Beam Energy Measurement using a Bayesian Approach with the Stacked Foil Method

TL;DR

This work develops a Bayesian stacked-foil approach to measure proton beam energy at a medical cyclotron without relying on current measurements. By irradiating Ti, Cu, and Nb foils and detecting $^{48}$V and $^{65}$Zn activities via HPGe spectroscopy, the entry energy $E_0$ is inferred through a probabilistic model that couples monitor cross-sections, stopping powers, and foil geometry. The method integrates uncertainties in cross-sections, stopping powers, and thickness as nuisance parameters, yielding robust energy estimates across configurations that degrade energy, and is validated with Monte Carlo simulations. The approach demonstrates accurate, configuration-flexible energy determination (8–19 MeV) in non-ideal setups, offering a practical tool for beam characterization, cross-section studies, and quality assurance in radiopharmaceutical production and radiobiology experiments.

Abstract

We present a practical method to measure the energy of proton beams at a medical cyclotron using the stacked foil technique in combination with a Bayesian inference method. By measuring the $^{48}$V activity induced in a stack of irradiated titanium foils, the proton energy can be inferred without relying on direct current or charge measurements, making the method suitable even for low-vacuum environments or air-exposed setups. This technique is further extended to configurations where the beam energy is degraded to levels around 8 MeV. A Bayesian fit of the measured activity profile allows not only for a robust energy estimation but also for a consistent treatment of uncertainties and nuisance parameters. Monte Carlo simulations are employed to validate the underlying assumptions, including the impact of energy dispersion or cross-section uncertainties. Our results demonstrate that this method provides accurate beam energy measurements across several typical experimental setups used at the Bern Medical Cyclotron. Additionally, we evaluate the sensitivity of the method to the choice of nuclear cross-section data and assess how the number of foils in the stack affects the uncertainty in the inferred beam energy.

Figures (11)

• Figure 1: A sketch of the different positions where the beam energy has been measured for the BTL set-up. The coin holding the foil stack has been placed at the positions indicated in red. At the first indicated position, the pristine beam energy is measured (①BTL Energy). At the second position, the beam energy has been degraded by an aluminium scatterer and an aluminium extraction window (②After Scatterer). At the third position, the beam energy is measured after having additionally passed through air, an ionisation chamber and a cell flask's wall (③Cell level). ① and ② have been performed using the target holder shown in Figure \ref{['fig:target holder']}. In another measurement (④ STS Energy) the beam energy of another beamline outport of the BMC was characterized. On this beamline a commercial Nirta solid target station (STS) is installed in combination with a 50 Ti exit window.
• Figure 2: The target coin has a lid (left) with a circular opening, attached to its backside (right) with small magnets. Between the two a single titanium foil is shown.
• Figure 3: The target station that was used for measurements ① and ②. The incident proton beam (indicated with labelled arrow) hits the target foil stack at the target position after being collimated by the grounded collimator and passing by the electron suppressor ring, which is only needed if one is interested in a target on current measurement. The depicted aluminium attenuators are optional, depending on the desired experimental configuration. The current measurement and bias voltage components have not been used for the measurements in this publication. The picture is adapted with modifications from dellepiane_research_2021.
• Figure 4: The composition of all four foil stacks. ① shows the composition of the 'pristine BTL energy' measurement's foil stack, ② the 'After Scatterer' measurement's composition, ③ the 'Cell Level' measurement's composition, and ④ the 'STS measurement' composition of the foil stack. The direction of the incoming proton beam is indicated by the arrows. The different colours of the layers indicate the material of the layer, where grey is indicating titanium, yellow is for copper and blue is for niobium. In stack ② the two shades of grey indicate active layers (dark) and beam degrading layers (pale).
• Figure 5: The above plot shows the IAEA-recommended cross-sections (hermanne_reference_2018) for the $^\text{nat}\text{Ti}(\text{p,x})^{48}\text{V }$ reaction in blue and the $^\text{nat}\text{Cu}(\text{p,x})^{65}\text{Zn }$ reaction in yellow. The bottom plot shows the expected foil energies for each measurement’s foil stack. These energies are calculated from the best fit initial energies listed in Table \ref{['tab:energy results']}.