## From the eMail Bag: Carbon Isotopes, Part 2: The Delta Notation

#### Posted on 28 December 2017 by David Kirtley

We occasionally receive excellent questions and/or comments by email or via our contact form and have then usually corresponded with the emailer directly. But, some of the questions and answers deserve a broader audience, so we decided to highlight some of them in a new series of blog posts.

In Part 1, we learned about carbon isotopes: how ^{14}C forms in the atmosphere, how different isotopes move through the Carbon Cycle, and how isotopic measurements reveal clues about our changing climate. In this post we will look at how measurements of changing isotopic ratios are described.

Here, again, is the IPCC graph (Figure 1) illustrating the rise in atmospheric CO2 (panel a, black saw-toothed line) and the decreasing ^{13}C:^{12}C ratio in the same CO_{2} (panel b, red line).

One of our readers was puzzled by this graph (which appears in our rebuttal "How do human CO2 emissions compare to natural CO2 emissions?") and emailed us these questions about it:

Can you help me interpret the red line? Does it indicate a decline (negative change) in the C13 isotope (i.e. -7.7 = -7.7 parts per thousands) or are the values showing a ratio of C13/C12 because of the delta symbol? If the red values do represent a ratio can you illustrate with a hypothetical example how a negative value was calculated; for instance how do you get a negative value by comparing C13 with C12 (I would have thought a ratio would produce a fraction of some sort)?

The concentration of atmospheric CO2 is most commonly measured in parts per million (ppm) as seen in the top (a) panel of the graph. The current value is about 405 ppm, which means that for every million molecules in a sample of air there are 405 CO2 molecules. This can be shown as a fraction or a ratio:

or as a decimal: 0.000405

We can also turn this into a percentage:

The carbon isotope ratios are measured in something quite different: the δ-value or notation: δ^{13}C(CO2)‰, or "delta ^{13}C (of CO2) per mil". The "per mil" (symbol: ‰) might lead you to think that this too is a measure of concentration, except, instead of a percentage ("per cent" or "per 100") the carbon isotopes might be a measure of "per mil" or "per 1000". *But this is not the case.** *(Or try this link if that last one doesn’t work.)

Here is the equation used to calculate the δ-value:

Notice that this equation contains a multiplication by 1000, which is where the "per mil" comes from. The values are multiplied by 1000 because they are very small numbers and this multiplication trick makes the values more "user-friendly". Look again at the IPCC graph which gives the δ-value in 1981 as -7.6‰, the "per mil" symbol tells us that the original value was multiplied by 1000, thus the original value was -0.0076.

But what does -0.0076 mean? Has the ^{13}C decreased by 0.0076...somethings?

Look again at the δ-value equation. You can see that the top part of the fraction within the bracket is the ratio of ^{13}C:^{12}C from some sample containing carbon. The bottom part of the fraction is another ^{13}C:^{12}C ratio from a *standard* sample which has a known, unchanging ratio of ^{13}C:^{12}C. For carbon isotopes, the standard used is a limestone formation from South Carolina called the Pee Dee Belemnite (or PDB)^{1}, which has an unusually high amount of ^{13}C.

The δ-value is basically a ratio of ratios and can be thought of as a scale to compare different isotope ratios (Figure 2). The standard sample is the zero point of this scale. If there is more ^{13}C in our sample than in the standard, then the δ-value will be positive; if there is less ^{13}C in our sample than in the standard, then the δ-value will be negative. The δ-value doesn't give us a *specific* number about our sample, as in* x* ppm of ^{13}C, rather it tells us the *relative* difference between the sample and the standard.

Why not just give us the specific numbers of carbon isotopes? (Show me the data!) Isotope ratios are measured by mass spectrometers but it is impossible for these devices to perfectly measure the ^{13}C to ^{12}C ratio in a sample. Lauren Shoemaker, in her in-depth NOAA website on isotopes, explains:

Isotope ratio mass spectrometers measure

relativeisotopic ratios much better thanactualratios. By comparing to a standard, the precision of the data values are much, much better since all values are relative to a given standard.

She also points out that δ-values make it "easier to compare results both among isotope laboratories and within a single laboratory over a long time period".

Delta-values also make the numbers associated with isotopic ratios much more "user-friendly". To see this let's work through some examples using the δ-value formula. The IPCC graph shows that in 1981 the δ-value for atmospheric CO_{2} was -7.6‰. The PDB standard ratio is 0.011237. With these two numbers we have enough information to calculate what the ^{13}C:^{12}C ratio was in 1981:

This works out to a ratio of 0.0111516 for the 1981 sample. For 2002 the δ-value was -8.1‰, which gives a ratio of 0.0111459.

Let’s broaden our view out a bit further than that twenty year time span. This graph in Figure 3 (from CSIRO, the Australian agency for scientific research) shows that before the Industrial Revolution the δ-value was -6.5‰. In today's atmosphere, the ^{13}C:^{12}C ratios give a δ-value of -8.5‰.

The table below shows the δ-values for various times along with the corresponding isotopic ratios, expressed both as decimals and as percentages of ^{13}C and ^{12}C in the atmospheric samples. You can see why scientists use δ-values rather than the actual ^{13}C:^{12}C ratio numbers, which only show changes far to the right of the decimal points! These ratios change by very small amounts over time, but they clearly illustrate big changes in the atmosphere's composition of ^{13}C and ^{12}C, pointing to the fossil fuel origins of more and more of the atmosphere's CO2.

Here is one final comparison to help make the δ-values more understandable. Annual global average temperatures are usually presented as *anomalies* with reference to some "base period". Figure 4 is a familiar graph of this from NASA-GISS with data from 1880 to 2016. The "base period" for this graph is 1951-1980^{2}, the annual temperatures for these thirty years are averaged together and this is set as the zero point of the anomaly scale. So, the base period is like the standard sample used as the zero point in the δ-notation scale. Then, each individual year's data point is compared to that zero point. If a year's temperature was warmer than the base period, then the anomaly for that year is positive, such as 2016's record high anomaly of 0.99°C. This is comparable to positive δ-values. Years with colder values than the base period would be negative, like 1904's record low value of -0.5°C. This is similar to the negative δ-values described above.

In both instances, δ-values and temperature anomalies, cumbersome numbers are converted into more meaningful and useful values. In the case of δ-values, very very small changes in isotopic ratios in the natural environment are more easily described, and we can see more clearly how the Earth's climate system works and changes over time.

1. "The original PDB sample was a sample of fossilized shells of an extinct organism called a belemnite (something like a shelled squid) collected decades ago from the banks of the Pee Dee River in South Carolina. The original sample was used up long ago, but other reference standards were calibrated to that original sample. We still report carbon isotope values relative to PDB but now use the term "VPDB" ["Vienna Pee Dee Belemnite"] to indicate that the data are normalized to the values of that standard." (USGS).

2. Any time period, and any length of time, may be used. A thirty year period is often used because thirty years is a long enough time to describe "average" climate variables.

Rob Honeycuttat 23:30 PM on 18 October, 2023Rabelt... You really should go read the comments policy. Moderation complaints are also against the rules here.

You've been asked for supporting materials to better explain your arguments here and, thus far, all I've seen is a belligerent inability to comprehend what's being explained to you followed by repeating what you'd previously stated. That doesn't advance the discussion.

Rabeltat 23:31 PM on 18 October, 2023I see no problem with having 1 of the participants being also a moderator capable of modifying and deleting comments, next time I will we more careful and not comment in privileged blogs

[more moderation complaints deleted]

Moderator Response:[BL] Clearly you have no interest in reading or understanding the Comments Policy.

Rob Honeycuttat 23:32 PM on 18 October, 2023My suggestion, Rabelt, is that you find a constructive way to move the conversation forward. And I think that's going to require that you spend some time better understanding the science surrounding carbon isotopes. Here's a good place to start.

Rabeltat 23:34 PM on 18 October, 2023Its been a pleasure partaking in the mainstream debate and seen the consequences, have a nice day and I hope you would be more interested in a free impartial discussion next time Mister Bob

[Moderation complaints deleted]

Moderator Response:[BL] Clearly unable to follow simple instructions.

Rob Honeycuttat 23:35 PM on 18 October, 2023"...next time I will we more careful and not comment in privileged blogs."

Or, perhaps, just read the commenting policies beforehand and play by the house rules. It's not that difficult.

Moderator Response:[BL] As a side note to general viewers, the only comments from Rabelt that are no longer visible are content-free complaints about moderation. Other recent comments that violated the Comments Policy had "warning snips" applied, which strikes them through but leaves them visible to all readers (including casual readers that have not registered or logged on). This is the usual process, in the hope that the offending participant will step back, modify their behaviour, and follow the rules.

Rabelt rapidly devolved into full-blown violations of the Comments Policy. Continuing on this path will lead to his posting privileges being rescinded.