# “Nuclear disintegrations” vs. “spallation”

Here I reproduce another email exchange about some very obscure details of the exposure age calculator for those who may care about this issue. If you have no idea what I am talking about, you are in the large majority of calculator users who are not affected by this issue.

**Email received by Balco:**

Hi Greg,

I wonder if you could explain to me how you have calculated the production rate by high-energy nucleogenic spallation in the Stone scheme (P_St = 4.49).There is a difference between “nuclear disintegrations” in the Stone/Lal and Dunai schemes and “nucleogenic spallations” in the other production schemes. I checked your code, made some small calculations with your production rates, and saw that the fast muon induced contribution is potentially added twice in the Stone/Lal scheme. Could you please check my considerations below and give me your opinion?

First, I always assumed that you have applied the same approach to all scaling schemes:

Get P_total_local

Subtract P_smu_local

Subtract P_fmu_local

Scale the rest to SLHL

Then the values in Table 6 of your “calculator” article (or Table 1 in update 2.2) would be nucleogenic spallations only. However, my calculations give me the impression that P_St consists of the nucleogenic spallations plus fast muon interactions (so, +0.1 atom/g/yr).

The calculations referred to here are in this PDF.

**Balco’s response:**

Yes, you are right, there is an inconsistency there. One problem I ran into in putting this together is that in addition to the surface production rates, the calculator needs to calculate the depth dependence, *P(z)*, for all the different production pathways. If the Lal polynomials describing “nuclear disintegrations” include both spallation and fast muon interactions in an unknown proportion, then there is no way to know how this combined production rate decreases with depth. It is not clear: i) if all fast muon interactions are included as part of “nuclear disintegrations,” or ii) what proportion of “nuclear disintegrations” represent fast muons as opposed to spallation. I suppose this is similar to asking if fast muon interactions produce disintegration stars in film, which is another question that I don’t know the answer to.

Therefore, it seemed that the most sensible thing to do was to pretend that the Lal polynomials describe spallation only, have this “spallation” production rate decrease with depth according to a spallation length scale, and then to have the fast muon production be separate so that it can have the correct depth dependence. So, yes, in a way fast muons are included twice in this part of the code. However, they are not “added twice” in the sense that the production rate is systematically too high, because the fast muon production rate computed according to Heisinger is removed before determining the best-fitting “spallation” production rate, if that makes any sense. Instead, the effect of this simplification is that the altitude dependence of the production rate calculated with the St and Lm scaling schemes is slightly weaker than Lal may have originally intended. Whether this is more or less correct, or good or bad for calculator users, is not clear. First of all, remember that the original Lal scaling scheme gave much more importance to negative muon capture than we now think is correct, so the (incorrect) Balco implementation of Lal actually has a *stronger* altitude dependence of the total production rate than Lal originally intended. Also, for example, the St scaling scheme, even including the inconsistency we are talking about here, fits the 2008 calibration data set better than the other scaling schemes that are more “correct.” Of course, this could be because the scaling scheme and the calibration data are both wrong in offsetting ways, but there is no way to determine this.

So the summary is that I am not sure what the best thing to do here is — it seemed like the only other option was to guess how much of “nuclear disintegrations” is accounted for by fast muons, and I didn’t want to guess.

One other thing is that in the calculations you sent, you noted that the total high latitude, 1013.25 hPa production rate for the Stone scaling scheme comes out higher than the value of 5.06 used in Stone (2000). Those values are not comparable — the calibration data set used in the 2008 paper is similar but not exactly the same as the data set used in the 2000 paper, and the methods of averaging are also different. So you should not expect those two values to agree.

**Response to response:**

Hi Greg,

Thank you for this explanation.

I see that my initial understanding of your procedure was correct:

Get P_total_local

Subtract P_smu_local (Heisinger)

Subtract P_fmu_local (Heisinger)

Scale the rest to SLHL

The average of these “rests” gives you the SLHL production by the nucleogenic spallation.

There is a potential danger here that a simple user of the production rates may mix your values and the procedures from the Stone/Lal papers, because the names of scalings are still the same. I can imagine that someone takes Stone PR from your update 2.2 (“Aha, good, the production rate corrected to the new standards!”) and puts it into his old Excel file for the Stone2000 scaling (“Like in old good days! I still understand how I calculate!”). Do you think it would help if you advise people to use Heisinger with the Stone production rate from your update? Or rename the scaling into Balco-Stone.

**Balco’s response to response to response:**

* I see that my initial understanding of your procedure was correct:
Get P_total_local
Subtract P_smu_local (Heisinger)
Subtract P_fmu_local (Heisinger)
Scale the rest to SLHL
The average of these “rests” gives you the SLHL production by the nucleogenic spallation*

Yes, (if you make sure to include the other stuff like thickness and geometric scaling) that is what I did to get the production rates in the 2008 paper. Now I have changed things a little bit — I do the calibration with a minimization method where I consider the calculator as a function with one input parameter (the SLHL production rate for “spallation”) and one output parameter (some sort of misfit statistic, like MSWD or chi-squared, that compares the true ages of calibration sites with the ages predicted by the calculator). Then for each scaling scheme, I choose the value of the SLHL production rate that minimizes the misfit between predicted and actual ages. This method allows me to only have one piece of code — the “forward” code that calculates ages from Be-10 concentrations — rather than also requiring a piece of “backward” code that computes production rates from calibration sites of known age. This reduces the potential for errors.

As you may have noticed, I am working on a web page that actually does this — you input a calibration data set, it finds the best-fitting production rates, and then lets you compute ages that are consistent with the calibration data set. It is on the ‘developmental’ page.

*There is a potential danger here that a simple user of the production rates may mix your values and the procedures from the Stone/Lal papers, because the names of scalings are still the same. I can imagine that someone takes Stone PR from your update 2.2 (“Aha, good, the production rate corrected to the new standards!”) and puts it into his old Excel file for the Stone2000 scaling (“Like in old good days! I still understand how I calculate!”). Do you think it would help if you advise people to use Heisinger with the Stone production rate from your update? Or rename the scaling into Balco-Stone.*

Well, there are many dangers. The world is a dangerous place. Seriously, you are right, people do get confused between the as-published scaling scheme, that includes negative muon production as well, and the online calculator that only uses the “spallation” polynomials. However, the important thing — and one of the main purposes of the calculator — is that if you think an author has made this mistake, you can enter his measurements into the online calculator and check. In addition, you may notice that I try very hard not to talk about the “production rate” used to compute exposure ages. Instead I talk about the “calibration data set and scaling scheme.” Obviously it is important to focus people’s attention on the thing that is actually measured (Be-10 concentrations at calibration sites) and not the scaling-scheme-dependent parameter that no one can ever measure directly (the “SLHL production rate.”).

enjoy,

–greg

I thought “nuclear disintegration” was the name of a band. Yes. I’m pretty sure it’s the name of a band.