The Mystery of the Delta Value – Why ‰ isn’t Parts Per Thousand

I’m about to gripe. But it’s a science gripe. It’s a technical gripe. It’s about stable isotopes. If you aren’t interested in stable isotopes, I suggest you tune out now.

There’s this thing in stable isotopes, especially the so-called ‘light’ isotopes like hydrogen, oxygen, carbon, and nitrogen. It’s called the ‘delta’ value. When we say what the stable isotopic measurement of something is, we say ‘the delta value is blah-blah-blah.’ The value is always in the units of permil (‰).

“δ¹³C of warm-season grasses is -14‰.”

δ¹³C is our symbol for the delta value (in this case, for carbon). We also talk about δ¹⁸O, δ¹⁵N, δ²H (or sometimes δD). These delta values give us information about the amounts of different isotopes in a substance. δ¹³C compares 13-C with 12-C. δ18O compares 16-O with 18-O. It’s a ratio, really, or it wants to be. All of these are numbers followed by the permil symbol, ‰.

Here’s the confusing thing. In the ‘regular’ world permil, or ‰, is parts per thousand. It’s a measure of concentration. How much of a substance is made up of a particular thing. So 75‰ is 75 parts per thousand, just like 75% is 75 parts per hundred. And since we’re talking about the relative amounts of different isotopes in a substance, it would follow that the permil symbol on a delta value would mean parts per thousand.

It doesn’t

Look again at that example above.

“δ¹³C of warm-season grasses is -14‰”

Is it possible to have a negative concentration? No. Absolutely not. Concentrations must be positive numbers, or zero. But delta values are frequently negative. And some are very negative.

Delta values are not concentrations. Thus, to say “δ¹³C of warm-season grasses is -14 parts per thousand” is WRONG. What we say is “δ¹³C of warm-season grasses is -14 permil.”

But why?

As I mentioned before, the delta value is basically a ratio. It is a comparison between a substance of known isotopic composition and the substance for which we want to know the composition. The delta value is calculated this way:

[(R_sample-R_standard)/(R_standard)]X1000=δ_{sample-standard}

Terms:

R is the ratio between the heavy isotope and the light isotope. This number is usually very, very small, because heavy isotopes are rare. This is a ratio, so there’s no units.

R_standard is the ratio measured from a standard – something for which we know the relative amounts of the heavy and light isotopes already

R_sample is the ratio measured from our ‘unknown’ – the sample we want to know the delta value for.

The delta value is the difference between R_standard and R_sample

[(R_sample-R_standard)/(R_standard)]X1000=δ_{sample-standard}

Notice that these are all ratios. There are no units. It is not a concentration.

The ‰ part comes in when we multiply by 1000. We multiply by 1000 because the number we get is really very, very small. By multiplying by 1000, it makes it easier for us to talk about the results. We add the permil symbol to remind ourselves that we’ve multiplied by 1000.

“1.34 permil” is easier to say than “0.00134.”

It’s not a concentration. It’s a convenience.

What a delta value tells you is how much more or less the unknown sample has of the heavy isotope than the standard. If it has relatively more of the heavy isotope, the delta values are positive, and the more positive (larger) the number, the more heavy isotope there is! In this case, the unknown is said to be ‘enriched’ in the heavy isotope. If the unknown has less of the heavy isotope than the standard, then the delta value is negative, and the unknown is called ‘depleted’ in the heavy isotope.

My gripe is this: Every time I see someone write “parts per thousand” in a professional manuscript, I scream. I mean, I really do. I have to get up and go for a walk. What that tells me is that the author doesn’t understand one of the most basic fundamentals of isotope geochemistry. When I’m at professional meetings and speakers say “parts per thousand,” I groan audibly. After that, they’ve got their work cut out for them to convince me that they really know what they’re doing. One time, I heard a speaker say “parts per million,” and I just about left. It’s easy to understand the mistake in thinking, but it’s hard to understand how someone can get a Ph.D. in isotope science and not understand this.

Grumble.