# Optimizing $^{211}$At production cross section by studying the rise of $^{210}$At cross section: First measurement using Linac SPIRAL2

S Ansari-Chauveau, A-M Frelin, G de France, A Guertin, F Haddad, X Ledoux, J Mrázek, E Šimečková

► To cite this version:

S Ansari-Chauveau, A-M Frelin, G de France, A Guertin, F Haddad, et al.. Optimizing $^{211}$At production cross section by studying the rise of $^{210}$At cross section: First measurement using Linac SPIRAL2. Applied Radiation and Isotopes, 2025, 225, pp.112061. 10.1016/j.apradiso.2025.112061. hal-05212726

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# Optimizing <sup>211</sup>At production cross section by studying the rise of <sup>210</sup>At cross section: First measurement using Linac SPIRAL2

S. Ansari-Chauveau<sup>a, ID,*</sup>, A-M. Frelin<sup>a ID</sup>, G. de France<sup>a</sup>, A. Guertin<sup>b</sup>, F. Haddad<sup>b,c</sup>, X. Ledoux<sup>a</sup>, J. Mrázek<sup>d</sup>, E. Šimečková<sup>d</sup>

## ARTICLE INFO

**Keywords:**

a-particle induced reactions on <sup>209</sup>Bi

Radionuclides: <sup>210-211</sup>At

Production cross sections

γ-ray spectroscopy

## ABSTRACT

Targeted Alpha Therapy (TAT) offers a promising approach to treat cancer, particularly micrometastases, by utilizing the short range and high linear energy transfer of alpha particles emitted by radionuclides. <sup>211</sup>At (half-life 7.2h) is one of the promising alpha emitters (only one alpha emitted during decay) that has been identified for nuclear medicine applications. It belongs to the halogen family and shares chemical properties with iodine, an element used for imaging (<sup>123</sup>I, <sup>124</sup>I and <sup>131</sup>I) and also widely used to treat thyroid cancer (<sup>131</sup>I). This chemical similarity enables the use of Iodine as an analogue for biodistribution and dosimetry studies while using <sup>211</sup>At for treatment in a theranostic approach.

In this study, an alpha beam accelerated by SPIRAL2, was used to produce <sup>211</sup>At via the reaction <sup>209</sup>Bi(<i>α</i>, 2n)<sup>211</sup>At on the NFS beam line. The production cross section of <sup>211</sup>At increases with increasing alpha energy up to 31 MeV. However, above 28.6 MeV, the production of <sup>210</sup>At occurs via the <sup>209</sup>Bi(<i>α</i>, 3n)<sup>210</sup>At reaction. <sup>210</sup>At decays to <sup>210</sup>Po, a highly toxic alpha-emitting radionuclide with a half-life of 138.3 days which cannot be separated chemically. Therefore, it is crucial to have a thorough understanding of the rise in <sup>210</sup>At production to optimize the generation of <sup>211</sup>At while minimizing the production of <sup>210</sup>At

To achieve this, <sup>209</sup>Bi targets were irradiated at various alpha beam energies between 28 to 31 MeV with high precision thanks to the characteristics of SPIRAL2 accelerator and <sup>210,211</sup>At cross sections were measured by using γ-ray spectroscopy. The incident particle flux was monitored using an instrumented Faraday cup. This flux measurement combined with the number of detected γ-rays allowed to determine the production cross sections of <sup>210,211</sup>At as a function of energy. The results are in good agreement with experimental values recommended by the International Atomic Energy Agency (IAEA) for <sup>211</sup>At and provide supplemental data for <sup>210</sup>At between 28.6 and 31 MeV. The data collected in this study will help optimize the energy range of interest for the production of <sup>211</sup>At and give <sup>211</sup>At its rightful place as a radionuclide for TAT.

## 1. Introduction

Targeted Alpha Therapy (TAT) has garnered significant attention in recent years due to the promising efficacy of alpha particles in eradicating tumor cells. This is largely attributed to their short penetration depth and associated high linear energy transfer (LET). Alpha emitters can be attached to molecules or compounds that specifically target cancer cells or their close environment. These targeting agents are designed to bind to proteins or antigens present on the surface of cancer cells, facilitating the direct delivery of alpha particles to the malignant tissue.

TAT is particularly effective against small tumors due to its high energy deposition, which ranges from 4 to 9 MeV for 1 alpha decay. The high linear energy transfer of alpha particles (approximately 100 keV/μm) results in significant biological damage, including the DNA double-strand breaks. Furthermore, hypoxic cancer cells such as in glioblastoma which is an aggressive form of malignant primary brain cancer, exhibit increased sensitivity to the destructive effects of high LET, enhancing the therapeutic efficacy of TAT in these environments (Sabri et al., 2024).

With a half-life of 7.2 h, <sup>211</sup>At is sufficiently stable to facilitate preparation and transport to cancer treatment facilities. It decays into

<footnote><sup>a</sup> GANIL, Grand Accélérateur National d'Ions Lourds (GANIL), CEA/DSM-CNRS/IN2P3, Boulevard H. Becquerel, Caen 14076, France</footnote>

<footnote><sup>b</sup> SUBATECH, Nantes Université, IMT Atlantique, CNRS/IN2P3, 3 Rue Alfred Kastler, Nantes 44300, France</footnote>

<footnote><sup>c</sup> GIP ARRONAX, 1 rue Aronnax, Saint-Herblain 44817, France</footnote>

<footnote><sup>d</sup> Nuclear Physics Institute CAS, Řež 25068, Czech Republic</footnote>

<footnote>* Corresponding author.
E-mail address: ansari-chauveau@ganil.fr (S. Ansari-Chauveau).</footnote>
<figure><img src="image_4.png" /><figcaption>**Fig. 1.** $^{211}$At decays to $^{207}$Bi through direct alpha decay and to $^{211}$Po via electron capture, with both nuclei ultimately decaying into stable $^{207}$Pb, values taken from ENSDF.</figcaption></figure>

$^{211}$Po through electron capture with a decay intensity of 58.2% and to $^{207}$Bi via alpha decay with a decay intensity of 41.8%, releasing 5.9 MeV alpha particles, both ultimately decaying into stable $^{207}$Pb (see Fig. 1). Importantly, both $^{211}$Po and $^{207}$Bi do not pose significant radiation concerns, as discussed in Feng and Zalutsky (2021). Each decay chain of $^{211}$At results in the emission of a single alpha particle, simplifying radiation dosimetry calculations for $^{211}$At-labeled TAT agents (Feng and Zalutsky, 2021). Additionally, $^{211}$At holds potential for imaging applications (Johnson et al., 1995), as its decay produces X-rays in the range of 77–102 keV, which can help evaluate the biokinetics of $^{211}$At in patients. The isotope $^{211}$At shows significant promise in TAT due to the cytotoxic properties of alpha particles, making it an ideal candidate for radiotherapeutic applications. However, depending on the energy of the alpha beam used to produce $^{211}$At through reaction $^{209}$Bi($\alpha$, 2n)$^{211}$At, $^{210}$At can also be produced via the reaction $^{209}$Bi($\alpha$, 3n)$^{210}$At. This may be problematic because $^{210}$At, which has a half-life of 8.1 h, decays with a 99.825% probability to $^{210}$Po, a highly radiotoxic isotope (Fellman et al., 1994). Therefore, optimizing the alpha beam energy is crucial to balance the high production of $^{211}$At and at the same time limit that of $^{210}$At.

Cross-section studies have been conducted numerous times in the past. Fig. 2 illustrates the cross sections of $^{210}$At and $^{211}$At, provided by the JANIS (Java-Based Nuclear Information Software) program (Soppera et al., 2014; Otuka et al., 2014), plotted against the incident alpha energy, revealing a distinct overlap between the two cross sections. However, earlier experiments primarily utilized cyclotrons, which have significant uncertainties in beam energy. As a result, they are not well-suited for cross-section studies at threshold energies or for examining cross-sections at small energy intervals. To investigate cross sections near threshold energies with cyclotron-generated beams, past experiments employed the stacked foil method, which involves assembling a series of thin foils to create a target. Although this method enables the measurement of the entire excitation function using fewer irradiation energies and experiments, it has notable drawbacks. The primary issue is the inaccuracy in calculating irradiation energy, as the beam passes through multiple thin foils, the thickness of which is challenging to measure accurately. This leads to energy straggling and increased energy uncertainties when an already high-uncertainty beam

<figure><img src="image_5.png" /><figcaption>**Fig. 2.** Production cross section of $^{210,211}$At as a function of the beam energy from the previous measurements (Tárkányi et al., 2022; Kelly and Segrè, 1949; Ramler et al., 1959; Deconninck and Longrée, 1974; Lambrecht and Mirzadeh, 1985; Hermanne et al., 2005; Stickler and Hofstetter, 1974; Patel et al., 1999; Rizvi et al., 1990; Singh et al., 1994).</figcaption></figure>

<table><caption>**Table 1**

Important characteristics related to the production of different At isotopes.</caption><thead><tr><th>Isotope</th><th>Reaction channel</th><th>Energy threshold [MeV]</th><th>T<sub>1/2</sub></th></tr></thead><tbody><tr><td><sup>212</sup>At</td><td><sup>209</sup>Bi(<i>α</i>,n)</td><td>15.6</td><td>0.314 s</td></tr><tr><td><sup>211</sup>At</td><td><sup>209</sup>Bi(<i>α</i>,2n)</td><td>20.7</td><td>7.214 h</td></tr><tr><td><sup>210</sup>At</td><td><sup>209</sup>Bi(<i>α</i>,3n)</td><td>28.4</td><td>8.1 h</td></tr><tr><td><sup>209</sup>At</td><td><sup>209</sup>Bi(<i>α</i>,4n)</td><td>35.9</td><td>5.41 h</td></tr></tbody></table>

of cyclotron passes through the target material and a beam degrader material (Ziegler and Biersack, 1985). In this study, however, due to the precise energy capabilities of SPIRAL2, a single target can be utilized. This article focuses on cross section measurements of $^{211}$At and $^{210}$At produced through the reactions $^{209}$Bi($\alpha$, 2n)$^{211}$At and $^{209}$Bi($\alpha$, 3n)$^{210}$At, employing mono-energetic alpha beams from SPIRAL2 LINAC at various energies. The nuclear data on γ-ray energies, branching ratio and half lives are taken from ENSDF.

In this study, the irradiation beam energy range for the Bi target was set between 28.6 and 31 MeV, as the energy threshold for $^{210}$At production occurs at 28.4 MeV, while $^{211}$At reaches maximum cross section value around 31 MeV. Beyond $^{210,211}$At, other atstatine isotopes may be produced depending on the beam energy (see Table 1).

## 2. Materials and methods

The standard equation for calculating the production cross section of an isotope is given by (1) and (2)

$$ \sigma = \frac{Act}{\phi \cdot \chi \cdot (1 - e^{-\lambda t_{irr}})} \cdot \frac{A}{N_A \cdot M_s} \quad (1) $$

where,

$$ Act = \frac{\lambda}{(e^{-\lambda t_1} - e^{-\lambda t_2})} \times \frac{M}{r \times \epsilon} \quad (2) $$

Each term in the above two equations are defined as follows:

* Act: Activity at the end of the irradiation

* λ: Radioactive decay constant of the isotope [s⁻¹]

* t₁, t₂: Time between end of irradiation and acquisition start (t₁) and stop (t₂) [s]

* M: Number of detected γ-ray at a given alpha energy

* r: Intensity of the measured γ-ray

* ε: Detection efficiency for the corresponding γ-ray energy

* σ: Production cross section [b]

* φ: Average incident particle flux over the irradiation time [s⁻¹]
* χ: Chemical purity of the target

* t<sub>irr</sub>: Irradiation duration [s]

* A: Target atomic mass [g mol<sup>-1</sup>]

* N<sub>A</sub>: Avogadro constant [mol<sup>-1</sup>]

* M<sub>s</sub>: Target surface mass [g b<sup>-1</sup>]

From the expression of sigma in Eq. (1), its uncertainty is composed of three components, as given in Eq. (3)

$$ \left(\frac{\delta\sigma}{\sigma}\right)^2 = \left(\frac{\delta Act}{Act}\right)^2 + \left(\frac{\delta\phi}{\phi}\right)^2 + \left(\frac{\delta M_s}{M_s}\right)^2 \quad (3) $$

The error on the activity is calculated using standard error propagation formula using Eq. (4) and the activity calculation given in Eq. (2)

$$ \delta Act^2 = f(\delta\lambda, \delta M, \delta r, \delta\epsilon) \quad (4a) $$

$$ \delta Act^2 = \underbrace{\left(\frac{\partial Act}{\partial \lambda}\right)^2}_{\text{1stterm}} \delta\lambda^2 + \underbrace{\left(\frac{\partial Act}{\partial M}\right)^2}_{\text{2ndterm}} \delta M^2 + \underbrace{\left(\frac{\partial Act}{\partial r}\right)^2}_{\text{3rdterm}} \delta r^2 + \underbrace{\left(\frac{\partial Act}{\partial \epsilon}\right)^2}_{\text{4thterm}} \delta\epsilon^2 \quad (4b) $$

$$ \delta Act^2 = \underbrace{\frac{Act^2}{\lambda^2} \left( 1 - \lambda \frac{-t_1 e^{-\lambda t_1} + t_2 e^{-\lambda t_2}}{e^{-\lambda t_1} - e^{-\lambda t_2}} \right)^2}_{\text{1stterm}} \delta\lambda^2 + \left(\frac{Act}{M}\right)^2 \delta M^2 + \left(\frac{Act}{r}\right)^2 \delta r^2 + \left(\frac{Act}{\epsilon}\right)^2 \delta\epsilon^2 \quad (4c) $$

where, 1st term is related to the radioactive decay constant of the isotope, 2nd term is related to the number of counts in the peak, 3rd term is related to the intensity of the measured γ-ray and 4th term is related to the detection efficiency for the corresponding energy.

## 2.1. Experimental setup

There are several accelerators in the world that are capable of generating alpha particle beams for producing astatine isotopes (Feng and Zalutsky, 2021). Nevertheless, few are capable of delivering variable energies, which are better suited to cross-section studies. One such accelerator is SPIRAL2 at Ganil, France. It is a superconducting linear accelerator renowned for delivering particle beams of exceptionally high intensity. This experiment took place within the Neutrons For Science (NFS) facility of SPIRAL2 (Ledoux et al., 2021) that contains the ion beam line, a converter for neutron production, a Time-of-Flight (TOF) area, and an Irradiation Chamber (IC) for studying ion-induced reactions. This IC, developed by the Nuclear Physics Institute (NPI) of the Czech Academy of Sciences (CAS) (Mrazek et al., 2020), is coupled to a Pneumatic Transfer System (PTS) that transports the samples to/from a storage in front of a High Purity Germanium (HPGe) detector placed in the TOF area in less than 1 min.

The beam is accelerated by 26 superconducting cavities whose phase and voltage are adjusted to achieve the desired beam energy with high precision. A first complete tuning based on an acceleration scheme simulation allows to reach the planned energy with an error under 1%. Then, a time-of-flight measurement at the end of the linac allows to reach the wanted energy with an uncertainty smaller than 1/1000 by adjusting the voltage level of the last accelerating cavity (Le Coz et al., 2014; Jamet et al., 2016, 2019).

The incident particle flux was monitored using a Faraday cup, with the current recorded every second. The beam current ranged

between approximately 110 and 140 nA for all irradiations. To validate the reliability of this measurement system, four copper targets were irradiated individually at beam energies of 28, 29, 30 and 31 MeV. This allowed for a comparison between flux measurements obtained using the Faraday cup system and those reported by the IAEA for the reactions natCu(α, x)<sup>67</sup>Ga and natCu(α, x)<sup>65</sup>Zn (Hermanne et al., 2018).

Copper targets (purity 99.9%, thickness 25 µm) have been obtained from Goodfellow whereas Bismuth targets with a diameter of 1.2 cm were fabricated through evaporation under vacuum at GIP Arronax. High-purity bismuth pellets, from Sigma-Aldrich, purity >99.9%, thickness 100 µm, were placed in a boat positioned in front of an aluminum support and were precisely weighted. The distance between the boat and the aluminum foil was set at 13 cm to ensure a uniform deposit. Once the system reached a vacuum of 5.0E-5 mbar, a current was circulated through the boat to heat the bismuth, facilitating its evaporation. After all the bismuth had evaporated from the boat, the device was halted, and the target was weighed to determine the bismuth mean thickness. Depending on the initial mass of bismuth (Bi) in the boat, this thickness ranged from 7 to 12 µm and led to energy losses of about 900 keV in the target. Thus, this variation is much more significant than the uncertainty in the beam energy. Each bismuth target was irradiated only once at a specific energy. The surface mass uncertainty has been estimated to 4% from the mass measurement and the surface measurement uncertainties.

The gamma-ray spectroscopy setup consisted of two EXOGAM clovers (Azaiez et al., 2000), which were used remotely to detect gamma rays emitted during the de-excitation process of the irradiated Bi target, as depicted in Fig. 3. EXOGAM is a highly efficient germanium gamma-ray spectrometer used in nuclear physics, particularly at GANIL, to study rare phenomena. Thanks to its high efficiency, EXOGAM system was adapted for this study, in which low cross sections, close to the production threshold, were measured for <sup>210</sup>At and the branching ratio of gamma rays emitted by <sup>211</sup>At is very low. Each EXOGAM clover comprises 4 detectors, hence for the experiment, 2 clovers with a total of 8 detectors were used. Due to the short half-lives of <sup>210</sup>At and <sup>211</sup>At, as well as the expected low counting rate, one target was placed in front of each clover simultaneously to maximize the acquisition time for each target. Remote and offline acquisition also provided a significant gain in statistics. Calibration source runs were interspersed between data runs.

Irradiation and spectrometry acquisition main characteristics are summarized in Table 2.

Fig. 4 illustrates the γ-ray spectrum obtained from EXOGAM. Most of the peaks in the spectrum are due to background noise (e.g., concrete), as the detection station was not shielded. Among them, the figure features the peaks of interest from both <sup>210,211</sup>At. These energy peaks along with their branching ratio are summarized in Table 3.

## 2.2. Data analysis

### 2.2.1. Energy calibration

Energy calibration is an essential step in the γ-ray spectroscopy data analysis which allows to match the centroid of the electronic signal to the actual energy of the γ-ray decay. In this case, each EXOGAM consisting of 4 segmented detectors, the energy calibration was carried out independently for each detector within the two clovers using <sup>60</sup>Co, <sup>133</sup>Ba, <sup>22</sup>Na and <sup>137</sup>Cs sources. In Fig. 5, the fit between energy and channel number for one of the detector, demonstrates linearity, as for all the detectors. This process was repeated for each detector to establish energy calibration coefficients.
<table><caption>**Table 2**

Irradiation, cooling (before measurement with EXOGAM) and acquisition durations for different irradiated Bi targets.</caption><thead><tr><th>Alpha energy (MeV)</th><th>Target index</th><th>Irradiation time</th><th>Cooling time</th><th>Acquisition duration</th></tr></thead><tbody><tr><td rowspan="2">28</td><td>1</td><td>1 h</td><td>1.68 h</td><td>14.60 h</td></tr><tr><td>2</td><td>1 h</td><td>4.25 h<br>6.77 h</td><td>2.47 h<br>14.60 h</td></tr><tr><td rowspan="4">28.6</td><td></td><td></td><td>4.58 h<br>6.78 h</td><td>2.16 h<br>1.96 h</td></tr><tr><td></td><td></td><td>8.78 h<br>1.45 d</td><td>17.25 h<br>13.74 h</td></tr><tr><td>3</td><td>5 h</td><td>2.45 d</td><td>14.49 h</td></tr><tr><td>4</td><td>40 m</td><td>3.38 h<br>6.83 h</td><td>5.11 h<br>5.11 h</td></tr><tr><td>29.5</td><td>5</td><td>35 m</td><td>14.87 h</td><td>2.47 h</td></tr><tr><td>30</td><td>6</td><td>30 m</td><td>1.63 d</td><td>1.96 h</td></tr><tr><td>30.5</td><td>7</td><td>20 m</td><td>10.85 h</td><td>2.16 h</td></tr><tr><td>31</td><td>8</td><td>20 m</td><td></td><td></td></tr></tbody></table>

<figure><img src="image_6.png" /><figcaption>**Fig. 3.** (a) Schematic of the Exogam setup: A support system was used to bring the target/source (T/S) closer to the clover. The T/S remained fixed in position for all runs. (b) The segmented Germanium clover, Exogam.</figcaption></figure>

<table><caption>**Table 3**

Energies and branching ratios of γ-ray emissions for the At isotopes from ENSDF, from which the cross section is measured (Singh et al., 2013, Shamsuzzoha Basunia, 2014).</caption><thead><tr><th>Nuclei</th><th>Energy [keV]</th><th>Branching ratio</th></tr></thead><tbody><tr><td><sup>210</sup>At</td><td>245</td><td>0.79(4)</td></tr><tr><td><sup>210</sup>At</td><td>1181</td><td>0.99(4)</td></tr><tr><td><sup>210</sup>At</td><td>1436</td><td>0.290(15)</td></tr><tr><td><sup>210</sup>At</td><td>1483</td><td>0.465(23)</td></tr><tr><td><sup>211</sup>At</td><td>687</td><td>0.00261(12)</td></tr></tbody></table>

### 2.2.2. Efficiency calibrations

Efficiency calibration was also performed using <sup>60</sup>Co, <sup>133</sup>Ba, <sup>22</sup>Na and <sup>137</sup>Cs sources which had known activities. Intensity for each energy peak was taken from ENSDF (Singh et al., 2013; Kondev, 2008). Efficiency was measured using the following equation:

$$ \epsilon = \frac{M}{r \cdot A_{cl} \cdot t} \qquad (5) $$

where $\epsilon$ is the detection efficiency, M is the number of counts in the peak, r is the intensity of the peak, $A_{cl}$ is the activity of the source in [Bq] and t is the acquisition time in [s].

The efficiency curves for two of the detector are illustrated in Fig. 6. The function used to fit the efficiency curves is given by the following

<figure><img src="image_7.png" /><figcaption>**Fig. 4.** Energy spectrum from one of the EXOGAM clover. The energies of the peaks of interest are written on the spectrum.</figcaption></figure>

<figure><img src="image_8.png" /><figcaption>**Fig. 5.** Linear fit between energy and channel.</figcaption></figure>

equation:

$$ \epsilon(E) = \frac{1}{E} \cdot \left[ P0 + P1 \cdot \text{Log}(E) + P2 \cdot \text{Log}(E^2) + P3 \cdot \text{Log}(E^3) + P4 \cdot \text{Log}(E^4) + P5 \cdot \text{Log}(E^7) \right] \qquad (6) $$

where, $\epsilon(E)$ is the fitting function, P0–P5 are the fit parameters and E is the energy in keV of γ-ray peaks. The efficiency of the peaks of interest of <sup>210</sup>At and <sup>211</sup>At are then interpolated from the curves in Fig. 6 for each detector.
<figure><img src="image_9.png" /><figcaption>Fig. 6. The efficiency curve of two of the detectors. (a) Efficiency Curve, detector-4, (b) Efficiency Curve, detector-8.</figcaption></figure>

In addition to the detection efficiency of each detector, a cross γ-ray contamination between the two EXOGAM clovers (less than 0.5% of the primary detection) was identified. This detection efficiency was also calibrated and the experimental count rates were corrected for this bias. This bias should be corrected in future experiments by moving both EXOGAM clovers farther away and by using additional shielding.

### 2.2.3. Dead time correction

There was dead time in the system and two methods were evaluated to handle the dead-time correction, with both approaches utilizing a constant background peak at 1462 keV of $^{40}$K. Because the spectrometry runs were long enough to observe changing count rates due to decreasing activity, and consequently an evolution of dead time, the runs were cut into sub-runs. Thus, both dead time correction methods were applied to each sub-run.

The first method employed a generalized approach to address the dead time of a non-paralyzable system (Pritchard et al., 2021), with EXOGAM being close to such a detector system. In this approach, the overall count rate for each sub-run was measured and the dead time ($\tau$) was individually determined for each detector using Eq. (7).

$$ \tau[\text{non-paralyzable system}] = \frac{1}{M_{\text{obs}}} \times \left(1 - \frac{M_{\text{obs}}}{M}\right) \qquad (7) $$

where, $M_{\text{obs}}$ is the total observed count rate in a given data run and M is the total count rate that would be observed without dead-time. M is unknown but $\frac{M_{\text{obs}}}{M}$ can be measured by gating on the 1462 keV line which is a constant background to get the ratio $\frac{M_{\text{obs},1462}}{M_{1462}}$ (equivalent to the ratio $\frac{M_{\text{obs}}}{M}$). $M_{1462}$ is taken from the calibration run for which dead time is considered negligible because of the low activity. Subsequently,

to establish the dead time factor of our system, all the aforementioned $\tau$ values (from Eq. (7)) were averaged. This constant dead time value ($\tau_{\text{average}}$) was then utilized to calculate the correction factor $C_{\text{m1}}$ using Eq. (8).

$$ C_{\text{m1}} = 1 - \tau_{\text{average}} \times M_{\text{obs}} \qquad (8) $$

However, EXOGAM proved to be not an ideal non-paralyzable system. After applying the first method, a perfect radioactive decay curve could not be achieved from the decreasing activity over time. This issue arose for any sub-runs exhibiting high rates associated with a specific target. As a result, a customized approach was developed, which also used the number of counts in the 1462 keV peak for each data sub-run along with a calibration run file where the activity level is sufficiently low to disregard dead time. In this method, the dead time correction $C_{\text{m2}}$ was performed using the following equation:

$$ C_{\text{m2}} = \frac{M_{\text{obs},1462}}{M_{1462}} \qquad (9) $$

where $M_{\text{obs},1462}$ and $M_{1462}$ are the counts per second in the 1462 keV peak of the considered sub-run and the calibration run respectively. This factor was calculated individually for each sub-run and for each detector, contrary to the dead time factor of method 1 which is averaged on all the runs.

Subsequently, the correction was applied to the peaks of interest for each data sub-run of each detector. The measured number of counts in the peak of interest can be divided by either $C_{\text{m1}}$ or $C_{\text{m2}}$ factor to get an estimation of the true number of counts.

A weighted mean or constant fit was then used to determine the activities and the cross section values for each beam energy. The average values and the activity uncertainty were estimated over all the data points through Eqs. (10) and (11) respectively.

$$ \overline{Act}_n = \frac{\sum Act_i / \delta Act_i^2}{\sum 1 / \delta Act_i^2} \quad \text{and} \quad \overline{\sigma}_n = \frac{\sum \sigma_i / \delta \sigma_i^2}{\sum 1 / \delta \sigma_i^2} \qquad (10) $$

$$ \overline{\delta Act}_n = \sqrt{\frac{1}{\sum 1 / \delta Act_i^2}} \qquad (11) $$

where, $\overline{\sigma}_n$ and $\overline{Act}_n$ are the mean cross section and activity, $\overline{\delta Act}_n$ is its uncertainty and $Act_i$ and $\sigma_i$ are the individual measurements. However the goodness of fit test for this mean often yielded a $\chi^2$ value smaller than the number of degrees of freedom $n_{\text{df}}$. This is because several parameters, such as branching ratio and efficiency, were common across several individual points, leading to correlated values making the individual errors larger than the dispersion itself. Ignoring existing positive correlations leads to Eq. (11) underestimating (sometimes significantly) the total cross section uncertainty.

To address this, the approach outlined in Schmelling (1995) was employed to manage correlated data without knowing the details of the covariance matrix. The idea is to chose an arbitrary correlation factor that is used to construct an effective covariance matrix, which allows to derive a $\chi^2$. The correlation factor CF is then refined through an iterative algorithm. This algorithm continuously updates CF until the resulting $\chi^2$ value closely matches the number of degrees of freedom, $n_{\text{df}}$. The final CF is then used to calculate a new estimate for the mean uncertainty.

In some other cases, the $\chi^2$ value was above $n_{\text{df}}$, which usually means that the individual uncertainties have been underestimated. In such a case the uncertainty of the mean activity was estimated by simply scaling Eq. (11) as follows:

$$ \overline{\delta Act}_n = \sqrt{\frac{1}{\sum 1 / \delta Act_i^2} \cdot \sqrt{\chi^2 / n_{\text{df}}}} \qquad (12) $$
<figure><img src="image_10.png" /><figcaption>**Fig. 7.** Comparison of copper cross section (CS) values recommended by IAEA with those measured using EXOGAM for prolonged remote analysis (Exp) for different alpha irradiation energies ($E_{irr}$).</figcaption></figure>

### 2.2.4. Measurement of flux

The flux measurements were conducted using a Faraday cup, where potential sources of error could come from various factors including the Faraday cup structure itself, the collimator, beam intensity fluctuations, and different electronic components. For this reason, the literature values recommended by IAEA in Hermanne et al. (2018) for $^{63}\text{Cu}(\alpha, x)^{67}\text{Ga}$ and $^{63}\text{Cu}(\alpha, x)^{65}\text{Zn}$ reaction were used to validate the flux measured by the Faraday cup.

Cross sections were determined by analyzing the activity measured in the γ-ray peaks at 184.5 keV and 300.2 keV for $^{67}\text{Ga}$ and 1115.5 keV for $^{65}\text{Zn}$. This analysis utilized Eq. (1) and dead time corrections, as detailed in the previous section, were applied to the copper targets analysis.

Fig. 7 illustrates the comparison between the measured values and the ones recommended by IAEA for the three aforementioned reactions. The current experimental results confirm, within the error, the copper cross section values for the $\text{Cu}(\alpha, x)^{67}\text{Ga}$ and $\text{Cu}(\alpha, x)^{65}\text{Zn}$ reactions.

The uncertainty on the flux was determined using a similar approach to the error propagation for cross section, as outlined in Eq. (4), with the flux and cross section parts interchanged. For this purpose, the copper cross section values from literature were used as input, while the remaining parameters such as number of counts and efficiency were used from EXOGAM measurements. The calculated flux uncertainties

<table><caption>**Table 4**
Uncertainty on the flux for the two alpha induced reactions on copper across different irradiation energies.</caption><thead><tr><th>Reaction</th><th>Energy [MeV]</th><th>percentage error on flux</th></tr></thead><tbody><tr><td rowspan="4"><sup>63</sup>Cu(α, x)<sup>67</sup>Ga</td><td>28</td><td>2.5</td></tr><tr><td>29</td><td>4.3</td></tr><tr><td>30</td><td>4.4</td></tr><tr><td>31</td><td>3.2</td></tr><tr><td rowspan="4"><sup>63</sup>Cu(α, x)<sup>65</sup>Zn</td><td>28</td><td>7.5</td></tr><tr><td>29</td><td>4.8</td></tr><tr><td>30</td><td>3.9</td></tr><tr><td>31</td><td>5.3</td></tr></tbody></table>

<table><caption>**Table 5**
Mean energy, activity after irradiation and cross section results of <sup>210,211</sup>At across different alpha irradiation energies between 28–31 MeV.</caption><thead><tr><th>Isotope</th><th>E [MeV]</th><th>Act [Bq]</th><th>CS [mb]</th></tr></thead><tbody><tr><td rowspan="7"><sup>210</sup>At</td><td>27.54(46)</td><td>16.6(5)</td><td>0.017(1)</td></tr><tr><td>28.14(46)</td><td>95.9(50)</td><td>0.023(2)</td></tr><tr><td>28.55(45)</td><td>19.0(8)</td><td>0.142(9)</td></tr><tr><td>29.06(44)</td><td>92.8(10)</td><td>1.70(12)</td></tr><tr><td>29.56(44)</td><td>982(41)</td><td>13.4(13)</td></tr><tr><td>30.06(44)</td><td>4020(307)</td><td>30.8(56)</td></tr><tr><td>30.57(43)</td><td>7483(1282)</td><td>54.6(40)</td></tr><tr><td rowspan="7"><sup>211</sup>At</td><td>27.54(46)</td><td>758 997(39 059)</td><td>669(53)</td></tr><tr><td>28.14(46)</td><td>3 184 252(99 558)</td><td>796(58)</td></tr><tr><td>28.55(45)</td><td>811 604(39 295)</td><td>811(65)</td></tr><tr><td>29.06(44)</td><td>592 711(31 551)</td><td>897(69)</td></tr><tr><td>29.56(44)</td><td>578 417(27 568)</td><td>986(82)</td></tr><tr><td>30.06(44)</td><td>333 066(18 913)</td><td>986(175)</td></tr><tr><td>30.57(43)</td><td>268 150(44 711)</td><td>854(73)</td></tr></tbody></table>

for the two reactions of alpha on Cu are listed in Table 4 at irradiation energies of 28 MeV, 29 MeV, 30 MeV, and 31 MeV. A mean value of, 4.5% was adopted for the flux uncertainty and was used to calculate <sup>210,211</sup>At cross section uncertainties.

## 3. Results and discussion

The analysis for <sup>211</sup>At was performed using the only γ-ray energy peak at 687 keV, while for <sup>210</sup>At, four distinct energy peaks at 245 keV, 1181 keV, 1436 keV, and 1483 keV were utilized, as outlined in Table 3. The measurement was conducted independently for each EXOGAM detector while employing both dead time correction methods.

The cross section results obtained using the two dead time correction methods were consistent with one another, however, only the results from method 2 are presented in this work as it provided smaller uncertainties and more accurate decay curves. As previously discussed, variations in dead time among the different EXOGAM detectors necessitated corrections at the sub-run level and resulted in multiple data points for each cross section measurement. This is illustrated in Fig. 8, which shows the cross section measurements for <sup>210</sup>At at an irradiation energy of 28.14 MeV, each point corresponding to a γ-ray energy peak and a sub-run. A polynomial (0-degree) fit performed on unweighted measurements produced a cross section value of 0.0244 mb, consistent with the weighted average value given by Eq. (10). The corresponding uncertainty given by Eq. (3) is around 7.7%. Fig. 8 also shows that there were fewer measurement points for the 245 keV γ-ray energy peak due to a higher background in the low-energy region (refer to Fig. 4). This background interference made it impractical to fit this peak effectively. There are also larger uncertainties associated with the 1436 keV peak because of its low branching ratio. This approach was applied to all other irradiation energies for both <sup>210</sup>At and <sup>211</sup>At and the results are presented in Table 5.

Figs. 9 and 10 illustrate the cross section results of <sup>211</sup>At and <sup>210</sup>At, respectively, from the present work, which are then compared with previous experiments values within our energy range (Tárkányi et al.,
<figure><img src="image_11.png" /><figcaption>**Fig. 8.** Cross section (CS) measurement at 28.6 MeV of $^{210}$At. Different colored area represents different γ-ray energy peak. Multiple data points for each γ-ray peak comes from several data run for each Exogam detector.</figcaption></figure>

2022; Kelly and Segrè, 1949; Ramler et al., 1959; Deconninck and Longrée, 1974; Lambrecht and Mirzadeh, 1985; Hermanne et al., 2005; Stickler and Hofstetter, 1974; Patel et al., 1999; Rizvi et al., 1990; Singh et al., 1994).

For $^{211}$At, the cross section matches rather well with most of the literature data across all irradiation energies. Despite the slight discrepancy between the current experimental value and the findings of Lambrecht and Mirzadeh (1985) (which could be explained by the different sources for nuclear data used for the cross sections calculations eg: for branching ratio) it remains in close agreement with the values of Hermanne et al. reported in Hermanne et al. (2005, 2018). It can also be noticed that the errors at 30.5 MeV are relatively large. This is primarily due to the substantial statistical error associated with the number of counts for the 687 keV peak. This high error is attributed to the relatively short run duration of approximately 2 h recorded around 39 h after irradiation (see Table 2), which is notably late for $^{211}$At, given its half-life of 7.2 h. Nevertheless, it must be noted that the experimental values are in agreement with the literature values considering the large error bars.

For $^{210}$At, it should be noted that the experimental results show discrepancies with the literature data below 30 MeV. These differences can be largely attributed to the large errors associated with the measured beam energies or even the absence of error estimates for the beam energies in previous studies, which were conducted using a cyclotron with less efficient detection methods. In contrast, this study benefits from the use of the SPIRAL2 linear accelerator combined with the high-efficiency gamma detector EXOGAM, leading to more precise results.

Our $^{210}$At curve not only demonstrates the expected trend but also exhibits a higher precision compared to other datasets and aligns well with the findings of Lambrecht and Mirzadeh (1985). Fig. 10 shows that $^{210}$At production is increasing slowly as a function of incident alpha energy as previously anticipated. This is a promising result for the production of $^{211}$At, enabling us to consider operating around 29 MeV while maintaining a low $^{210}$At/$^{211}$At ratio.

The uncertainties for the cross section values ranged from about 6% to 10% for $^{210}$At and $^{211}$At, except at 30.06 MeV where the uncertainty is higher (about 18%). These weighted uncertainties have several origins, which are detailed in Eqs. (3) and (4). It should be noted that the uncertainty on the count rate is very variable as it depends strongly on the acquisition conditions (cooling time and acquisition time) and the peak fitting when the peak area is very small (in case of small cross sections). However, it is largely reduced by the total acquisition time allowed by the remote γ-ray spectroscopy setup and the

<figure><img src="image_12.png" /><figcaption>**Fig. 9.** Cross section results from $^{211}$At (in black) plotted as a function of the mean alpha energies in the targets and compared with several literature values.</figcaption></figure>

<figure><img src="image_13.png" /><figcaption>**Fig. 10.** Cross section results from $^{210}$At (in black) plotted as a function of mean alpha energies in the targets and compared with several literature values.</figcaption></figure>

weighted calculation. On the contrary, the uncertainty associated with the incident particle flux is determined by the uncertainty of the cross sections used in this work to monitor the flux. Thus, its contribution significantly limits the achievable measurement uncertainty.

## 4. Conclusion and perspectives

This study presents the first cross-section results for $^{210}$At and $^{211}$At measured using SPIRAL2, a linear accelerator that delivers an alpha beam with minimal energy dispersion. Previously, similar measurements have been conducted using cyclotron facilities. This unique characteristic of SPIRAL2, combined with the high detection efficiency of the EXOGAM HPGe detector, enables a precise investigation of the rise in $^{210}$At production near the threshold, a crucial factor for optimizing high-purity $^{211}$At production.

For the first time, this study utilized single targets instead of stacked foil targets, complemented by a pneumatic transport system that allowed for rapid target changes with each energy adjustment. Our results align well with values reported in the literature. The primary objective was to investigate the cross-section of the $^{209}$Bi($\alpha$,3n)$^{210}$At reaction close to the threshold, allowing for a detailed scan of $^{210}$At production in this region. Our results corroborate the literature and compel the IAEA to reconsider their recommended dataset near the threshold. Additionally, the data on $^{211}$At also show good agreement with the literature, taking into account error bars and the evolution of nuclear data over time.
This work also demonstrates the effectiveness of the irradiation
station, combined with the adjustable beam energies of SPIRAL2. This
is promising for conducting further studies, particularly in optimiz-
ing the production of alpha-emitting radioisotopes for radiotherapy
applications and performing homogeneous-target irradiation tests.

CRediT authorship contribution statement

**S. Ansari-Chauveau:** Writing – original draft, Investigation, Formal analysis. **A-M. Frelin:** Writing – original draft, Investigation, Formal analysis. **G. de France:** Writing – review & editing, Supervision, Investigation, Funding acquisition, Conceptualization. **A. Guertin:** Writing – review & editing, Validation, Investigation. **F. Haddad:** Writing – review & editing, Validation, Resources, Investigation, Funding acquisition, Conceptualization. **X. Ledoux:** Writing – review & editing, Validation, Investigation, Conceptualization. **J. Mrázek:** Writing – review & editing, Validation, Investigation. **E. Šimečková:** Writing – review & editing, Investigation.

Declaration of competing interest

The authors declare the following financial interests/personal rela-
tionships which may be considered as potential competing interests: G.
de France reports financial support was provided by French National
Research Agency. J. Mrázek reports financial support was provided by
MEYS, Czech Republic. If there are other authors, they declare that they
have no known competing financial interests or personal relationships
that could have appeared to influence the work reported in this paper.

Acknowledgments

This study was financially supported by the REPARE ANR project (Projet-ANR-19-CE31-0013) and was partly supported by MEYS, Czech Republic, under the project SPIRAL2-CZ. The supporting data for this article are registered (de France, 2025) following the GANIL data policy.

Data availability

Data will be made available on request.

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