Low energy elastic scattering of H, D and T on $^{3}$He and $^{4}$He

Motivated by the needs of atomic tritium sources for neutrino mass experiments, we present calculations of energy-dependent elastic scattering cross sections of hydrogen isotopes (H, D and T) on helium isotopes ($^3$He and $^4$He) in the temperature range 1mK to 300K. The tritium-on-helium cross sections are found to be enhanced over their hydrogen-on-helium counterparts by a near-threshold resonant s-wave bound state at low energy, similar to that predicted in the triplet T-T system. While the energy-dependent cross sections span a wide range at low energy due to this s-wave enhancement, they tend toward a common value at high energy where the scattering becomes effectively geometric in nature.

💡 Research Summary

The paper presents a comprehensive theoretical study of low‑energy elastic scattering between the hydrogen isotopes (H, D, T) and the helium isotopes (³He, ⁴He), motivated by the requirements of atomic tritium sources for next‑generation neutrino‑mass experiments such as Project 8, KATRIN++ and QTNM. The authors calculate energy‑dependent elastic cross sections over a temperature range from 1 mK to 300 K, a regime relevant for cryogenic buffer‑gas cooling, supersonic expansion sources, and the continuous production of ³He from tritium β‑decay within the source.

Because neither helium isotope carries electronic spin, and nuclear‑spin interactions are negligible, only elastic scattering needs to be considered. The scattering problem is reduced to a one‑dimensional radial Schrödinger equation for the nuclear motion under the Born‑Oppenheimer approximation. The key input is the interatomic potential V(R), taken from the high‑quality Meyer‑Frömmhold ab‑initio potential. To improve agreement with experimental diffusion data, the authors also employ a modified version of this potential (mod‑MF) that introduces a steeper short‑range repulsive core, following Chung and Dalgarno. Both potentials retain the long‑range van‑der‑Waals −C₆/R⁶ attraction.

Partial‑wave analysis yields phase shifts δₗ(E) for each angular momentum ℓ. The total elastic cross section is σ(E)=4πk⁻²∑ₗ(2ℓ+1)sin²δₗ(E). At ultralow energies the s‑wave (ℓ=0) dominates, giving σ₀=4πa_s² where the scattering length a_s=−lim_{k→0}tanδ₀(k)/k. By solving the Schrödinger equation for a range of reduced masses µ, the authors map a_s(µ) (Fig. 2). They find that when µ≈3.9 µ_H‑H a near‑threshold bound state appears, causing a dramatic increase of a_s for the heavier isotopic combinations. This is analogous to the well‑known near‑threshold resonance in triplet T‑T scattering, but occurs at a slightly larger reduced mass because of differences in the potential shape.

The calculated s‑wave scattering lengths (Table I) show that T‑³He and T‑⁴He have a_s values of several thousand picometres, leading to low‑energy cross sections that are 10⁴ times larger than those for H‑³He or H‑⁴He. Energy‑dependent cross sections (Fig. 4) reveal that at temperatures below ≈10 mK the resonant s‑wave enhancement produces a wide spread of values, whereas at higher temperatures (tens of kelvin) the contributions from higher partial waves dominate and the cross sections converge to a common “black‑disc” limit σ≈2πr²≈2.2×10⁻¹⁴ cm², where r is the sum of the van‑der‑Waals radii of H (100 pm) and He (140 pm). The modified potential yields slightly higher cross sections at low energies, especially for the lightest pair H‑³He, but the differences become negligible for the heavier T‑He systems.

From an experimental perspective, the enhanced T‑He cross sections affect several aspects of tritium source design: (1) the continuous presence of ³He in the source leads to non‑negligible T‑³He scattering, influencing vapor dynamics and loss rates; (2) cryogenic buffer‑gas cooling using He will be less efficient for tritium than for hydrogen because of the larger cross section; (3) supersonic expansion beams rely on energy transfer from H or T to He, and the ratio of H‑He to T‑He cross sections determines how well hydrogen can serve as a proxy for tritium in test experiments; (4) recent proposals for cryogenic dissociation sources have used He‑He cross sections as placeholders for T‑He, an approximation now shown to be inadequate.

The authors provide the full set of energy‑dependent cross sections as supplementary tables and release an open‑source code to reproduce the calculations, facilitating immediate use by the community. In conclusion, the work delivers the first quantitative predictions of H/D/T‑He elastic scattering across the temperature range relevant to cold‑atom tritium technologies, highlights the crucial role of a near‑threshold s‑wave bound state in amplifying T‑He interactions, and establishes a reliable high‑energy geometric limit. These results are essential for accurate modeling of tritium source performance and for guiding the design of future neutrino‑mass experiments.