The depth distribution of X-ray interactions can be quantitatively analyzed by plotting event frequency as a function of waveform duration. For each event, duration was measured between the 1%-rising edge and the 50%-falling edge thresholds relative to waveform amplitude. Prior to this measurement, a moving average filter was applied to waveforms to reduce high-frequency noise. Using this method, our definition of event duration encompasses the interval from the initiation of the scintillation process to approximately the moment when the electron cloud’s centroid reaches the anode. These two limits are also illustrated in Figure 3.

Figure 5 presents an example distribution of event durations obtained with 5.9-keV X-rays. The data exhibit the expected exponential absorption law of X-rays in matter, with notable deviations observed only for longer-duration waveforms (leading to apparent drift path lengths exceeding the detector’s physical length of 2.5 cm). These deviations from theoretical predictions are attributed to two primary factors: (1) the weaker electric field near the detector window (12 μ m-thick Kapton foil, which bends outward due to differential gas pressure) and (2) statistical fluctuations in the electroluminescence photon burst duration.

To convert event duration into mean electron drift path length, we first fit the event duration distribution with an exponential function over the range of approximately 1.2–2.2 cm, where the data closely follow an exponential trend. We then apply a scaling procedure to determine this conversion factor. Since the maximum physical drift path is 2.5 cm, measured events with apparent path lengths exceeding this limit were actually produced below the 2.5-cm threshold. Accordingly, the conversion factor is scaled such that the excess area (events measured beyond 2.5 cm) equals the deficit area (between the measured and fitted distributions below 2.5 cm). This ensures event count conservation and maps zero duration to zero drift path length (the PMT’s transit time spread, ∼ 10 ns, is negligible when compared to electron drift times). The uncertainty in the measured drift path length is estimated to be ∼ 10%.

The X-ray attenuation coefficient extracted from the exponential fit after conversion factor adjustment provides an independent validation of our calibration method. The experimental value (i.e., an attenuation length λ = 0.233 ± 0.023 cm) agrees well with theoretical attenuation expected for 5.9-keV X-rays in xenon at a pressure of ∼ 1.2 bar and a temperature of ∼ 305 K ( λ = 0.226 cm), showing only a 10% relative deviation.

With this calibration established, we can classify all PMT waveforms according to their electron drift path length. This classification method was used to create the color-coded waveform representation in Figure 4 (right).

The PMT charge generated by each X-ray interaction can be obtained by integrating the PMT waveform from 100 ns before the start of the event to 150 ns after the event termination. We present these charge values in the natural units of the integral (mV ⋅ μ s), since the voltage across the oscilloscope’s 50- Ω input resistor is directly proportional to the PMT current. This integration window excludes most PMT afterpulsing, which would otherwise degrade the detector’s energy resolution. By analyzing events with similar drift path lengths, we can compute the charge distribution for specific X-ray interaction depths. Figure 6 (left) presents an example charge distribution for interactions occurring at a path length around 2.34 cm.

The centroid of the 5.9-keV full-absorption peak in these distributions is directly proportional to the number of EL photons collected by the PMT for events at the corresponding depth. To characterize the EL gain dependence on interaction depth, we fitted Gaussian functions to the full-absorption peaks of charge distributions obtained at various path lengths. Figure 6 (right) shows the resulting relationship between the EL gain and drift path length, which is well-described by a fourth-degree polynomial function.

This depth-dependent gain variation is primarily governed by geometric factors, specifically the solid angle subtended by the PMT as electrons drift toward it, and remains independent of both X-ray energy and electric field, as confirmed in our experimental measurements (see chapters 5 and 6 in (Henriques, 2019)). The fitted polynomial function is then used to determine the appropriate EL gain correction applied to each waveform charge value, based on its measured drift path length.

The evolution of the electron cloud along the drift path can be described by the hydrodynamic approximation of the Boltzmann equation (Gonzalez-Diaz et al., 2018): ∂ N e ∂ t + v d ∂ N e ∂ z ′ − D T ∂ 2 N e ∂ x ′ 2 + ∂ 2 N e ∂ y ′ 2 − D L ∂ 2 N e ∂ z ′ 2 + η v d N e = 0 (2) where N e is the electron density per unit volume, D T and D L are the transverse and longitudinal diffusion coefficients respectively, v d is the drift velocity, and η is the attachment coefficient. Solving this equation yields: N e x ′ , y ′ , z ′ , t = e − x ′ − x 2 + y ′ − y 2 4 D T t − t 0 4 π D T t − t 0 ⋅ e − z ′ − z + v d t − t 0 2 4 D L t − t 0 4 π D L t − t 0 1 / 2 ⋅ n ̄ e e − η v d t − t 0 (3) here, ( x ′ , y ′ , z ′ , t ) represent the coordinates from the observer’s point of view, while ( x , y , z , t 0 ) denote the initial position and time of the ionization cloud containing n ̄ e electrons.

By projecting the electron density ( N e in Equation 3) along the z -dimension (the drift direction) and neglecting electron attachment, which is a reasonable approximation for pure xenon, Equation 3 can be simplified into a one-dimensional Gaussian function. The standard deviation of this function is directly proportional to the square root of time ( σ = 2 D L Δ t ) , and the centroid is directly proportional to time (centroid = v d Δ t ).

Based on this simplification, we simulated the evolution of a Gaussian-distributed electron cloud along the drift axis to analyze the shape of PMT waveforms. Our simulation incorporated the solid angle subtended by the PMT window and the exponential decay rate of the xenon second continuum, which we assumed to be 100 ns (Henriques et al., 2024). The radial charge distribution of the initial electron cloud was obtained from (Dias et al., 1993).

Section 3.1 introduced a method to determine the conversion factor between the waveform duration and path length drifted by the electron cloud. However, calculating the electron drift velocity ( v d ) directly from this factor would overestimate the result due to several broadening effects: the Xe triplet state deexcitation time ( ∼ 100 ns), the initial spatial distribution of the electron cloud, and longitudinal electron diffusion. Using the electron transport simulation described in Section 3.3.1, we quantitatively assessed these broadening mechanisms and derived appropriate correction factors. Though modest (typically < 5%), these corrections were applied to all v d measurements.

Figure 8 presents the experimental v d values measured in pure xenon across various reduced electric fields, alongside expected values from Magboltz simulations (Biagi, 2025; Biagi, 1999). The tight alignment between the two datasets validates our approach.

To investigate the influence of longitudinal electron diffusion on detector response, we employed the electron cloud simulation detailed in Section 3.3.1. Figure 9 presents a comparison between simulated (left) and experimental (right) PMT waveform trailing edges across various drift path lengths. As shown in simulated waveforms, the effect of longitudinal diffusion is particularly noticeable near the maximum amplitude and in the falling edge. For longer path lengths (red curves), the electron cloud becomes more elongated due to diffusion, resulting in longer fall times and lower waveform maximum amplitudes. Simulation results are closely matched by our experimental observations, confirming that the electron cloud expands longitudinally as it drifts through the EL region. This relationship between waveform fall time and drift path length provides a robust method to quantify the longitudinal electron diffusion coefficient from experimental waveforms.

The majority of afterpulsing in our PMT occurs within the first 900 ns after the emission of the photoelectron. Therefore, the afterpulsing effect on waveform trailing edges remains constant across different drift path lengths, provided waveforms are not shorter than 900 ns. Consequently, while the absolute fall time values are influenced by both afterpulsing and the initial electron cloud spread, only longitudinal diffusion determines how fall time varies with drift path length. As an additional precaution, we restricted our fall time measurements to the region between 90% and 60% of the waveform maximum amplitude, further reducing any residual afterpulsing contributions.

Using our electron cloud simulation framework (Section 3.3.1), we constructed a model that predicts waveform fall times as a function of drift path length. The model incorporates three key parameters: electron drift velocity, longitudinal diffusion coefficient, and the initial spatial spread of the electron cloud. To extract the diffusion coefficient from experimental data, we fitted this model to the measured fall times across various drift path lengths. To minimize the impact of high-frequency noise, average waveforms were used instead of individual waveforms in the fall time measurements. In this fitting procedure, we fixed the drift velocity to our experimentally determined values (Section 3.3.2) while allowing both the diffusion coefficient and initial electron cloud spread to vary as free parameters. To avoid the boundary effects, we excluded data points with path lengths exceeding 1.7 cm from our analysis.

The results of our longitudinal diffusion coefficient measurements are summarized in Figure 9 (bottom), which displays the parameter D L * = 2 D L / v d (representing the spatial spread per unit square root of drift length) for pure xenon at 1.23 bar. Experimental data show reasonable agreement with theoretical predictions from Magboltz simulations (black curve), further validating our waveform-based measurement approach.

Electron attachment can be quantified by examining average PMT waveforms at different interaction depths, since the number of emitted EL photons is directly proportional to the surviving electron population. Figure 10 illustrates this effect by comparing waveforms from pure Xe and Xe- CF 4 mixtures. Pure xenon waveforms maintain consistent amplitude profiles regardless of interaction depth, indicating negligible electron loss. In contrast, Xe- CF 4 mixtures exhibit a progressive amplitude reduction with increasing drift path length, a clear signature of CF 4 -induced dissociative attachment capturing electrons during their drift.

To determine electron attachment coefficients, we developed a method based on the EL signal analysis at a fixed reference position. For each average waveform, we integrated the signal when the electron cloud was between 6.5 and 7.5 mm from the anode. This integration measures the reduced EL yield ( Y / p ( 7  mm ) ) at this reference position. The shaded green region in Figure 10 (top left) denotes the 6.5–7.5 mm integration window, whose bounds were determined from the waveform-duration to mean drift-path calibration (Section 3.1.1). By plotting these yield values against the path length ( Δ x ) traveled by electrons before reaching this position (Figure 10 top right), we observe the expected exponential decay curve. The decay constant from exponential fit provides the attachment coefficient ( η ) , which increases with CF 4 concentration while remaining effectively zero for pure xenon.

Figure 10 (bottom) shows the experimental attachment coefficients obtained with three Xe- CF 4 mixtures across several reduced electric fields, alongside theoretical values from Magboltz simulations. There is general good agreement between simulations and experimental data, except for 0.029% CF 4 . For this particular concentration, measurements were performed approximately 1 month after the initial detector filling, suggesting possible contamination with additional electronegative species (particularly O 2 , H 2 O, or CO 2 ) that may have accumulated over time despite the continuous gas purification.

When evaluating how electron attachment affects EL yield, we must first account for other molecular additive effects that simultaneously influence the signal. Processes such as quenching of atomic and dimer excited states and electron cooling also contribute to EL yield degradation. To isolate the effect of these other processes, we applied the method described in Section 3.3.4, where the EL yield at zero drift path length ( Δ x → 0 ) , before attachment occurs, is given by parameter a in the exponential fit function f ( x ) = a e − η Δ x .

The ratio between a -values for Xe- CF 4 mixtures and pure Xe can be used to rescale the mixture waveforms to effectively remove non-attachment-related degradation effects. This normalization, visible in Figure 10 (top left), allows direct observation and quantification of yield reduction specifically caused by electron attachment.

Figure 11 presents our key experimental results quantifying electron attachment effects on electroluminescence, obtained with 5.9-keV X-rays for pure Xe and various Xe- CF 4 mixtures at different electric fields. The left panel presents the relative EL yield under electron attachment conditions ( Y att / Y 0 ) plotted against the corresponding electron attachment coefficients ( η ) . These yield ratios were obtained by integrating rescaled Xe- CF 4 waveforms (with attachment) and comparing them to integrated pure Xe waveforms (without attachment), with η values determined as described in Section 3.3.4. The right panel shows the experimental Q -factors plotted as a function of η . These Q -factors, quantifying statistical fluctuations in the EL process, were measured using the methodology detailed in Section 3.2.

For small attachment values, analytical theory predicts Q = η g / 3 and Y att / Y 0 = 1 − η g / 2 (derived in Ref. (Azevedo et al., 2018)), where g represents the EL region length. However, our data show significant deviations from these approximations, particularly at higher attachment coefficients, as demonstrated in Figure 11. To understand these deviations, we developed a Monte Carlo simulation of the electron drift and attachment process.

The simulation follows individual electrons through the EL region, where each electron’s attachment position is generated according to an exponential probability distribution governed by the attachment coefficient. We integrate the photon yield along each electron’s trajectory, weighted by the solid angle subtended by the PMT from each position, until either attachment occurs or the electron reaches the anode. The simulation outputs both the mean detected photon yield and its statistical distribution, enabling direct calculation of Y att / Y 0 and Q -factor.

Our simulation results (also shown in Figure 11) demonstrate excellent agreement with experimental data, validating our understanding of the underlying physical processes. Importantly, the simulation revealed that detector geometry significantly influences attachment effects. The sharply increasing solid angle as electrons approach the PMT causes the Q -factor (Equation 1) to degrade approximately twice as much as it would be expected in a constant-solid-angle configuration when attachment coefficients exceed 0.1 cm − 1 . This degradation stems from increased variability in the number of detected photons when electrons are attached at different positions. While less pronounced than its effect on the Q -factor, the solid angle variation also degrades the Y att / Y 0 value. This geometric effect explains the observed deviations from analytical predictions and highlights an important consideration for optimal detector design.

The electroluminescence yield in noble gases exhibits an approximately linear relationship with the reduced electric field, expressed as Y / p = a ( E / p − E t h / p ) (Henriques et al., 2024; Monteiro et al., 2007; Oliveira C. A. B. et al., 2011), a behavior that seems to persist even in the presence of molecular additives (Henriques et al., 2019). In this relation, the EL amplification parameter a directly correlates with the scintillation probability (SP), while the x-intercept parameter E t h / p reflects the EL threshold required to initiate electroluminescence.

Molecular additives influence this relationship through three distinct mechanisms: (1) electron cooling, which primarily affects the E t h / p value by shifting yield curves toward higher fields; (2) quenching of xenon excited states, which reduces the SP and consequently the a -parameter; and (3) electron attachment, which, due to its characteristic dependence on electric field strength, can also modify the Y / p ( E / p ) slope or even slightly degrade its linearity. Additional electroluminescence production processes such as Neutral Bremsstrahlung are not considered here as they are negligible compared to excimer-based electroluminescence within our working conditions (Henriques et al., 2022). Therefore, to accurately determine the intrinsic SP, we must first compensate the yield curves for attachment effects using Y att / Y 0 values derived from the Monte Carlo simulation described in Section 3.3.4, given known attachment coefficients.

Figure 12 presents both raw and attachment-compensated EL yield curves for two Xe- CF 4 mixtures (0.012% and 0.029% by concentration) alongside pure Xe results, plotted against the reduced electric field. After correcting for attachment effects, the mixture yields recover to nearly pure Xe levels, indicating minimal degradation in both SP and EL threshold.

Linear regression analysis (also depicted in Figure 12) of these corrected yield curves resulted in E t h / p values of 0.69 ± 0.04 kV cm − 1 bar − 1 for pure xenon, compared to 0.57 ± 0.55 and 1.02 ± 0.40 kV cm − 1 bar − 1 for the 0.012% and 0.029% CF 4 mixtures, respectively. From the ratio between the slope parameters ( a ) of corrected mixture yield curves and that of pure Xe (assuming a SP of 100% for pure xenon), we calculated scintillation probabilities of (93 ± 18)% and (112 ± 13)% for the 0.012% and 0.029% CF 4 mixtures, respectively. These results support theoretical predictions that CF 4 at these sub-percent concentrations has negligible impact on SP (Azevedo et al., 2018). Although our data follow expected behavior, the pronounced uncertainties arising from both the small number of field points and the larger CF 4 attachment-correction errors (versus pure xenon) hinder more definitive interpretations of subtle effects.

Prior to developing the waveform-based methodology described in this paper, we conducted extensive studies on the effects of molecular additives ( CO 2 , CH 4 , and CF 4 ) on xenon electroluminescence (Henriques et al., 2017; Henriques et al., 2019). Those earlier investigations employed conventional signal shaping instrumentation that did not preserve temporal waveform information. While such data cannot directly yield parameters like electron drift velocity or longitudinal diffusion, a reasonable estimation of attachment coefficients can be obtained through careful analysis of the recorded pulse-height distributions.

As illustrated in Figure 4 (left), the detector pulse-height distribution can be parameterized by a sum of Gaussian functions, each corresponding to a particular X-ray interaction depth. Electron attachment affects both the EL yield and Q-factor (Figure 11), causing distinctive distortions in the shape of the pulse-height distribution. These attachment-induced changes to the centroids (Figure 6) and sigma values of the Gaussian components can be accessed through the simulation framework described in Section 3.4.1. Using this approach, we can construct a fit function with the attachment coefficient being left as a free parameter. By fitting this model to the experimental pulse-height distributions, the EL yield, energy resolution and attachment coefficient can be obtained. A more detailed description of this methodology can be found in (Henriques, 2019; Henriques et al., 2019).

Using these roughly estimated attachment coefficients, we applied appropriate corrections to the previously measured EL yield curves using the simulated relationship between Y att / Y 0 and η established in Figure 11 (left). The corrected yield curves then allowed determination of scintillation probabilities using the methodology described in Section 3.4.2. Figure 13 presents the results of this re-analysis alongside theoretical predictions from the microscopic simulation framework described in (Azevedo et al., 2018). The concentration uncertainty stems from the Residual Gas Analyzer measurements, whose methodology was extensively detailed in (Henriques, 2019). In Xe- CF 4 and Xe- CO 2 mixtures, the attachment coefficient measurement is the main contribution for SP uncertainty, resulting in larger error bars.

The uncertainty analysis reveals that for CF 4 mixtures, where electron attachment is substantial, the error in the derived attachment coefficient dominates the uncertainty budget for scintillation probability. The concentration uncertainty stems from the Residual Gas Analyzer measurements, whose methodology was extensively detailed in (Henriques, 2019).

The results confirm our theoretical expectations: quenching of excited xenon states is substantial in both CO 2 and CH 4 mixtures but minimal or absent in CF 4 mixtures. While simulation predictions generally match experimental findings within uncertainty bounds, we observe that the model tends to underestimate scintillation probabilities in low-concentration CH 4 mixtures, suggesting potential refinements to the simulation framework may be necessary. These results align well with our current measurements presented in Section 3.4.2, which are also included in Figure 13 for direct comparison. Although the scintillation probabilities determined in Section 3.4.2 benefit from more precise attachment coefficient measurements, it is ultimately limited by low statistics in the EL yield data points, resulting in comparable error margins in the final scintillation probability values.