The $β$-decay spectrum of Tritiated graphene: combining nuclear quantum mechanics with Density Functional Theory

TL;DR

The paper tackles how a graphene hosting substrate alters the $\beta$-decay spectrum of Tritium, a problem with direct implications for neutrino-mass measurements. It introduces a hybrid, multi-method framework that combines density functional theory to compute Tritium/He interaction potentials with a full quantum treatment of the nuclear decay, exploring sudden, semi-sudden, and adiabatic limits for the final state. Key results include detailed Tritium orthogonal and parallel potentials, multiple Helium final-state schemes, and predicted end-point shifts and bound-state features that distinguish substrate effects from vacuum. The authors discuss limitations and lay out a roadmap for improved non-adiabatic treatments and scalable environmental sampling, highlighting how substrate-induced signatures could be leveraged by PTOLEMY-like experiments to constrain the neutrino mass while guiding future theoretical developments.

Abstract

We present the results of a multi-methodological study aimed at investigating the interaction between graphene and Tritium during its $β$-decay to Helium, under different levels of loading and geometrical configurations. We combine Density Functional Theory (DFT), to evaluate the interaction potentials, with calculations of the decay rate, in order to study the consequences that the presence of the substrate has on the $β$-decay spectrum of Tritium. We determine the shape of the event rate, accounting for the effects of (part of) the corresponding condensed matter degrees of freedom. In the context of future neutrino experiments, our results provide important information aimed at the optimization of hosting material, as well as the determination of the physics reach. Furthermore, our work outlines a novel theoretical and computational scheme to address a question at the boundary between high and low energy physics. This requires non-conventional declinations of DFT combined with full quantum treatments of the nuclear configuration involved in the decay process.

Figures (6)

• Figure 1: Scheme of the theoretical framework and calculations performed in this work. Panel (a): Calculation of the matrix element between initial and final states to evaluate the decay rate spectrum. Panel (b): DFT calculation of the Tritium potential on graphene in the initial ground state. Panel (c): Sudden approximation for the calculation of the potential felt by Helium just after the decay. Panel (d): DFT calculation of the electronic relaxation via the Kohn-Sham (KS) scheme. Panel (e): Born-Oppenheimer dynamics (BO-MD) of the system after the electronic relaxation.
• Figure 2: Model system and calculation setup. The structures include 32 C atoms and up to 32 T atoms and are extended using periodic boundary conditions. The supercell (whose boundaries are in blue in (a)) is the 4$\times$4 repeat of the unit cell (in red in (a)). The periodicity in the $z$-direction is 22 Å. The exchange and correlation functional is the Perdew-Burke-Ernzerhof PBE parametrization of the Generalized Gradient Approximation. The core electrons are treated with the Projector Augmented Wave approach and with pseudopotentials Lejaeghere_2016, and the cutoff for the plane wave expansion is $k^2/2=65$ Ry. Grimme Van der Waals empirical corrections Grimme_2006 are included. The Brillouin zone is sampled with $18\times18\times 1$ symmetrically distributed $k$-points Monkhorst_1976. When structure relaxation is needed (e.g., panel (a) of Fig. \ref{['fig:scheme']}), we reduce the forces down to 10$^{-3}$ Ry/Bohr. Panel (a): Top view of the fully loaded chair structure. Panel (b): Side view of the same. The coordinates along which the Tritium is moved are also reported. In the first prescription for the Tritium potential, all other atoms are allowed to relax, in the second one the carbon ones are kept frozen (gray shaded region), while in the third prescription the other Tritium atoms are also kept frozen (orange shaded region). Panel (c): Side view of the 50% loaded structure. Panels (d) and (e): Models of the 6.3% and 3.1% loaded systems. The distance between two Tritium atoms is approximately 9.7 Å and 5.6 Å, respectively. Panel (f): Tritium loaded in "dimers", in ortho, meta and para configuration, each considered in cis and trans conformations, as defined in the figure itself.
• Figure 3: Sample orthogonal and parallel potentials. Panel (a):$U_z$ at 100% loading. Solid and dashed lines correspond to different magnetization of the system, as indicated in the legend, while different colors correspond to different conditions for the relaxation of nuclei. Specifically: red and green are obtained slowly dragging the Tritium under consideration, and relaxing the rest of the system during dragging, either keeping fixed only the Carbon nucleus right below (red), or all Carbon nuclei (green). Magenta lines are obtained fixing the C-T distance at evenly separated values along the path of the Tritium and relaxing all the rest, while for the blue line all Carbons are kept fixed in their starting position (more details on the calculation are reported in SI.III.A). Side views of the structures in the minimum, barrier and detached states are reported under the plots. A top view of the system (unbound state) is also reported, colored according to the local magnetization (red for spin up, blue for spin down). Panel (b):$U_z$ computed at 50% loading with symmetrically distributed Tritium, evaluated with free magnetization, which turns out to be $M=16\mu_{\rm B}$ ($1\mu_{\rm B}$ per unit cell), and with the full relaxation prescription. The structures, located near their corresponding place along the curve, illustrate the distribution of magnetization, with coloring as in panel (a), and the arrows indicating the direction of the spin. Panel (c):$U_z$ for the case with lowest loading. Side and top views of the structures are also reported, colored according to magnetization as in the previous panels. We also show the definitions of the various energies and special positions discussed in the main text. Panels (d)-(f): Parallel potentials for $100\%$ loading, with as a function $x$, $y$, and a direction at $30^\circ$ from the horizontal one, as indicated in the structures shown in each panel. The different colors correspond to different values of the $z$-coordinate, fixed at different heights, as indicated in the legend of panel (e). The potentials are evaluated keeping fixed all nuclei except the Tritium under consideration.
• Figure 4: Orthogonal Helium potential in sudden and semi-sudden approximation (respectively, SA and SSA, in blue and red), and in the adiabatic (BO) approximation (green). The system is considered electrically isolated during decay, and therefore the total charge is +1, although in the three cases the Helium is extracted in different ionic states, as indicated. The corresponding parallel potentials are reported in Fig. SI.4. For comparison, the Tritium potential is also reported, aligned to a common energy frame as explained in the text and in the SI, section SI.I.F.
• Figure 5: Electron $\beta$-decay spectra obtained for each of the three schemes employed to define the final Helium potential. The initial Tritium potential is taken to be the one corresponding to 100% loading, computed in the prescription where all other nuclei are kept fixed (see Section \ref{['sec:initial']}). The total Tritium mass is taken to be $M_{\rm tot} = 1 \text{ } \upmu\text{g}$. The electron's kinetic energy is measured with respect to the end-point of a Tritium nucleus in vacuum and for a massless neutrino, $K_0$. Upper panels: Total rate for $m_\nu = 0.2$ eV. We also show the partial contribution for some selected final states (or groups of final states), as indicated directly in the figures. Lower panels: Total rates in a region much closer to the end-point, as computed for different values of the neutrino mass.