# What is the 26/10 production ratio anyway?

This rather long post covers the issue of the Al-26/Be-10 production ratio. The value of this ratio is important for several reasons.

First, it gives some information about whether a surface sample has experienced a single period of exposure at the surface. The basic, although slightly oversimplified, concept here is that if you measure a ratio that is a lot lower than the production ratio, your sample has spent a significant fraction of its life buried far enough below the surface to partially or completely stop the cosmic ray flux and allow previously produced nuclide inventories to decay…so if you compute the exposure age of the sample based on the assumption that it hasn’t been buried, you’ll be wrong.

Second, and more importantly, if you have a sample that is buried below the surface now, and you measure this ratio, the difference between the measured ratio and the production ratio tells you how long the sample has been buried. This concept, although also slightly oversimplified here, is the foundation of the method of burial dating.

The currently accepted value of the 26/10 production ratio, when appropriately normalized to the “07KNSTD” and “KNSTD” isotope ratio standardizations for Be-10 and Al-26, respectively, is 6.75. Interestingly, an uncertainty is almost never quoted for or applied to this number. Also interestingly, as far as I can tell, this number is just taken from a paper written in 1989 by Kuni Nishiizumi and colleagues. They stated it as 6.1; restandardizing Be-10 measurements from the then-current “KNSTD” standardization to the now-current “07KNSTD” corrects it to 6.75. In the widely utilized 2008 paper describing the online exposure age calculators formerly known as the CRONUS-Earth online exposure age calculators, we just accepted this number, mainly because we concluded that production rate calibration data that were available at the time agreed with the Nishiizumi estimate, and we didn’t propose any change. Subsequently, most papers about burial dating (which is where this number is really important) adopted this value or something very close to it (exactly how you do the KNSTD–>07KNSTD conversion affects the actual number at rounding error level). To summarize, we have all basically ignored this issue for 25 years. Why? Because 6.75 works. Most measurements on surface samples seem to more or less agree with this canonical value, and we have not ever found that burial ages computed using a ratio of 6.75 have been inconsistent with other age constraints.

Unfortunately, it may now be time to revisit this issue, because in recent years there has been (i) a lot more collection of production rate calibration data, as well as (ii) a lot more attention to first-principles modeling of production processes and their spatial variation. So in the rest of this post I’ll try to exploit this new information to figure out what the 26/10 production ratio really is, and, more importantly, to determine if we can just keep on not worrying about this issue as we have done successfully for the last 25 years.

To clarify what exactly we are talking about, in this post I am talking about the ratio of total Al-26 and Be-10 production rates at the surface, that is, including both spallation and muon production. In some work (including lots of mine) there is some confusion about whether an “Al-26/Be-10 production ratio” refers to spallogenic production only, or to total production. Fortunately, most of the time these are very close to the same thing, so any consequences of this confusion are small. But here, I am talking about total production.

The most straightforward way to determine the production ratio is simply to do what Nishiizumi and others did. Find some samples that have the following properties: they have been exposed at the surface for a single period of time; at the beginning of this period their nuclide concentrations were zero; surface erosion during the exposure period was negligible; and the duration of exposure is much less than the half-life of either Al-26 or Be-10. Then measure the concentration of both nuclides, and take the ratio. In the Nishiizumi paper these are samples of glacially polished bedrock from the California Sierra Nevada that have been exposed since deglaciation ca. 13,000 years ago.

The easiest way to find a larger set of samples that have these properties is, in general, to look at samples collected for production rate calibration purposes. Selecting samples suitable for measuring the 26/10 production ratio, of course, is less restrictive than selecting ones suitable for production rate calibration, because to measure the production ratio, you don’t have to know the exact duration of exposure, just that the exposure time has been a lot shorter than the Al-26 half-life. So lots of other samples that aren’t suitable for production rate calibration could be used to estimate the 26/10 ratio. However, existing production rate calibration data sets are a convenient way to find a set of samples that have been carefully selected to have simple exposure histories and negligible erosion. Looking in the ICE-D production rate calibration database and removing one extreme outlier measurement (26/10 = 14; not sure what went wrong here), one finds 90 measurements of both Al-26 and Be-10 on samples collected for calibration purposes. Each of these is a distinct measurement in which both nuclides were measured in a particular aliquot of quartz. Here is the distribution of 26/10 ratios in these samples:

On the left this is shown as a histogram and on the right as a normal kernel density estimate. A couple of points about this:

The average of all these data (shown as a red line on both plots) is 6.76. OK, we’re done. Back to sleep for another 25 years.

Just kidding. We’re not quite done. One initial thing that is notable about this data set is that it is skewed to the low side. This observation is consistent with the easiest way to make an error when measuring the 26/10 ratio in a sample. Specifically, one measures the Al-26 concentration by multiplying a measured total amount of Al by a measured Al-26/Al-27 ratio. The total Al measurement is made by taking aliquots of a solution of quartz dissolved in HF, evaporating the HF in the aliquots, redissolving the Al in a dilute acid, and measuring the Al concentration by ICP. If you are not careful in this process, you can retain insoluble Al fluorides in the final step, which means that the Al concentration in solution underestimates the total amount of Al in the sample, and leads to an underestimate of the 26/10 ratio. The mild negative skewness of these data suggests that this may have been a problem for some of the measurements, in which case the mean value of these data might slightly underestimate the true production ratio (the median value is 6.84).

For a slightly different perspective on this, then, let’s try estimating the production ratio from calibration data using a slightly different workflow, as follows:

- Start with the “CRONUS-Earth primary calibration data sets” for Be-10 and Al-26, again as represented in the ICE-D calibration database.
- Using these data and the production rate calibration code from the version 3 online exposure age calculator, determine best-fitting values of reference production rates for Be-10 and Al-26. I’ll use two scaling methods: the non-time-dependent “St” scaling and the time-dependent “LSDn” scaling.
- Also compute production rates due to muons for Al-26 and Be-10; add production rates by spallation and muons together; divide to get an estimate of the 26/10 production ratio.

The main difference between this method and simply averaging all direct measurements of the 26/10 ratio as I did above is just that it allows us to incorporate data from calibration samples where only one nuclide was measured: we can utilize Be-10 data that lack corresponding Al-26 measurements, and vice versa. In addition, all measurements represented in the CRONUS-Earth “primary” data set are relatively recent, meaning that the folks who did them certainly should have been careful about insoluble fluorides. In any case, this workflow uses somewhat different data and should yield an estimate of the 26/10 production ratio that is somewhat independent from the simple average above.

On the other hand, it brings up something new. The “St” scaling method, as implemented in the various online exposure-age calculators, assumes that Al-26 and Be-10 have the same latitude/elevation scaling, but the proportion of total production that is due to muons will vary with elevation and latitude (mostly with elevation). In addition, the relative proportions of fast muon production and negative muon capture change with elevation. As these three production pathways (spallation, fast muons, negative muon capture) all have different 26/10 production ratios, this means that the total 26/10 production ratio must vary with elevation (and a little with latitude as well). The point is that what we know about production rate scaling predicts that the 26/10 production ratio isn’t a constant value as we have been assuming, but will vary with elevation.

This variation is predicted to be close to negligible for the “St” scaling method: muon production makes up at most 2 or 3 percent of total production, so variations in the smallness of this very small fraction result in only per-mil-level variations in the production ratio. Thus, for this scaling method, mostly we disregard this issue completely. The “LSDn” scaling method, in contrast, predicts not only variation in the muon contribution, but also, more importantly, that spallogenic production of Al-26 and Be-10 will scale differently with magnetic field position and elevation. The reason for this is that this scaling method uses the results of an atmospheric particle transport model to predict the energy spectrum of fast neutrons at your location, and then multiplies this spectrum by a set of cross-sections for Al-26 or Be-10 production. These cross-sections depend on energy, and they’re different for the two nuclides. The following shows the neutron cross-section estimates used in LSDn scaling, as a function of neutron energy:

Green is the neutron interaction cross-section for production of Al-26 from Si; blue is Be-10 from O; red is Be-10 from Si. The threshold energy for Al-26 production is lower than that for Be-10, so in parts of the atmosphere where neutron energy is lower (that is, at greater atmospheric depths or lower elevations), Al-26 production is favored relative to Be-10 production, and the 26/10 production ratio is predicted to be higher. So once we know that Al-26 and Be-10 have different threshold energies for production (we’ve known this for a while), this implies that the spallogenic production ratio has to vary with location; it can’t not do so. Similar work using an atmospheric particle transport model by Dave Argento (here) also highlighted the prediction that the 26/10 production ratio should change with elevation; he calculated that the spallogenic production ratio should be near 7 at sea level and 6.75 at high altitude.

The main point here is that first-principles physics and calculations based thereon predict that the 26/10 production ratio should vary with location. So let’s look at the data set of measured ratios (the data presented in histogram form above) as a function of elevation, to see if this variation is evident. Here are measured ratios plotted against elevation:

Here are the same data with (i) the average (6.76), and (ii) a linear regression in elevation (6.94 – 1.022e-4 * elevation) plotted in red:

Although the correlation between elevation and measured 26/10 is weak (correlation coefficient is -0.25; p = 0.02), it’s not zero. And a simple linear regression through these data agrees very closely with the prediction in the Argento paper.

This shows that the data are consistent with an elevation dependence to the production ratio, even though they don’t require it at very high confidence. But to back up a bit, this whole discussion of elevation dependence interrupted what we were trying to do: estimate the 26/10 production ratio by separately calibrating the Al-26 and Be-10 production rates. Now having shown that that no matter how we do this, we always predict some geographic variation in the ratio, we’ll continue to look at this as a function of elevation. Here is the result for the ‘St’ scaling method:

Although it is a bit hard to tell this from the figure, there are separate green and blue dashed lines showing predictions for low latitude (high magnetic cutoff rigidity) and high latitude (low cutoff rigidity), respectively. As discussed above, although geographic variation is predicted for this scaling method, it’s very small. Reference production rates fit to the calibration data predict a 26/10 production ratio of 7, which is, in fact, a bit higher than the estimate we obtained above from simply averaging the direct measurements. Again, what seems to be the most likely explanation for this is just that there are more data involved, and we may have excluded some of the low outliers by choosing a more recently generated data set.

Here is the result for the “LSDn” scaling method (again, this is total production including muons):

As discussed, this scaling method predicts a larger elevation dependence as well as a larger difference between high (green) and low (blue) cutoff rigidity. Again, it’s in decent agreement with the simple linear regression on these data as well as the predictions in the Argento paper (the latter isn’t surprising; they’re based on similar physics). To summarize, put all these predictions together with the data:

My summary of this is as follows. The data set of direct ratio measurements on individual samples is, by itself, consistent with the canonical value of 6.75, although there is some evidence that this number might be skewed a bit low due to measurement errors. Basic physics of Al-26 and Be-10 production, on the other hand, indicates that the production ratio should be elevation-dependent. Scaling models that use this physics (e.g., LSDn), when fitted to calibration data, predict that the ratio should be in the range 7-7.1 at low elevation and in the range 6.5-6.75 at high elevation; these predictions are also completely consistent with the data set of direct measurements. Because the data set is rather scattered, mainly just due to measurement uncertainty, it does not appear to provide a means of choosing between scaling models at high confidence. So at this point, it appears that (i) simply assuming 6.75 always, and (ii) using an elevation-dependent value as predicted by LSDn scaling, are both consistent with available data. Choose (ii) if you like physics, and (i) if you like simplicity. Overall, I think I lean towards (ii), that is, physics.

For perspective, the difference between 6.75 and 7 is 3.7%. That’s not insignificant — it maps to age differences of 5-15% in typical burial-dating applications — but we’re haggling over fairly small differences here.

Finally, for completeness I’m going to cover one more source of indirect evidence for what the value of the production ratio is. This stems from the observation that if the production ratio at low elevation is 7 rather than 6.75, we should probably recalculate a bunch of previously published burial ages. Thus, it might be a good idea to look into whether changing the production ratio causes any serious problems with existing burial-dating results. One could pursue this either by recalculating a bunch of burial ages and seeing if you find any conflicts with independent age constraints, or by a more explicit method in which if (i) you have a buried sample, (ii) you know its age independently, and (iii) you know the decay constants for Al-26 and Be-10, you can measure both nuclides in your sample and back-calculate the surface production ratio. This latter method features in a recent paper by Zhou and others, who burial-dated a fluvial gravel whose age was bracketed independently by argon-dated lava flows. They back-calculated the production ratio in this way to be between 6 and 7.2 (at 68% confidence), which is not particularly precise, but shows good agreement with predictions and surface measurements. Good. Pass. On the other hand, in a 2010 paper about glacial stratigraphy in the central US, we dated a paleomagnetically normal unit three times by the burial-isochron method (this is the “Fulton” till in that paper). Magnetically normal means it must be younger than the Brunhes-Matuyama reversal at 0.78 Ma; we dated it at 0.8 +/- 0.05 Ma, which is just barely consistent with the constraint. Assuming the 26/10 ratio to be 7 rather than 6.75 would make that age estimate older, 0.87 Ma, which would be badly inconsistent with the paleomagnetic stratigraphy. But — hold the presses — in the time period between that paper and now there’s also been a change in best estimates for muon interaction cross-sections, which acts to bring that age estimate back down to 0.81 (still +/- 0.05) Ma, and this remains consistent with the paleomagnetic boundary. So this highlights that there are a lot of moving parts involved in recalculating old burial age data, but for this one case I can think of immediately in which a burial age estimate bumps up against an an independent age closely enough to provide a serious constraint on the 26/10 production ratio, the hypothesis that the production ratio is near 7 at sea level passes.

So what ratio to use for burial dating? As noted above, physical arguments appear rather compellingly in favor of an elevation dependence for the ratio. If this is true, then if the average ratio computed from a set of samples that span a range of elevations is 6.75, the true production ratio must be higher at low elevation (conveniently, Nishiizumi and others made their measurement at middle elevations). I think it seems sensible to include this effect in burial-age calculations.

P.S.: I still haven’t addressed the issue of what the uncertainty in the production ratio is.

Beel et al, 2016 (QSR) report mostly samples that were intermittently covered by ice in western Greenland, but the few samples consistent with more or less continuous exposure also report 26/10 ratios close to or greater than seven, consistent with your synopsis. We also observed this to a large degree in the CRONUS New Hampshire calibration samples. Both of these sites would fall in the category of a more low energy dominated spectrum.

Very interesting but It is very important to make clear what we are talking about: the spallation production ratio or the total (muons + neutrons). This latter can take a value up to 12 at depth (4-6 m). Such ratios were related in the past to bad 27Al measurements but now with the better knowledge of muon contributions such high ratios can be found, especially in glacial environments where glacier abrasion is important.

This is true, but the total surface production ratio is the important parameter for burial dating, which is really the main point here. For Al-26 and Be-10, obviously, muons are a very small contribution at the surface.

Was not the statistical analysis you performed on the databases the opportunity to determine the uncertainties associated to the deduced 26/10 ratios ?

This issue is fairly complicated. If we accept that the ratio varies with elevation, then we cannot make an uncertainty estimate directly from a population of measurements that come from different elevations. Instead we have to compare the measurements to a scaling model that predicts a geographic variation in the ratio. We can do this in two ways.

First, we could calibrate Be-10 and Al-26 production rates separately, determine the uncertainties in each calibrated production rate, and then propagate these uncertainties into the ratio. Unfortunately, as pointed out very clearly in the Borchers paper about production rate calibration, the scaling models do not fit the calibration data at acceptable confidence, so it is not possible to make a justifiable uncertainty estimate for the production rate estimates, which in turn means that this approach would not yield a justifiable uncertainty estimate for the ratio either.

Alternatively, we could use the LSDn scaling model to predict the 26/10 ratio for each of the samples where the ratios were measured, and then compare that to the measured ratios. This comparison yields a reduced chi-squared value of 2.8 for, I think, 86 DOF, which has effectively zero probability-of-fit. Thus we have the same problem as in the Borchers paper, which is not surprising because it is basically the same calculation, only using slightly different data.

The standard deviation of the measured ratios with respect to the LSDn predicted ratios is 9% (which is basically the same as the SD of the ratios considered by themselves at 9.5%). If we argue that there is still some unquantified analytical error in the measurements, then 9% is probably a useful upper bound for the uncertainty in the ratio. However, 9% uncertainty means that the ratio is only constrained to be between 6.4 and 7.6 at 68% confidence, and, as discussed briefly above, that large an uncertainty range is incompatible with many burial-dating results. I don’t think anyone thinks the ratio is 6.4 or 7.6.

Basically, the problem is that if we are making Be-10 measurements at 3% and Al-26 measurements at 6% precision, the standard deviation of measured 26/10 ratios from simply exposed samples should be only 6.5%. The fact that it is 9% indicates that there is some unquantified analytical uncertainty present, which makes it hard to estimate the true uncertainty in the ratio.

If we just ignore the reported measurement uncertainties and the question of whether or not the measurements are consistent with the predicted ratios at measurement uncertainty, then we could compute the standard error of the mean (9% / sqrt(90)) to be 1%. So that is probably a good lower bound on the uncertainty in the ratio. To summarize, I think it’s clear that the uncertainty in the ratio is less than 9% and more than 1%, but it is not clear (at least to me) how to make a better uncertainty estimate. Unfortunately, this is not super helpful.

I think the most sensible strategy to improve this situation would be to make (or compile existing) measurements on higher-elevation and/or older samples where there is more Al-26 to measure. Most of the measurements in the existing calibration data set are from low-elevation sites with late-glacial ages where Al-26 concentrations are relatively low. Alternatively, we could try to estimate the ratio from near-saturated surfaces in low-erosion-rate environments such at Antarctica, where it is possible to make very precise measurements. A handful of such measurements have been done. That might be a good subject for a future blog post.