertainty in the Fm_large-blank, Fm_cont, and m_cont values cannot be uniquely determined. In our case, the uncertainty in Fm_large-blank was taken to be:

(8) $$\delta _{F{m_{large - blank}}}^{} = {1 \over 2}F{m_{large - blank}}$$

This value encompasses about two-thirds of the variation seen in individual Fm values of large-mass “radiocarbon-free” samples.

For the error in Fm_cont, and m_cont, an error was assigned to each value such that, after propagation, approximately 68% of the corrected Fm (i.e., Fm_unkown) values were within 1 standard deviation of the consensus value resulting in a normal distribution of results. This error determination was done after χ² minimization. The darker gray lines in Figures 2 and 3 show the sensitivity of expected Fm_measured to uncertainty in the Fm of the contaminant. Light gray lines show the sensitivity of expected Fm_measured to uncertainty in the mass of the contaminant.

Figure 4 Uncorrected and corrected data from TIRI K and FIRI C (Turbidite Carbonate). The solid line is the consensus value of Fm = 0.1043.

Figure 4 shows uncorrected and corrected data from measurements of the inorganic TIRI K / FIRI C (Turbidite Carbonate) sample material using the Fm_large-blank, Fm_cont, and m_cont values for inorganic samples listed in Table 1. Likewise, Figure 5 shows uncorrected and corrected data from measurements of the C-8 (Oxalic Acid) sample materials using the Fm_large-blank, Fm_cont, and m_cont values for organic samples listed in Table 1. Error bars for corrected Fm values (i.e., Fm_unknown) are much larger than the original Fm_measured error bars due mainly to the uncertainty of both Fm_cont and m_cont. To assess the quality of the blank values, Table 2 lists χ²’s of two indicative sub-samples calculated before and after blank correction. These χ²’s are not the same as those used to derive the three-parameter model (i.e., Equation 7). This is a separate calculation and these χ² values should be judged quantitatively only in relative sense in that the corrected values listed in Table 2 are artificially reduced by the boost in the uncertainties of Fm_large-blank, Fm_cont, and m_cont.

Figure 5 Corrected and uncorrected data from the organic sample C-8 (Oxalic Acid). The solid line is the consensus value of Fm = 0.1503.

Table 2 Reduced chi-squared statistics calculated before and after blank and mass balance blank correction. See text.

While the model works to the extent that broad bias in reported Fm_unknown values at low sample mass is removed, it fails to describe the substantial variability observed in the raw (i.e., Fm_measured) data. While this variability is not totally understood, the assigned uncertainties in the blank values (i.e., δ_{Fmlarge-blank}, δ_Fmcont, and δ_mcont) significantly increases the reported uncertainties on unknowns (i.e., δ_Fmunknown) and compensates for the variability observed in the raw data.

For samples with masses >100 μg carbon we do not apply a mass balance correction. For those samples the mass dependent correction is relatively small, and we find that more accurate results are obtained by applying only a blank that is measured concurrently with the unknown sample as opposed to a long-term average blank. Additionally, the values listed in Table 1 are not valid for all sample types. For example, ¹⁴C analysis of tree rings involves cellulose extraction chemistry, and would require a process specific determination of the associated blank.

NOSAMS also measures “ultra-small” samples. Ultra-small samples are typically defined as those samples having a carbon mass less than ~25 μg carbon. A similar exercise for estimating Fm_cont, m_cont, and Fm_blank for these mass samples was conducted. Although not presented here, results similar to those listed in Table 1 were obtained. (Shah-Walter Reference Shah-Walter, Gagnon, Roberts, McNichol, Lardie-Gaylord and Klein2015).

SUMMARY

We have developed a formula that fits the observed mass dependency in measured Fm values. Using long-term measurements of quality control samples, we have modeled the Fm and mass of the blank added during sample processing and AMS measurement. Long term measurements also incorporate short term (wheel-to-wheel) variability in the blank. Different sample types were used to assess the blank added for different sample preparation methods. Applying the mass dependent and mass independent blank values, we have corrected measured Fm values of secondary standards to significantly improve their agreement with consensus values in the mass range 25 μg < mass < 100 μg.