The Technical Details: Determining Delta Values

Stable Isotopes

When we talk about the isotopic ratio in a sample, we talk about the delta value. Let's look at how a delta value is actually calculated:

1. The first step in figuring out the δ¹³C for a sample is to find the ratio of ¹³C to ¹²C within the sample.
2. Next compare (by dividing) this ratio to the ratio of ¹³C to ¹²C in a standard.
   • There is a specific standard, with a known, unchanging ratio of ¹³C to ¹²C that all laboratories use in their comparison. For the stable carbon isotopes, this standard is a limestone (called Pee Dee Belemnite—or PDB) from South Carolina. Although PDB is no longer run as the standard, other carbonates (with a known, unchanging ¹³C to ¹²C ratio) are used and compared on the PDB scale. The carbonate standard is reacted with an acid to create gaseous CO₂, so that the sample and standard are both in the same phase.
   • Often the sample and standard may have very similar ratios of the two stable isotopes, which will give you a value very close to (but not exactly) 1.
   

   Two sample bellows are used so that the sample is compared to a standard
3. Many samples that actually have different ratios of ¹³C to ¹²C will give what seems like similar values (say, for example, 0.99 and 0.98). These samples do have different isotopic ratios, but this is hard to see when they only differ after the decimal point. To make this difference easier to see, 1 is subtracted from this value, and then this new calculation is multiplied by 1,000 to give the actual δ¹³C of the sample.
   • This makes it much easier to see the difference between two samples. For the ratios of 0.99 and 0.98, the delta values are -10‰ and -20‰ respectively.

The equation for this is:

Why Go Through This Much Work?

This seems like an awful lot of calculations when you can just look at differences among samples in their ¹³C to ¹²C ratios and ignore all of the calculation steps. The reason that it is conventional to compare to a standard (and then continue on to the next steps in order to get a more ‘friendly’ value) is so that it is easier to compare results both among isotope laboratories and within a single laboratory over a long time period. It is impossible to have an isotope ratio mass spectrometer that perfectly finds the ratio of ¹³C to ¹²C in a sample.

Isotope ratio mass spectrometers measure relative isotopic ratios much better than actual ratios. By comparing to a standard, the precision of the data values are much, much better since all values are relative to a given standard. For example, if the ratio for both the sample and standard are overestimated (or underestimated) by the same relative amount, then dividing the two values will account for this, making it possible to compare δ¹³C among laboratories all across the world.

Radiocarbon Δ¹⁴C

The formula for determining the Δ¹⁴C of a sample is similar to δ¹³C:

The difference is in the term F_N[x], which is still a comparison of the sample to a standard. However, after this comparison, several other calculations occur to find F_N[x].

• The ratio is corrected for “background” ¹⁴C counts, where atoms or molecules that were accidentally and incorrectly identified as ¹⁴C are no longer included.
• The ratio is additionally corrected for the small amount of radioactive decay between the time the sample was collected and the time it was measured, so that the Δ¹⁴C at the time of collection rather than the time of analysis is reported.
• The final difference is that Δ¹⁴C is normalized, where the effect of fractionation is removed. That is, we know from the ¹³C measurements that, for example, when carbon dioxide is photosynthesized by plants, it fractionates, resulting in proportionately less ¹³C in the plant. The same thing happens to ¹⁴C, so plants have proportionately less ¹⁴C than the atmosphere does. If we know how much ¹³C fractionation occurs, we can calculate precisely how much ¹⁴C fractionation there is. We then calculate how much ¹⁴C would have been in the sample if it had not fractionated. This is the Δ¹⁴C. Why go to all this trouble? The main reason is that for radiocarbon dating, scientists want to study how much ¹⁴C has decayed, not how much has fractionated, and this normalization allows them to do just that. The second reason is that it makes it easier to understand the ¹⁴C in the atmosphere – now when plants photosynthesize CO₂, the Δ¹⁴C value in the atmosphere does not change. Of course, we can always reverse the calculations to discover the amount of ¹⁴C without applying this normalization, and this is written as δ¹⁴C.