Based on questions I’m asked and evidence of confusion observed in papers and in conversations, one of the most confusing aspects of Be-10 measurements and their interpretation is the relationship between the 10/9 ratio in a Be AMS measurement standard and the half-life of Be-10. AMS measurements of the amount of Be-10 in a sample are always normalized to a particular AMS standard, so when these results are reported someplace they are described as being, for example, “normalized to the NIST SRM 4325 standard material with a Be-10/Be-9 ratio of 2.68e-11.” The part that causes confusion is that they could alternatively be described as being “normalized to the NIST SRM 4325 standard material with a Be-10 half-life of 1.34 Myr.” It turns out that these two statements are equivalent, but why they are is confusing. There are two main issues here that don’t make a lot of sense to many people: first, it’s not obvious why the 10/9 ratio in the standard is related to the half-life of Be-10, and  second, it’s not obvious what the relationship is between the half-life that is used to describe normalization of AMS measurements and the half-life of Be-10 that one will later use to compute an exposure age or an erosion rate from the measurement.

So I’m going to make one more attempt to clear this up, from the beginning. First, there will be a long section on basic concepts, then I will try to answer the two questions above.

Long section on basic concepts.

Definition of half-life and decay constant.  The rate at which Be-10 decays can be described by either a decay constant or a half-life, which are related as follows:

$t_{1/2,10} = \frac{-\ln\left( 1/2 \right)}{\lambda_{10}}$                   (1)

where $t_{1/2,10}$ is the Be-10 half-life (1.39 x 10^6 yr) and $\lambda_{10}$ is the Be-10 decay constant (5.00 x 10^-7 /yr).

How does one measure the decay constant? One actually measures the decay constant by obtaining a large quantity of Be-10 and using a beta counter to measure the number of decays per time interval. The decay rate, also called activity, is related to the amount of Be-10 and the decay constant by:

$A = N_{10}\lambda_{10}$                        (2)

where, to keep the units consistent, $N_{10}$ is the number of atoms of Be-10 and A is the activity in decays per year.

It will therefore be obvious that to determine the decay constant, one must know how many atoms of Be-10 one has. Usually, this is done by obtaining a sample of beryllium that has been neutron-irradiated so that unnaturally large amounts of Be-10 are present. One uses any standard chemical analytical method to determine the total amount of Be that is present, and a mass-spectrometric measurement of some sort to determine the Be-10/Be-9 ratio. So the sequence of events to determine the Be-10 half-life is as follows: determine how much Be you have, determine its 10/9 ratio, use this information to compute the amount of Be-10 present ($N_{10}$ above), measure its activity by decay counting (A above), and then apply equation (2) above to compute the decay constant/half-life. All the steps of this process are fairly easy except for determining the 10/9 ratio — making absolute isotope ratio measurements with a mass spectrometer at high accuracy is very difficult, because different isotopes of the same element are usually transported through the machine and detected with different efficiencies. So this is the hard part of the measurement.

How does one actually make AMS measurements of the amount of Be-10 in a sample? In analyzing a geological sample that we want to exposure-date using Be-10, we add a measured quantity of Be-9 (the “carrier”), dissolve the sample, extract the now-mixed carrier Be-9 and sample Be-10, then measure the Be-10/Be-9 ratio of the Be that we extracted. Because we know how much Be-9 we added (and because the sample contributes a negligible amount of Be-9), we can multiply this amount (in number of atoms) by the measured ratio to get the number of atoms of Be-10 that were present in the sample.

What is an “AMS standard” and why do you need it? As mentioned above, it is very difficult to measure the absolute isotope ratio of anything with any mass spectrometer, and AMS technology is no different — it is probably worse, in fact, because beams of Be-9 and Be-10 are not even detected in the same type of detector on an AMS. Thus, what we actually do in an AMS measurement is compare the apparent Be-10/Be-9 ratio of a sample with the apparent Be-10/Be-9 ratio of a standard material for which we already know the actual isotope ratio. I will call a material like this an “AMS standard.” Let’s say we have a Be AMS standard that is known to have an absolute isotope ratio of Be-10/Be-9 = 1 x 10^-13. We alternate measurements of this and a sample, and we find that the apparent Be-10/Be-9 ratio of the sample is twice that of the standard. We can use these two pieces of information to determine that the absolute isotope ratio of the sample is 2 x 10^-13. The point of all this is that all Be-10 measurements depend on an AMS standard whose true Be-10/Be-9 ratio is already known. You can’t analyze samples unless you have an AMS standard.

How do  we know the absolute 10/9 ratio of the AMS standard? The obvious question, then, is how we determine the absolute 10/9 ratio of the standard. Well, making an absolute isotope ratio measurement by mass spectrometry is really hard, but beta counting is easy, and we already know the half-life for Be-10, right? So if we obtain a sample of Be-10-enriched Be, we can determine how much Be-10 is present by measuring the activity by beta counting, and applying Equation (2) to determine the number of atoms of Be-10 that are present. We can then dilute our sample with Be-9 to make an AMS standard with any 10/9 ratio that we want.

This describes how most AMS standards are prepared. The actual measurements involved in determining the 10/9 ratio of the AMS standard are i) how much Be is present, and ii) what Be-10 decay rate is observed. One then must assume a value for the Be-10 half-life to come up with an estimate of the absolute 10/9 ratio.  If one assumes a different half-life, then one comes up with a different ratio. Thus, defining an absolute 10/9 ratio for an AMS standard that will then be used as a basis for measuring Be-10 concentrations in samples is the same as choosing a value for the Be-10 half-life. The actual measurements that were made on the standard material — the Be concentration and the Be-10 activity — don’t change. However, if one updates, or revises, the Be-10 half-life, this implies a corresponding update or revision of the absolute isotope ratio of an AMS standard prepared in this way.

Summary so far: the absolute 10/9 ratio of the AMS standard is what we really need to know to make the measurements of Be-10 concentration in samples that we want to make. However, in most cases, the absolute 10/9 ratio of the AMS standard involves an activity measurement, which can only be interpreted as a 10/9 ratio if you know the Be-10 half-life. So for AMS standards that are based on activity measurements, the half-life of Be-10 and the absolute isotope ratio of the standard are linked — if you know one, you know the other, and if you change one, you have to change the other.

OK, now to an example. One commonly used AMS standard was prepared by Kuni Nishiizumi by diluting a sample of Be-10-enriched Be called the “ICN standard.” This dilution is called the “01-5-4” dilution. When used as an AMS standard this material is usually called “KNSTD3110.”
It has a measured Be-10 activity of 97.3 decays/yr per mg Be.  This activity measurement is related to the Be-10 half-life and to the absolute 10/9 ratio of the Be in the solution by:

$R = \frac{A}{\lambda_{10} N_{9Be}}$

where R is the 10/9 ratio, A is the number of Be-10 decays per year per mg Be, and $N_{9Be}$ is the number of atoms of Be-9 per mg Be (6.68 x 10^19). If we assume that the Be-10 half-life is 1.5 Myr (i.e. $\lambda_{10}$ is 4.62e-7), then the 10/9 ratio of this material is 3.153 x 10^-12, and when comparing samples to this standard during AMS measurements we would compute the 10/9 ratio of samples based on this value. For example, if we observed that the apparent 10/9 ratio during AMS measurement of a sample was 0.1 times that of this standard, we would conclude that that 10/9 ratio in our sample was 3.153 x 10^12 x 0.1 = 3.15 x 10^-13.

What if you’re not sure what the Be-10 half-life is? The fact that the half-life of Be-10 is linked to the absolute isotope ratio inferred from activity measurements on AMS standards has been important because, until relatively recently, there was some doubt about what the half-life of Be-10 actually was. Another measurement of the half-life had concluded that it was actually not 1.5 Myr but 1.34 Myr (i.e.,  $\lambda_{10}$ is 5.17e-7). If we assume this half-life when computing the 10/9 ratio of the KNSTD3110 standard from the measured activity, we would conclude that the absolute 10/9 ratio of this standard was 2.81 x 10^-12. Then given the same AMS measurements on our sample, we would conclude that the 10/9 ratio in our sample was 2.81 x 10^-13, or 11% lower than we concluded before.

Confusion about the Be-10 half-life = confusion about how much Be-10 is in your sample. As shown in the example above, because the absolute isotope ratio of an AMS standard is computed from an activity measurement and an assumed half-life for Be-10, what half-life you choose affects the absolute isotope ratio of the standard and thus the absolute isotope ratio of your sample. Doubt about the Be-10 half-life implies equal doubt about how many atoms of Be-10 are actually in a sample.

Attempt to summarize and clear up the main areas of confusion.

What is the relationship between the half-life of Be-10 and the Be-10 concentration in my sample, i.e. how can my measurements be described as being normalized to a particular value of the half-life?

This should be clearer now. Really, your measurements are normalized to an AMS standard whose absolute 10/9 ratio is known. But that “absolute ratio” is (in most cases…see more below) not directly measured.  The Be-10 activity is what is actually measured, and that measurement plus the value of the Be-10 half-life is used to compute the absolute 10/9 ratio. So if you change the half-life, you change the absolute ratio for the standard (which doesn’t sound so “absolute” now…this is part of the confusion).

This is why stating that a measurement is “normalized to the NIST SRM 4325 standard material with a Be-10/Be-9 ratio of 2.68e-11.” is the same as saying that it is “normalized to the NIST SRM 4325 standard material with a Be-10 half-life of 1.34 Myr.” A critically important point here, however, is that there IS NO WAY for a user to know that these two statements are equivalent, unless he or she carries out additional research into what the actual measured Be-10 activity in the standard is. In my view, the first statement is much more clear and much more meaningful for the user: if you know the sample-standard relationship measured by AMS, you can use the information in the first statement to compute how much Be-10 is in your sample. You can’t use the second statement to do this unless you carry out additional research.

What is the relationship between the Be-10 half-life used to compute the absolute 10/9 ratio in a standard and the Be-10 half-life that I will use to compute exposure ages and erosion rates?

Well, they should be the same. Whatever you think the Be-10 half-life is, it can’t be two different things at the same time, so if you use one value of the half-life to compute the number of atoms of Be-1o present in your sample, you must use the same value of the half-life to compute its exposure age or erosion rate. Extensive discussion of this is beyond the scope of the present post, but if you use inconsistent values in these two parts of an exposure age calculation, you create systematic errors that are often quite important. Thus, if you are working from an AMS standard whose absolute isotope ratio is defined by reference to a particular value of the Be-10 half-life, you must use the same value of the half-life in computing exposure ages or erosion rates.

This statement is pretty simple, but there are two important aspects to it that are sometimes confusing. First, to apply this statement you need to determine whether or not your AMS standard is defined in relation to an activity measurement and a half-life, and if so, determine what this half-life is. Second, even if you correctly apply this statement and use the same half-life to define your AMS standard and to interpret your measured Be-10 concentrations, if this value of the half-life is incorrect, you will obtain the wrong answer.

There is one other very important point here: if you are using an AMS standard whose absolute 10/9 ratio does NOT involve any assumptions about the half-life, then you can choose any value of the half-life you think is most accurate to carry out further calculations. I will cover this issue in the next post.

Summary remarks

This post attempts to explain why the half-life of Be-10 is related to the number of atoms of Be-10 that you think are present in your sample based on an AMS measurement. This issue is confusing. In a subsequent post I will try to explain how AMS specialists and users of Be-10 measurements can best clear up confusion and make this issue easier to deal with. The main advance that makes this possible is a recent paper by Kuni Nishiizumi and others in which they created a series of AMS standards whose absolute isotope ratios are NOT determined by reference to any particular value of the Be-10 half-life. This advance separates the question of “how many atoms of Be-10 are in my sample” from that of “what is the half-life of Be-10,” which in turn i) makes things a lot simpler, and ii) makes most geological applications of Be-10 measurements more accurate.