Thomas-Imel Box Framework

• Thomas-Imel Box Framework is a semi-analytic model that describes electron–ion recombination in liquefied noble gases using a 'box' approach for low-energy recoils.
• It quantitatively links deposited energy with scintillation (S1) and ionization (S2) yields by utilizing parameters like recombination probability, box size, and electron mobility.
• The model underpins calibration in TPCs for dark matter detection, enhancing sensitivity through precise energy reconstruction and background discrimination.

The Thomas-Imel Box Framework provides a semi-analytic model for electron–ion recombination in liquefied noble-gas detectors, crucial for energy reconstruction and event classification in rare-event searches such as direct dark matter detection. Developed by J. Thomas and D.A. Imel, the model describes recombination under the "short track" regime relevant to low-energy recoils, where all electron–ion pairs are formed in a compact spatial region. Its predictive capacity for the partition of deposited energy between scintillation (S1) and ionization (S2) signals underpins calibration methodologies in xenon and argon time projection chambers (TPCs).

1. Theoretical Structure and Key Equations

The Thomas-Imel "^^^^1^^^^" model assumes that an initial recoil in a noble liquid generates $N_i$ electron–ion pairs and $N_{ex}$ excited atoms. Each electron–ion pair is presumed to be confined within a notional box of size $a$, and the probability $r$ that a given ion recombines with its parent electron is governed by the dimensionless recombination parameter $\xi$. For liquid xenon, the recombination probability is given by: $r = 1 - \frac{1}{\xi} \ln(1+\xi)$ with

$\xi = \frac{N_i\, \alpha}{4\, a^2\, \nu}$

where $\alpha$ encapsulates details of electron and ion mobilities, $a$ is the box dimension, and $\nu$ is the mean speed of the ionized electrons (Boulton et al., 2017). The prompt scintillation yield is proportional to $n_\gamma = N_{ex} + N_i r$, while the ionization yield is $n_e = N_i (1 - r)$.

In applications to both xenon and argon, the model is integrated with Lindhard theory to relate recoil energy $E$ to the total number of excitation quanta. For liquid argon, the number of detected electrons from an energy deposition $E$ is

$N_e(E) = \frac{1}{\gamma} \ln(1+\gamma N_i)$

where $\gamma$ is the field-dependent box parameter, and $N_i$ in turn is computed using the fraction of energy going into ionization defined by the stopping powers and Lindhard partition (collaboration et al., 2021, Franco, 2 Jan 2026, Sorensen et al., 2011).

2. Application to Charge and Light Yields in Noble Liquids

The partition of energy between S1 and S2 is central to TPC operation and background rejection. The Thomas-Imel box model directly links observed quanta to underlying recombination dynamics:

• For xenon, S1 is proportional to $n_\gamma$, S2 to $n_e$ (Boulton et al., 2017).
• The model’s recombination law reliably predicts S1/S2 as a function of electric drift field $F$ and recoil energy. The recombination fraction decreases with increasing $F$, though suppression is slow—for example, for a $2.8$ keV event in LXe, $r$ drops from $\approx0.29$ at $0.1$ kV/cm to $\approx0.23$ at $2$ kV/cm (Boulton et al., 2017).
• In LAr, the charge yield per unit energy ($Q_y = N_e/E$) is described for both electronic recoils (ER) and nuclear recoils (NR) down to sub-keV scales by the generalization: $$Q_y(E) = \frac{F}{E\, C_{\rm box}} \ln\left[1 + \epsilon\,\beta\,C_{\rm box}\,F\,\frac{s_e(\epsilon)}{s_e(\epsilon)+s_n(\epsilon)}\right]$$ where $C_{\rm box}$, $\beta$, and the stopping power terms are calibrated to data (Franco, 2 Jan 2026).

3. Calibration in Dual-Phase TPCs: Methods and Results

Practical extraction of the box parameter relies on experimental measurements of S1 and S2. Typical procedures include:

• Conversion of S1 (measured in photoelectrons) to photon counts $n_\gamma$ using light-collection efficiency and photoelectron corrections.
• Conversion of S2 to electron counts $n_e$ via the single-electron gain and electron-extraction efficiency.
• Use of external constraints on $N_{ex}/N_i$, typically obtained from prior measurements or simulation frameworks such as NEST.
• Direct calculation of the recombination fraction $r$ via simultaneous S1/S2, numerical inversion of the model equation to obtain $\xi$ (or $\gamma$), and fits to empirical field dependences, e.g., $4\xi/N_i = A (F/1\,\mathrm{kV/cm})^{-\delta}$, with $A$ and $\delta$ extracted from data fits (Boulton et al., 2017).

The validity of the Thomas-Imel model is established within specific energy and field regimes:

• Excellent agreement with xenon data for energies below $\sim10$ keV (Sorensen et al., 2011, Boulton et al., 2017).
• For DarkSide-50 liquid argon, the model accurately describes charge yields for ER down to 0.18 keV and for NR to 0.435 keV, with global fits providing $C_{\rm box}$ and $\beta$ parameter values, and an empirical extension for ER above 3 keV (collaboration et al., 2021).

4. Screening Potential Choices and Model Selection

Refinement of nuclear recoil modeling in LAr is sensitive to the atomic screening function entering the calculation of stopping powers. The global fit in DarkSide-50/ARIS/SCENE/ReD combined datasets leverages three screening potentials:

• Ziegler–Biersack–Littmark (ZBL)
• Molière (Wilson parametrization)
• Lenz–Jensen (analytic Thomas-Fermi alternative)

A Bayesian model comparison robustly prefers the Lenz–Jensen function: odds of $10^4\!:\!1$ (LJ/ZBL) and $10^7\!:\!1$ (LJ/Molière), indicating decisive statistical evidence by the Jeffreys–Kass convention (Franco, 2 Jan 2026).

5. Experimental Implications, Validity Range, and Limitations

A precise Thomas-Imel calibration is indispensable for interpreting low-energy recoil signatures, especially for rare-event searches in dark matter and neutrino physics:

• Accurate knowledge of $Q_y(E)$ and its field/energy dependence informs energy thresholds and discrimination between ER/NR, directly impacting sensitivity to sub-GeV WIMPs and axions (Boulton et al., 2017, Sorensen et al., 2011, collaboration et al., 2021).
• In regimes where track length approaches the recombination/diffusion scale (beyond several keV), the model's assumptions fail, and additional empirical terms or transition to Birks-type (columnar) recombination is required (collaboration et al., 2021).
• Experimental data demonstrate that at very low energies (e.g., 0.27 keV in LXe), recombination becomes negligible, independent of applied field—a regime where essentially all ionized electrons escape (Boulton et al., 2017).
• Extrapolation beyond validated energy ranges (below direct data, above few keV for ER) is model-dependent and introduces systematic uncertainty, dominated by the limitations of the simplifying "box" picture and possible changes in $N_{ex}/N_i$ (collaboration et al., 2021, Franco, 2 Jan 2026).

6. Impact on Dark Matter Search Sensitivity

Implementations of the Thomas-Imel box model, especially as realized with updated screening potential choices, directly enhance the reach of noble-liquid dark matter searches:

• Adoption of the Lenz–Jensen screening in DarkSide-50 analyses leads to a $20$–$50\%$ increase in predicted signal for sub-5 keV NR, improving the 90% C.L. exclusion flows for $1.2$ GeV/$c^2$ WIMPs by up to a factor of 5 in specific fluctuation models (Franco, 2 Jan 2026).
• For next-generation detectors such as DarkSide-20k, model refinements enable nearly an order of magnitude improvement at the lowest WIMP masses with a 2 $e^{-}$ NR threshold.
• These results validate combining the Thomas-Imel framework with precise electronic/nuclear stopping models and full detector-response calibration, establishing a best-practice paradigm for low-mass dark matter direct detection and rare-event search calibration (Franco, 2 Jan 2026, collaboration et al., 2021).