Approximate evaluation of the total alpha contamination

by Silvia (February 2025)

Here I try to evaluate of the total alpha contamination of the crystal. Taking into account the decay chains of 232Th, 235U and 238U, I try to estimate the activity of each isotope, in order to reproduce the observed ADC spectrum.

First I start considering the 232Th chain. I assume that all the alpha decays of these chain are in equilibrium, with an activity equal to the value that I have previously found for the 224Ra→220Rn→216Po→212Pb triplet.

For each isotope decay, I consider the Q value, and in the case that only one alpha particle of a fixed energy is emitted, I reproduce the expected ADC distribution by constructing a Gaussian centered on Q and with the FWHM equal to the energy resolution (calculated by Lorenzo). Then I convert the energy to ADC, using the calibration curve found in my previous work.

However, for many isotopes, the decay table shows more than one alpha energy. This means that the daughter nucleus can be in an excited state and then deexcites with the emission of one or more gamma rays with a total energy Eg. In this case, for ANY possible alpha energy (emitted with probability > 0.1), I construct a Gaussian distribution as explained above, using the value Q-Eg for the alpha energy. Then, for each corresponding gamma energy Eg, I construct a second Gaussian distribution using the energy resolution and the calibration curve for gamma rays (found by Yingjie). Then I sum the two obtained Gaussian distributions, for alphas and gammas.

In Fig.1 the total ADC spectrum obtained with all the decays of the 232Th chain is compared with the experimental spectrum.

It can be seen that all the calculated peaks correspond to the observed peaks in the data (except for the 212Po, which, as we know, can only be seen with a dedicated data acquisition with a shorter time window due to its very short lifetime), so in principle all the decays of the 232Th chain could be present.

I do the same procedure for the U235 chain, using the activity found for the alpha triplet 223Ra→219Rn→215Po→210Pb. The results are shown in Fig.2

The figure shows that in principle all decays could be present, but the peak due to 232U + 231Pa is too high, so these decays could be absent or be present with a lower activity.

Fig.3 shows the sum of the previously obtained 232Th and 235U decay chains (red curve). It can be seen that the four peaks in the data, centered at about 33000, 38000, 43000 and 48000 can be completely explained by the decays of these two chains. No other contribution is needed. On the contrary, the first two high peaks at lower energies require additional contributions.

Let's try the 238U chain. Fig.4 shows the ADC obtained assuming an activity of 2 mBq/kg.

In principle the only possible contribution is from the first four isotopes of the chain, i.e. 238U, 234U, 230Th and 226Ra. Finally, I sum the contribution of the 3 chains, adjusting the activity of some of the isotopes in order to fit the data of the first 2 peaks and I obtain Fig.5 (red curve). Tab.1 gives the activity of the isotopes of the three chains that produce the red curve.

According to this approximate model, the first high peak is due to the decay of 238U and 232Th, that are the head of the corresponding chain, while the second high peak is due to 230Th and 234U, both from the 238U chain.

Note that in the 238U chain, there are only 3 isotopes contributing to the total activity, 238U, 234U and 230Th, the ones considered “primordial”, and can have an independent life with respect to the other components of the chain. Their activity is found to be about 6.6 Bq/kg.

Taking this value as true, I calculate the expected activity of 235U, knowing that in nature, for a given mass of uranium, the fraction of 238U is f=0.9928 and the fraction of 235U is f=0.0072. The activity of an isotope of atomic weight M and average life Tau is proportional to f/Tau/M. Thus, assuming an activity of 6.6 Bq/kg for 238U, the activity of 235U is 21.4 times smaller, i.e. 0.31 Bq/kg. This value is the one shown in green in Table 1 for 235U and its daughter 231Pa.

Also 232Th is primordial (the only primordial isotope of its chain) and indeed I found that it has a different activity from its daughters.

The activity values shown in red in the table were obtained empirically, without a precise fit. Since the “theoretical” curve is made by the sum of a large number of Gaussians (one for each alpha and gamma energy), I did not obtain a reliable fit by using Root.

I did not obtain a perfect agreement of the data with the “theoretical” curve, because in my simple calculations I did not take into account the possible loss of energy by the gammas escaping from the crystal. A more accurate calculation should be done with a Monte Carlo.