As noted in Section 1, tabulated transport coefficients and scattering cross-sections play a vital role in plasma modeling and in enabling transparent, quantitative comparisons across different experimental studies. Over the past decade, the expansion of experimental datasets has also enabled data-driven approaches, most notably deep learning techniques, to infer electron scattering cross-sections from measured transport properties (Stokes et al., 2020; Jetly and Chaudhury, 2021; Muccignat et al., 2024). To date, most databases (including LXCat and NIST) have focused primarily on electron transport in the gas phase. However, recent developments have begun to broaden this scope. For example, LXCat has announced an updated version of its database that will incorporate new states of matter and broader classes of incident particles (Boer et al., 2023).

In this section, we present representative data on the electron mobility and diffusion coefficients in LXe and LAr over a range of experimental conditions. The full dataset is available on GitHub ¹ and is structured in a format similar to LXCat. While CSV formats were initially used, we are exploring alternative text-based formats to better support transparency and long-term maintainability. A complete list of the data in the repository is available in Supplementary Material.

These representative samples highlight both the range of experimental behavior observed and the degree of consistency between early pre-2000 measurements and more recent studies motivated by large-scale liquid TPC development. Because published transport data are typically reported as functions of the absolute electric field at fixed temperatures and pressures, we have converted the data into reduced quantities (e.g., E / N ) using reported liquid densities where available. In cases where the density is not given, we estimate it along the saturated liquid line using interpolated data from the NIST Chemistry WebBook ² database of Thermophysical Properties of Fluid Systems (Lemmon, 2025).

Transport in these systems is often modeled using the drift-diffusion equation (Huxley and Crompton, 1974), which requires knowledge of drift velocity (mobility), swarm broadening (diffusion), and net particle production or loss (ionization or attachment). In noble liquids, at low fields, electron impact ionization does not occur and attachment is negligible in pure media. However, even trace impurities such as oxygen or water vapor can introduce attachment pathways, emphasizing the importance of high-purity conditions in experiments. For TPC applications, high mobility and low diffusion are especially desirable, motivating the use of liquid-phase detectors.

The top panel of Figure 1 shows the reduced electron mobility in LXe as a function of reduced electric field, spanning more than 50 years of measurements. The data are grouped by background temperature and compared to empirical fit models from the NEST simulation framework (Szydagis et al., 2022; 2025).

Recent measurements have largely been limited to low reduced fields ( E / N < 0.01  Td), making high-field behavior less well characterized. The TPC data suggest higher mobilities at similar conditions, e.g., when comparing the data of Huang and Freeman (1978); Miller et al. (1968), and that of Njoya et al. (2020). This may reflect differences in experimental design, particularly liquid purity and field uniformity. The mobility curves at 177K and 183K exhibit slightly different slopes compared to those at 163K, potentially pointing to temperature-dependent structural changes or impurity effects. The model included in NEST v2.3.8 (Szydagis et al., 2025) aligns well with all measurements, favoring the more recent measurements at low fields, although this agreement likely reflects the empirical basis of the model. As Baudis et al. (2023) notes, impurities such as water vapor and hydrocarbons can serve as additional energy-loss channels, leading to higher mobility and suppressed diffusion. This may help explain the elevated drift velocities reported in early studies such as Gushchin et al. (1982), where purification methods were not documented.

Figure 1 also presents reduced longitudinal and transverse diffusion coefficients in LXe, in the bottom panel. Although uncertainties are significant, the longitudinal diffusion coefficients decrease with increasing field strength, while the transverse diffusion coefficients appear largely field-independent within each dataset. The trends in the longitudinal diffusion are broadly consistent with NEST simulations, though further high-precision measurements would help constrain the full field dependence and reduce uncertainty. The NEST curve for the transverse diffusion coefficient has been determined from the experimental measurements of Albert et al. (2017), but underestimates the measurements at higher fields from Doke (1982) (which has an estimated uncertainty of approximately ± 10 % (Doke, 1981)). Clearly, a single model consistent with the full range of LXe transport coefficients is still elusive.

Figure 2 shows the reduced mobility of electrons in LAr, with data grouped by temperature over the range of 85 K–150 K. The results show considerable variation in both plateau values and field dependence, reflecting the underlying temperature sensitivity of the medium’s structure. In particular, the nonlinear temperature dependence of the pair correlation function and the static structure factor plays a key role at low fields.

Within each temperature group, measurements are generally consistent, with the exception of the early results reported by Gushchin et al. (1982), which again appear anomalously high when compared to recent experiments. These discrepancies are likely attributable to uncharacterized impurities.

Temperature clearly plays an important role in determining transport behavior, not just through thermal velocity distributions, but also via structural correlations in the liquid. This sensitivity presents challenges for empirical modeling. The NEST profiles do a sensible job of representing the mobility data at reduced fields above 0.001 Td, but break down at low fields. The Brookhaven National Laboratory (BNL) tool’s empirical fit (Li et al., 2016) performs reasonably well at 85 K and 120 K but fails to reproduce data at 145 K. This highlights a broader problem: empirical models often lack predictive power outside the narrow conditions for which they were calibrated, reinforcing the need for ab initio approaches that incorporate medium structure and density effects explicitly.

Although LAr and LXe are the most studied noble liquids, other systems have also been investigated. Measurements in liquid krypton are broadly consistent with LAr and LXe results (Nishikawa et al., 2007; Schnyders et al., 1966), showing intermediate behavior. In contrast, transport in neon and helium shows marked differences due to their unique electronic properties. Neon and helium have low polarizability and exhibit positive scattering lengths due to Pauli exclusion, resulting in fundamentally different scattering dynamics (Bruschi et al., 1972; Sakai et al., 1982; Borghesani et al., 1988; Hernandez, 1991; Schmidt et al., 2003; Borghesani, 2021). These liquids support the formation of electron bubbles, i.e., quasi-stable low-density cavities, due to strong short-range repulsive forces (Hernandez, 1991). As a result, mobility in neon and helium is often lower in the liquid than in the gas phase. By contrast, in argon and xenon, attractive interactions promote electron localization at atomic sites, preventing bubble formation and leading to higher liquid-phase mobility (Borghesani et al., 1992; Borghesani, 2006). Electron localization and self-trapping is discussed further in Section 3.3.

The behavior of electrons in gases and liquids under the influence of an external electric field E is governed by the BE, which describes the evolution of the phase-space distribution function f ( r , v , t ) . This function represents the probability density for finding an electron with velocity v at position r and time t . The BE is (Boltzmann, 1872; Robson et al., 2017): ∂ ∂ t + v ⋅ ∇ + e E m e ⋅ ∂ ∂ v f r , v , t = − J f , (1) where e and m e are the electron charge and mass, and J ( f ) is the collision operator describing interactions between electrons and the background medium.

In the dilute (or swarm) limit, where the electron population is sparse and does not perturb the background gas, the collision operator remains linear and can be decomposed into contributions from various interaction processes: J f = J e l f + J e x f + J a t f + J i o f + ⋯   . (2) with J e l ( f ) , J e x ( f ) , J a t ( f ) , and J i o ( f ) representing elastic scattering, excitation, attachment, and ionization, respectively. Boltzmann’s original formulation only included elastic processes (Boltzmann, 1872), but subsequent extensions have incorporated inelastic collisions, particle creation and loss, quantum effects, and even relativistic or condensed-phase considerations [see Robson et al. (2017) for a review].

For systems involving electrons it is common to simplify the collision operator using a mass-ratio expansion in m e / ( m e + m 0 ) , where m 0 is the mass of the neutral target (Kumar, 1967; Robson et al., 2017). For light projectiles, only the first-order term is typically required, which simplifies the analysis significantly (Chapman and Cowling, 1970; Frost and Phelps, 1962; Kumar et al., 1980).

Number-conserving processes, such as elastic collisions and electronic excitations, are described using the semi-classical Wang Chang–Uhlenbeck–de Boer operator (Wang Chang et al., 1964). For excitation processes, this is expressed as: J e x f = ∑ j , k ∫ d g ̂ ′ d v 0 g σ j , k ; g , χ × f r , v , t F 0 j v 0 − f r , v ′ , t F 0 j v 0 ′ , (3) where v and v ′ represent the electron pre- and post-collision velocities, v 0 and v 0 ′ represent the background atom pre- and post-collision velocities, g and g ′ are the relative pre- and post-collision velocities, and d g ̂ ′ is a unit vector in the direction of g ′ . The labels j and k describe the internal states of the background atoms, σ ( j , k ; g , χ ) is the differential cross-section for the transition ( j , v , v 0 ) → ( k , v ′ , v 0 ′ ) , and F 0 j ( v 0 ) is the Maxwellian distribution for background atoms in state j . Energy conservation links the pre- and post-collision velocities via 1 2 m e g 2 + U j = 1 2 m e g ′ 2 + U k , where U j and U k are the internal energies of the respective states. When j = k , the collision represents an elastic scattering event. Modifications of the elastic scattering process in a dense gas or liquid medium, such as coherent scattering, are discussed in Section 3.2.

In contrast, non-conserving processes like electron attachment and ionization remove or create electrons, respectively. Attachment processes are described by: J a t f = ∑ j f r , v , t ∫ d v 0 g σ a t j ; g F 0 j v 0 , (4) where σ a t ( j ; g ) is the attachment cross-section for the j th attachment process. Ionization is represented as: J i o f = ∑ j n 0 j v σ i o j ; v f r , v , t − 2 n 0 j ∫ d v ′ v ′ σ i o j ; v ′ B v , v ′ ; j ′ f r , v ′ , t , (5) where σ i o ( j ; v ) is the ionization cross-section for the j th process, and B ( v , v ′ ; j ′ ) is the probability distribution for the velocities of the outgoing electrons. Although ionization is fundamentally a three-body process, it is typically assumed that the heavy neutral remains stationary (valid to zeroth order in the mass ratio) so that energy and momentum are shared only between the incident and ejected electrons.

To solve the BE, a common strategy is to expand the distribution function in spherical harmonics, Y m [ l ] (Ness and Robson, 1986; Robson and Ness, 1986; Robson et al., 2017): f r , v , t = ∑ l = 0 l max ∑ m = − l l f m l r , v , t Y m l v ̂ . (6) where v ̂ is the velocity unit vector. In systems with a preferred direction, such as those with a uniform electric field, this simplifies to a Legendre polynomial P l expansion: f r , v , t = ∑ l = 0 l max f l r , v , t P l cos ⁡ θ , (7) with θ being the angle between the velocity vector and the electric field. Although the two-term approximation ( l max = 1 ) has been widely used for electrons, its limitations are well established (Robson and Ness, 1986; White et al., 2003; Petrović et al., 2009), and convergence must be tested by systematically increasing l max .

Various representations of the speed-space dependence of the coefficients in Equations 6 and 7 have been previously employed, including Sonine polynomials (Burnett, 1935; 1936), cubic splines (Pitchford et al., 1981; Penetrante et al., 1985), finite volume (Boyle et al., 2017), and finite difference schemes Maeda and Makabe (1994); Loffhagen and Winkler (1996); Sigeneger and Winkler (1996); Trunec et al. (2006); Stephens (2018). The combination of spherical harmonics and Sonine polynomials yields the well-known Burnett functions (Kumar, 1966; Burnett, 1935; 1936), which have played a critical role in the development of a unified solution to Boltzmann’s equation across all mass ratios.

Regardless of the exact speed-space representation, substituting the angular expansion into the BE yields a hierarchy of coupled differential equations for the expansion coefficients f m ( l ) ( r , v , t ) . In the spatially homogeneous steady-state case, these take the form: J l f m l − e E m e l + 1 2 l + 3 d d v + l + 2 v f m l + 1 − l 2 l − 1 d d v − l − 1 v f m l − 1 = 0 , (8) where J l denotes the spherical harmonic component of the collision operator.

Once these coefficients are known, key physical observables can be calculated. The electron number density n is given by: n = 4 π ∫ 0 ∞ d v v 2 f 0 0 v , (9) and is typically normalized to unity. The flux drift velocity W is obtained from: W = 4 π 3 ∫ 0 ∞ d v v 3 f 0 1 v , (10) while the mean energy ϵ is: ϵ = 4 π ∫ 0 ∞ d v 1 2 m e v 4 f 0 0 v . (11) Finally, the effective ionization coefficient (normalized to the neutral density N ) is expressed as: R N = 4 π ∫ 0 ∞ d v v 3 σ gain v − σ loss v f 0 0 v , (12) where σ gain and σ loss include all relevant cross-sections for particle-creating and particle-depleting processes, respectively.

To extend these results beyond spatial homogeneity, the distribution functions f m ( l ) ( r , v , t ) can be expanded in terms of spatial and temporal gradients. This formalism connects directly to hydrodynamic transport coefficients, including bulk and flux mobilities and diffusion tensors, thus providing a bridge between microscopic kinetics and macroscopic fluid descriptions (Robson et al., 2017; Robson and Ness, 1986; Ness and Robson, 1986; White et al., 2009).

There are often multiple common ways of presenting the same physical phenomena, which can depend on the exact type of measurement undertaken. For instance, the electron mobility μ = W / E (or reduced electron mobility N μ ) is often tabulated instead of the drift velocity, while the ionization coefficient α ≈ R / W is measured in a steady-state Townsend experiment Huxley and Crompton (1974); Dujko et al. (2008). The transverse diffusion coefficient D T is routinely combined with the mobility μ and presented as the characteristic energy, 3 2 D T / μ .

In liquid-phase systems, electron transport is not always governed solely by scattering and diffusion. Under certain conditions, excess electrons may become transiently localized in the medium due to spontaneous density fluctuations. This phenomenon, known as electron self-trapping, results in the formation of localized states, such as bubbles or solvated electron configurations, that significantly alter the transport dynamics (Nieminen et al., 1980; Maris, 2003; Rumbach et al., 2015; Abel, 2013; Gopalakrishnan et al., 2016).

The driving mechanism behind self-trapping lies in the nature of the electron–atom interaction in noble liquids. These interactions are typically dominated by short-range repulsion, resulting in a positive background potential V 0 ( N ) > 0 , where N is the local atomic density. An electron can lower its free energy by migrating into regions of reduced density, where the value of V 0 is lower. If the density depression is large enough, the energetic gain from a reduced potential energy can outweigh both the kinetic energy cost of localization and the mechanical work needed to expand the local cavity (Borghesani et al., 1992; Hernandez, 1991). Conversely, when V 0 ( N ) < 0 electrons are self-trapped in higher-than-average density fluctuations, as is the case for the heavier Nobel gases and highly polarizable molecular fluids (Mikulski et al., 1997; Krebs and Lang, 1996; Krebs and Wantschik, 1980). In all such cases, a metastable localized state may be formed. These states are weakly bound and have been observed in both theoretical studies and experimental measurements of atomic liquids (Borghesani, 2006).

Borghesani (2006) proposed a phenomenological model for mobility in systems exhibiting self-trapping, treating the total electron mobility as a weighted average of contributions from two populations: free (delocalized) electrons and localized electrons trapped in density fluctuations. The mobility of the free electrons is computed using a heuristic kinetic model, while the contribution from localized electrons is modeled using hydrodynamic descriptions of bubble transport. Although this approach qualitatively reproduces observed trends, it struggled to match absolute mobility values, primarily because the assumption of a single, well-defined localized state is an oversimplification. In reality, liquid systems support a broad distribution of fluctuation sizes and configurations, each contributing differently to the transport properties.

To address these limitations, more sophisticated approaches have been developed. Cocks and White (2016) performed ab initio calculations of the probability that an electron scatters into a low-density region and becomes trapped. Their model accounts for both the formation and energetic stability of these localized states and provides a physically grounded framework for studying electron localization in liquids.

Further progress was made by Stokes and collaborators (Stokes et al., 2016), who introduced a generalized BE capable of describing the interplay between localized and delocalized electron populations. Their framework incorporates scattering into and out of density fluctuations, allowing for both trapping and detrapping processes to be treated self-consistently. This hybrid transport model provides a more complete picture of electron dynamics in systems where localization phenomena are non-negligible.

Overall, electron self-trapping represents a key departure from conventional gas-phase transport behavior and introduces rich physics into the study of dense, disordered systems. Accurate treatment of this effect is essential for modeling electron mobility in noble liquids, particularly under low-field or low-temperature conditions where fluctuations are more prominent and localization becomes more probable.

Studies of ionisation, scintillation, and luminescence in noble gases and liquids, particularly in LAr and LXe, play a vital role in the development of modern detector technologies (González-Díaz et al., 2018; Bonivento and Terranova, 2024). When a charged particle traverses a noble liquid, energy dissipation occurs via ionization, excitation, and the generation of sub-excitation electrons. The average energy loss per ionization event slightly exceeds the ionization potential or the energy for the interband transition, as it encompasses multiple ionisation events. Notably, the average ionization energy is lower in the liquid phase compared to the gas, implying more efficient ionization. Among noble liquids, LXe exhibits a lower energy cost per ion pair than LAr, resulting in a higher ionization yield. The liberated electrons then drift and diffuse under an applied electric field, enabling both calorimetric and spatial reconstruction in TPC detectors (Aprile and Baudis, 2010).

Scintillation, the emission of light following atomic or molecular excitation, plays a central role in signal formation. More broadly, luminescence includes all photon emission processes (scintillation among them) arising from energy deposition and subsequent relaxation of excited states. In the context of liquid noble gases, these terms are often used interchangeably, and we refer to all prompt light emission as scintillation.

The scintillation and fluorescence in these dense and liquid states have been of great experimental interest. Key observations of near infrared emission in high density Xe gas and in Xe liquid has been observed across multiple experiments with strong VUV fluorescence characteristics (Bressi et al., 2000; Belogurov et al., 2000; Borghesani et al., 2001; Piotter et al., 2023). In LAr and LXe, scintillation is predominantly driven by the formation and subsequent de-excitation of dimers, also known as excimers, i.e., transient, bound states formed when an excited atom (Ar* or Xe*) interacts with ground-state atoms. These excimers exist in singlet and triplet configurations, which de-excite by emitting vacuum ultraviolet (VUV) photons: approximately 128 nm in LAr and 178 nm in LXe. Time-resolved spectroscopy confirms this mechanism, showing distinct fast and slow scintillation components that correspond to singlet and triplet state lifetimes (Kubota et al., 1978; Hitachi et al., 1983; Segreto, 2021). Further experimental evidence includes the dependence of scintillation yield and timing on factors such as electric field intensity, particle type, and energy deposition (Doke et al., 2002; Doke, 2006). These observations are consistent with theoretical models that describe the formation and decay of excimers.

Direct de-excitation of excited atoms also contributes to scintillation, though to a lesser extent. For example, in LXe, isolated Xe* atoms can emit photons upon returning to the ground state. In the dense liquid environment, this pathway is usually suppressed in favor of excimer ( Xe 2 *) formation via collisions between excited and ground-state xenon atoms. However, at low electric field strengths, recent studies show that neutral bremsstrahlung (NBrS) resulting from scattering of sub-excitation electrons on neutral atoms, can dominate the excimer scintillation in both gas- and liquid-phase xenon (Buzulutskov et al., 2018; Henriques et al., 2022; Bondar et al., 2023). NBrs is predicted to occur in all noble gases, with an intensity proportional to the elastic electron-atom scattering cross section (Buzulutskov et al., 2018), and is thus strongly dependent on the density effects described in Section 3. Although NBrs and the direct de-excitation of excited LXe atoms generally play a relatively minor role in the scintillation process compared to the excimer mechanism, they do contribute to the overall luminescence.

To model these processes quantitatively, accurate rate coefficients for inelastic collisions, such as atomic excitations and interband transitions, are required. Figure 9 illustrates rate coefficients computed using MC simulations and electron scattering cross-sections from Simonović et al. (2019). At low reduced fields, E / N , excitation and interband transitions are negligible. However, as E / N increases beyond 20 Td, the interband transition rate becomes dominant. The computed rate coefficients are smooth and featureless, reflecting the threshold-like onset of inelastic processes as electrons gain sufficient energy. These results are essential for refining empirical simulation tools such as NEST (Szydagis et al., 2011; 2021; 2025). In addition, the rate coefficients for various inelastic processes are required as input for the fluid modeling of discharges in LXe.

Doping is a powerful technique for enhancing detector performance. For example, introducing trace amounts of LXe into LAr shifts the emission spectrum from 128 nm to 175–178 nm. This shift improves photon detection efficiency by reducing Rayleigh scattering and matching the spectral sensitivity of common photodetectors. Even at concentrations as low as a few parts per million, xenon doping can significantly increase scintillation yield by enabling energy transfer from LAr to LXe excimers (Segreto, 2021; Fields et al., 2023; Abud et al., 2024). A first step towards accurate ab initio modeling of doped and simple liquid mixtures is detailed in Section 4.2.

In contrast, impurities such as nitrogen, oxygen, and water vapor degrade detector performance by quenching scintillation and capturing free electrons. These species either absorb VUV photons or serve as electron scavengers, reducing both light output and charge collection. Oxygen, in particular, has a high electron affinity and is highly disruptive to charge readout (Acciarri et al., 2010; Barrelet et al., 2002). Modeling studies have been conducted to understand impurity dynamics in LAr detectors, including their sources, transport, and removal mechanisms. To mitigate these effects, sophisticated purification systems and impurity monitoring tools, such as ultra-sensitive trace gas sensors developed for the DARKSIDE experiment, have been employed to maintain purity at sub-part-per-billion levels (Mount et al., 2012).

Overall, the interplay between excitation, ionization, and scintillation mechanisms, along with the impact of doping and impurity control, defines the performance of noble-liquid detectors. Accurate modeling of these processes, supported by both experimental data and advanced simulations, remains critical for optimizing detector response and enabling precision measurements in neutrino and dark matter experiments.

The ECFA DRD roadmap Colaleo et al. (2021) highlighted the potential benefits of using liquid mixtures and dopants as a detector medium. For instance, doping LAr with small concentrations of LXe has been shown to enhance charge amplification by over two orders of magnitude (Kim et al., 2002), offering a promising route for improved detector performance (Segreto, 2021; Fields et al., 2023; Abud et al., 2024). As discussed in Section 4.1, even trace amounts of LXe in LAr reducing Rayleigh scattering, matches the spectral sensitivity of common photodetectors, and enables energy transfer from LAr to LXe excimers.

In Section 3.2, we introduced the formalism for coherent elastic scattering in pure single-component liquids. Extending this framework to mixtures involves generalizing the collision operator to account for multiple species. In the case of a binary mixture (e.g., a doped LAr–LXe system), the coherent elastic collision operator generalizes to (Boyle et al., 2024): J 0 Φ 0 = m e m a v 2 v ν 1 a v v Φ l + k b T m e d d v Φ 0 + m e m b v 2 v ν 1 b v v Φ l + k b T m e d d v Φ l , (23) J l Φ l = ν ̃ l v Φ l , for  l ≥ 1 , (24) where ν l a ( v ) and ν l b ( v ) are the usual Legendre components of the binary collision frequencies for each species: ν l a , b v x a , b N = 2 π v ∫ 0 2 π σ a , b v , θ 1 − P l cos ⁡ θ sin ⁡ θ d θ . (25) The structure-modified collision frequency ν ̃ l ( v ) is now calculated from the combined differential cross-section: Σ v , θ = σ a v , θ S a a K + σ b v , θ S b b K + 2 σ a v , θ σ b v , θ S b a K . (26) where K = 2 m e v ℏ sin θ 2 , and S a a , S b b and S a b are the partial static structure factors describing intra- and inter-species correlations (see Boyle et al. (2024); Hansen and McDonald (1976) for details). In the dilute limit where K is large, the structure factors approach their respective mole fractions ( S a a ( K ) → x a , S b b ( K ) → x b and S a b ( K ) → 0 ) and the usual binary scattering results are recovered, as required. The formalism also correctly reduces to the single-species case when σ b → σ a , i.e., S a a + 2 S a b + S b b → S a , where S a represents the structure factor of the pure fluid.

The static structure factors and pair distribution functions required in these calculations can be obtained via molecular dynamics or MC simulations (van Gunsteren and Berendsen, 1990; Lindgård, 1994; Wedberg et al., 2011). However, for hard-sphere liquids, an analytic solution is available from the Percus–Yevick approximation (Percus and Yevick, 1958; Lebowitz, 1964; Hiroike, 1969). For binary mixtures, the structure is fully specified by ratio of hard-sphere diameters of the two species β = d b d a , the diameter and density fraction of species a ( d a , x a ) , and the total packing fraction Φ : Φ = π 6 N x a d a 2 + π 6 N x b d b 2 . (27) where x b = 1 − x a , and N is the total number density. These parameters define the extent of spatial correlation within the mixture and strongly influence electron transport. We now consider the benchmark model introduced in Boyle et al. (2024), shown in Figure 10.

Figure 11 illustrates the structure-modified collision components Σ 0 − Σ l for three different compositions: pure species a ( x a = 1 ) , pure species b ( x a = 0 ) , and a 50:50 mixture ( x a = 0.5 ) . Similar to the pure liquid case, the structure-modified profiles exhibit oscillations around the constant binary scattering value and show significant suppression of momentum-transfer at low energies. Interestingly, at energies near 0.1 eV, the 50:50 mixture profile exceeds those of both pure systems, indicating non-linear mixing behavior in the cross-sections.

These effects are directly reflected in the calculated transport coefficients. Figure 12 shows the variation of drift velocity W , mean energy ϵ , and reduced diffusion coefficients N D L and N D T , as functions of the reduced electric field. The results are shown for two packing fractions ( Φ = 0.2 and Φ = 0.4 ) with β = 2 and α = β 2 = 4 . Large variations spanning orders of magnitude are observed across different mixture compositions. Notably, at the higher packing density ( Φ = 0.4 ) , the transport coefficients for the 50:50 mixture fall outside the range bounded by the two pure species, a behavior not seen at lower density ( Φ = 0.2 ) . This non-linearity underscores the importance of particle correlations in dense mixtures and their impact on transport properties.

As demonstrated in Boyle et al. (2024), even modest amounts of dopant or secondary species can dramatically alter electron transport in noble liquid systems, sometimes enhancing and other times suppressing mobility and diffusion compared to the pure fluids. This is supported by earlier measurements of electron mobility in liquid Ar-Kr and Ar-Xe mixtures, which experimentally demonstrated a doping-induced reduction of the mobility (Borghesani et al., 1997). This opens new possibilities for tuning detector media by carefully selecting admixture compositions to optimize signal response, timing, or gain.

Although coherent elastic scattering dominates at low electron energies, a complete description of transport in mixtures also requires accounting for how dopants affect polarization screening and bulk potential contributions. Extending the framework developed in Section 3.2.2 to multi-component systems is a subject of ongoing work, and will be essential for predictive modeling of next-generation doped noble-liquid detectors.

Dual-phase TPCs take advantage of both the liquid and gas phases of noble elements, and are central to many state-of-the-art detectors Colaleo et al. (2021); Baudis (2024). In these systems, a well-defined gas-liquid interface is intrinsic to the detector geometry, typically formed by maintaining the liquid at its boiling point through controlled vapor-condensation mechanisms.

In a typical dual-phase detector, ionization electrons generated in the liquid drift under a modest electric field until they reach the interface. At this boundary, they are extracted into the vapor phase, where higher field strengths are used to generate secondary scintillation light or avalanche multiplication via amplification grids. The efficiency and fidelity of this extraction process are crucial for detector sensitivity, timing resolution, and energy reconstruction. However, the gas-liquid interface introduces several physical complexities that are not present in homogeneous systems.

One significant complication is the likely accumulation of surface charge at the interface in both the longitudinal and transverse directions, relative to the extracting field. Such charge buildup may lead to distortions of the local electric field, inhibit electron extraction, or produce spurious signals, especially in large-scale detectors where small distortions can impact performance over long drift distances.

Moreover, the microscopic structure and energy barrier of the interface, which is shaped by features such as atomic-scale density gradients and thermal fluctuations, further complicates the prediction of electron behavior near this region. For example, in the case of liquid He, excess electrons impinging on the interface have been computationally studied with density functional theory Ancilotto and Toigo (1994). By self-consistently computing the shape of the barrier, rates of electron tunneling could be compared to experimental results in a semi-quantitative manner. On the other hand, for very low energy electrons, precise modeling of the interface barrier is less important, because their wavefunctions extend so far in space that only “smeared” or averaged features of the interface affect the outcome, as shown for electron self-trapping in dense helium gas Borghesani et al. (2024) and also for liquid Ar at the normal boiling point Borghesani et al. (1991). This contrasts sharply with higher-energy electrons, where detailed modeling of the barrier is crucial to predict tunneling rates and interfacial interactions.

Accurately modeling electron transport across the gas–liquid boundary requires more than simply adding together the transport models from each phase. While simplified approaches often treat the interface as a sharp boundary layer with idealized transfer conditions, a more complete and physically consistent treatment must account for the continuous variation in density, potential energy, and scattering environment that electrons encounter as they approach and cross the interface.

A promising step in this direction comes from recent work by Garland et al. (2018a), who modeled the interface as a smooth density gradient characterized by an equilibrium profile spanning the two phases. Key to their approach was the relation of the spatial variation in V 0 , the quasi-free electron energy, to a varying effective electric potential which through the gradient presents as a spatially inhomogeneous effective electric field. This effective field may then functionally act to inhibit electron extraction across the interface from LAr or LXe to gas, or conversely accelerate electron transport from gas into liquid. Therefore, this framework may be applicable to the interpretation of dual-phase detector electron extraction efficiency from LAr or LXe into the gas phase in low-field or cryogenic conditions Xu et al. (2019). This approach is supported by molecular dynamics and MC simulations, which show that noble liquids such as LAr and LXe exhibit finite-thickness interfacial layers with well-defined thermodynamic properties. Early studies (Chapela et al., 1977; Trokhymchuk and Alejandre, 1999) employed Lennard-Jones potentials to simulate noble gas interactions and were able to reproduce key interfacial properties, including the surface tension, equilibrium vapor and liquid densities, and the interfacial width.

These simulations provide critical inputs for transport modeling, especially for defining the spatial variation of scattering cross-sections, background potential energy, and local electron density in the interfacial region. Incorporating this information into kinetic, fluid or MC models enables a more accurate and continuous treatment of electron transport from liquid to gas. It also provides a path towards predicting extraction efficiencies, timing characteristics, and possible sources of noise originating from interfacial phenomena. As dual-phase detectors continue to scale in size and complexity, a deeper understanding of interfacial electron dynamics grounded in molecular simulation and kinetic theory will be essential for ensuring the performance and reliability of future experiments.