research papers
A new imaging technology based on Compton Xray scattering
^{a}Instituto Galego de Física de Altas Enerxías (IGFAE), Rúa de Xoaquín Díaz de Rábago, s/n, Campus Vida, 15782 Santiago de Compostela, Spain, ^{b}Synchrotron Radiation Research and NanoLund, Lund University, Box 118, 221 00 Lund, Sweden, and ^{c}I3N, Physics Department, University of Aveiro, Campus Universitário de Santiago, 3810193 Aveiro, Portugal
^{*}Correspondence email: angela.saa.hernandez@usc.es
A feasible implementation of a novel Xray detector for highly energetic Xray photons with a large solid angle coverage, optimal for the detection of Compton Xray scattered photons, is described. The device consists of a 20 cmthick Geant4 simulations, by considering a realistic detector design and response, it is shown that photon rates up to at least 10^{11} photons s^{−1} onsample (5 µm waterequivalent cell) can be processed, limited by the spatial diffusion of the photoelectrons in the gas. Illustratively, if making use of the Rose criterion and assuming the dose partitioning theorem, it is shown how such a detector would allow obtaining 3D images of 5 µmsize unstained cells in their native environment in about 24 h, with a resolution of 36 nm.
filled with xenon at atmospheric pressure. When the Comptonscattered photons interact with the xenon, the released photoelectrons create clouds of which are imaged using the produced in a custommade multihole acrylic structure. Photonbyphoton counting can be achieved by processing the resulting image, taken in a continuous readout mode. Based onKeywords: Compton scattering; hard Xrays; cell imaging; time projection chamber; electroluminescence.
1. Introduction
Despite some Xray facilities and experiments making use of Compton scattering to probe for instance the electronic and magnetic structure of materials (Sakurai, 1998; Tschentscher et al., 1998), the limited and (brightness) that is currently available at the required high energies (≳20 keV) seem to have precluded the popularization of these techniques. With the advent of the fourth generation of synchrotron light sources, such as ESRFEBS (Admans et al., 2014), the projected APSU (APS, 2019), Petra IV (Schroer et al., 2019) and SPring8II (Asano et al., 2014), as well as the proposal of novel facilities based on Xray freeelectron lasers (Huang, 2013), which increase the and coherent for hard Xrays by at least two orders of magnitude beyond today's capability, a unique opportunity arises to use Compton scattering in ways that were not conceived before. An example of these new possibilities is scanning Compton Xray microscopy (SCXM) (VillanuevaPerez et al., 2018). This technique has the potential of obtaining tens of nanometre resolution images of biological or radiosensitive samples without sectioning or labelling. Thus, it bridges the capabilities of optical and electron microscopes. Exploiting Compton interactions for biological imaging is possible because, in spite of its inelastic nature, the SCXM technique makes an optimal use of the number of scattered photons per unit dose, i.e. the deposited energy per unit of mass. Generally speaking, an efficient use of Compton scattering implies, first and foremost, that a nearly 4πcoverage is required (Fig. 1), at an optimal energy around 64 keV if aiming for instance at resolving DNA structures (VillanuevaPerez et al., 2018). This poses a formidable challenge for current detection technologies, which are costly and have detection areas much below the required size. Conversely, at lower Xray energies (≲10 keV), imaging based on coherent scattering has benefited from the development of ultrafast pixelated silicon detectors, capable of performing photoncounting up to 10^{7} counts s^{−1} pixel^{−1}. A nowadays typical detection area is 40 cm × 40 cm, sufficient for covering the coherent forward cone at a distance of about 1 m, at near 100% (Förster et al., 2019). At higher energies, silicon must be replaced by a semiconductor with a higher to Xrays, e.g. CdTe. However, targeting a geometrical acceptance around 70% at 64 keV, while providing enough space to incorporate a compact setup (namely the sample holder, step motor, pipes, shielding and associated mechanics), would imply an imposing active area for these type of detectors, well above 1000 cm^{2}. For comparison, PILATUS3 X CdTe 2M, one of the latest highenergy Xray detectors used at synchrotron sources, has an active area of 25 cm × 28 cm (DECTRIS, 2019). Clearly, the availability of a 4π/highenergy Xray detector would soon become an important asset at any nextgeneration facility, if it can be implemented in a practical way.
In this work we have implemented a novel approach for the detection of 4π Comptonscattered photons based on a technology borrowed from particle physics: the Electroluminescent Time Projection Chamber (ELTPC), discussing its performance as an SCX microscope. TPCs, introduced by D. Nygren in 1974 (Nygren, 1974, 2018), are nowadays ubiquitous in particle and nuclear physics, chiefly used for reconstructing particle interactions at high track multiplicities (Alme et al., 2010), and/or when very accurate event reconstruction is needed (Phan et al., 2016; Acciarri et al., 2016; GonzálezDíaz et al., 2018). The main characteristics of the particular TPC flavour proposed here can be summarized as: (i) efficient to highenergy Xrays thanks to the use of xenon as the active medium, (ii) continuous readout mode with a time sampling around ΔT_{s} = 0.5 µs, (iii) typical temporal extent of an Xray signal (at midchamber): ΔT_{xray} = 1.35 µs, (iv) about 2000 readout pixels/pads, (v) singlephotoncounting capability, and (vi) an energy resolution potentially down to 2% FWHM for 60 keV Xrays, thanks to the mode (Kowalski et al., 1989), only limited by the Fano factor F.^{1} Importantly, the distinct advantage of using instead of conventional avalanche multiplication is the suppression of ion space charge, traditionally a shortcoming of TPCs operated under high rates.
Our design is inspired by the proposal of Nygren (2007), that has been successfully adopted by the NEXT collaboration in order to measure neutrinoless doublebeta decay (Monrabal et al., 2018), but we include three main simplifications: (i) operation at atmospheric pressure, to facilitate the integration and operation at present Xray sources, (ii) removal of the photomultiplierbased energy plane, and (iii) introduction of a compact allinone structure, purposely designed for photoncounting experiments.
In this paper we discuss, starting from Section 2, the main concepts and working principles leading to our conceptual detector design. Next, in Section 3, we study the photoncounting capabilities of a realistic detector implementation. We present the expected performance when applied to the SCXM technique in Section 4. Finally, we assess the limits and scope of the proposed technology in Section 5.
2. TPC design
2.1. Dose and intrinsic resolving power
In a scanning, darkfield, configuration, the ability to resolve a feature of a given size embedded in a medium can be studied through the schematic representation shown in Fig. 2 (top), that corresponds to an arbitrary step within a twodimensional (2D) scan, in a similar manner as presented by VillanuevaPerez et al. (2018).
Three main assumptions lead to this simplified picture: (i) the dose fractionation theorem (Hegerl & Hoppe, 1976), based on which one can expect threedimensional (3D) reconstruction capabilities at the same resolution (and for the same dose) than in a single 2D scan, (ii) the ability to obtain a focal spot, d′, down to a size comparable with (or below) that of the feature to be resolved, d, and (iii) a depth of focus exceeding the dimensions of the sample under study, l. A possible technical solution to the latter two problems was introduced by VillanuevaPerez et al. (2018), targeting a 10 µm depth of focus at a 10 nm focal spot, thanks to the combination of multilayer Laue lenses (MLLs; Bajt et al., 2018) with a stack of negative refractive ones. Since that technique would enable any of the scenarios discussed hereafter, we adopt the situation in Fig. 2 (top) as our benchmark case, and we use the Rose criterion (Rose, 1946) as the condition needed to discern case f (feature embedded within the scanned volume) from case 0 (no feature), that reads in the Poisson limit as
with N being the number of scattered photons. Substitution of physical variables in equation (1) leads directly to a required fluence of
and we will assume d′ ≃ d. Here λ_{w}, λ_{f}, λ_{a} are the Comptonscattering mean free paths of Xrays in water, DNA and air (or helium), respectively (Table 1), and dimensions are defined in Fig. 2 (top). Finally, we evaluate the dose that will be imparted at the feature in these conditions as
where σ_{ph} is the photoelectric and is the differential for Compton scattering, both evaluated at the feature. M_{f} is the feature molar mass, N_{A} the Avogadro number, ɛ the photon energy and θ its scattering angle. The dose inherits the approximate l/d^{4} behaviour displayed in equation (2).

Working with equation (3) is convenient because it has been used earlier, in the context of coherent scattering, as a metric for assessing the maximum radiation prior to inducing structural damage (Howells et al., 2009). By resorting to that estimate [black line in Fig. 2 (bottom)], the doses required for resolving a feature of a given size can be put into perspective. These doses, obtained using Geant4 (Agostinelli et al., 2003) for a DNA feature embedded in a 5 µm waterequivalent cell, are shown as continuous lines. Results resorting to NIST values (Berger et al., 2010) and Hubbell parameterization for (Hubbell et al., 1975) are displayed as dashed lines, highlighting the mutual consistency in this simplified case. Clearly, SCXM can potentially resolve 33 nmsize DNA features inside 5 µm cells, and down to 26 nm if a stable He atmosphere around the target can be provided.
Using equation (3) as a valid metric for intercomparison between SCXM and coherent scattering is at the moment an open question and will require experimental verification. In particular, the formula assumes implicitly that the energy is released locally. However, a 10 keV photoelectron has a range of up to 2 µm in water, while a 64 keV one can reach 50 µm. An approximate argument can be sketched based on the fact that the average energy of a for 64 keV Xrays (in the range 0–14 keV) is similar to that of a 10 keV photoelectron stemming from 10 keV Xrays, a typical case in coherent diffraction imaging (CDI). Given that at 64 keV most (around 70%) of the energy is released in Compton scatters, the situation in terms of locality will largely resemble that of coherent scattering. Hence, compared with CDI, only about 30% of the energy will be carried away from the interaction region by the energetic 64 keV photoelectrons. On the other hand, at 30 keV (the other energy considered in this study) the photoelectric effect contributes to 90% of the dose, so one can expect a higher dose tolerance for SCXM than the one estimated here.
Naturally, the shielding pipes, the structural materials of the detector, the detector efficiency, the instrumental effects during the reconstruction, and the accuracy of the counting algorithms can limit the achievable resolution, resulting in dose values larger than the ones in Fig. 2. These effects are discussed in the following sections.
2.2. Technical description of the TPC working principle
When Xrays of energies of the order of tens of keV interact in xenon gas at atmospheric pressure, the released photoelectron creates a cloud of σ) size of 0.25–1 mm [Fig. 3 (top)]. If the Xray energy is above that of the xenon Kshell, characteristic emission around 30–34 keV will ensue, in about 70% of the cases. At these energies, Xray interactions in xenon take place primarily through the photoelectric effect, with just a small () probability of Compton scattering.
(containing thousands of electrons) with a typical (1The ionization clouds (hereafter `clusters') drift, due to the electric field E_{drift} of the TPC, towards the electroluminescence/anode plane, as shown in Fig. 4 (top), following a diffusion law as a function of the drift distance z,
where and are the longitudinal and transverse diffusion coefficients, respectively (McDonald et al., 2019). In fact, diffusion is impractically large in pure noble gases, given that the cooling of ionization electrons is inefficient under elastic collisions only. Addition of molecular additives, enabling vibrational at typical electron energies, is a well established procedure known to improve the situation drastically, and can be accurately simulated with the electron transport codes Magboltz/Pyboltz (Biagi, 1999; Al Atoum et al., 2020). In particular, a small (0.4%) addition of CH_{4} is sufficient to reduce the cluster size well below that in pure xenon [Fig. 3 (bottom)], as required for An essential ingredient to the use of Xe–CH_{4} admixtures is the recent demonstration that the signal is still copious in these conditions (Henriques, 2019).^{2} Hence, for a drift field E_{drift} = 110 V cm^{−1}, the cluster's longitudinal size can be kept at the σ_{z} = 4 mm level even for a 50 cmlong drift, corresponding to a temporal spread of σ_{t} = 0.75 µs, while the transverse size approaches σ_{x, y} = 10 mm.^{3} The electron drift velocity is v_{d} = σ_{z}/σ_{t} = 5 mm µs^{−1}.
The proposed detection concept is depicted in Fig. 4 (top), with Fig. 4 (bottom) displaying a closeup of the pixelated readout region, that relies on the recent developments on largehole acrylic multipliers (GonzálezDíaz et al., 2019). Provided sufficient field focusing can be achieved at the structure, as shown in Fig. 4 (bottom), the ionization clusters will enter a handful of holes, creating a luminous signal in the corresponding silicon photomultiplier (SiPM) situated right underneath, thus functioning, in effect, as a pixelated readout. In summary: (i) Xrays that Comptonscatter at the sample interact with the xenon gas and give rise to clusters of characteristic size somewhere in the range 1–10 mmσ, depending on the distance to the plane; (ii) given the relatively large Xray of around 20 cm in xenon at 1 bar, one anticipates a sparse distribution of clusters, that can be conveniently recorded with 10 mmsize pixels/pads, on a readout area of around 2000 cm^{2} (N_{pix} = 2000).
From the FWHM per Xray cluster at about midchamber: Δ_{x,y}_{xray} = = 16 mm, an average multiplicity M of around 4 per cluster may be assumed if resorting to 10 mm × 10 mm pixels/pads. The temporal spread, on the other hand, can be approximated by: ΔT_{xray} = = 1.35 µs.
Heuristically, by taking as a reference an interaction probability of P_{int} = 2.9 × 10^{−4} (5 µm waterequivalent cell, 10 mm of air), a 70% ε, and an m = 20% pixel occupancy, this configuration yields a plausible estimate of the achievable as
compatible a priori with the beam rates for hard Xrays foreseen at the new generation of light sources (Admans et al., 2014). However, in order to have a realistic estimate of the actual counting performance, it is imperative to understand which level of occupancy/pileup can be really tolerated by the detector, before the photoncounting performance deteriorates above the Poisson limit or proportionality of response is irreparably lost. We address this problem specifically in Section 3.
2.3. Geometry optimization with Geant4
The suitability of the TPC technology for SCXM depends primarily on the ability to detect ∼60 keV photons within a realistic gas volume, in the absence of pressurization. Given that the πgeometry adapting to this case is a hollow cylinder with a of around half a meter. On the other hand, the geometrical acceptance is a function of , with L being the length and R_{i} the inner radius of the cylinder. In order to place the sample holder, step motor, optics, pipes and associated mechanics, we leave an R_{i} = 5 cm inner bore.
of 60 keV Xrays in xenon is 20 cm, the most natural 4Finally, the xenon thickness (R_{o} − R_{i}), that is the difference between the outer and inner TPC radii, becomes the main factor for the detector efficiency, as shown in Fig. 5. We discuss two photon energies: 30 and 64 keV. The latter represents the theoretical optimum for SCXM in terms of dose, while the former, sitting just below the Kshell energy of xenon, is a priori more convenient for counting due to the absence of characteristic (Kshell) Xray reemission inside the chamber. The is similar for the two energies, therefore no obvious advantage (or disadvantage) can be appreciated in terms of detector efficiency, at this level of realism.
We consider now a realistic geometry, opting for an inner cylinder shell made out of 0.5 mmthick aluminium walls, with 2 mm HDPE (highdensity polyethylene), 50 µm kapton and 15 µm copper, sufficient for making the field cage of the chamber, that is needed to minimize fringe fields (inset in Fig. 6). The HDPE cylinder can be custommade and the kapton–copper laminates are commercially available and can be adhered to it by thermal bonding or epoxied, for instance. The external cylinder shell may well have a different design, but it has been kept symmetric for simplicity. We consider in the following a configuration that enables a good compromise in terms of size and flexibility: L = 50 cm and R_{o} = 25 cm. The geometrical acceptance nears in this case 80%. An additional 10 cm would be typically needed, axially, for instrumenting the readout plane and taking the signal cables out of the chamber, and another 10 cm on the cathode side, for providing sufficient isolation with respect to the vessel, given that the voltage difference will near 10 kV. Although those regions are not discussed here in detail, and have been replaced by simple covers, the reader is referred to Monrabal et al. (2018) for possible arrangements. With these choices, the vessel geometry considered in simulations is shown in Fig. 6, having a weight below 10 kg.
The necessary structural material of the walls and the presence of air in the hall reduce the overall efficiency from 62.8% to 58.5% (64 keV) and from 64.5% to 40.0% (30 keV). The beam enters the experimental setup from the vacuum pipes (not included in the figure) into two shielding cones (made of stainless steel and covered with lead shields) and from there into the sample region. Our case study is that of a 33 nm DNA feature inside a 5 µm cell, and 5 mm air to and from the shielding cones. The conical geometry is conceived not to crop the angular acceptance of the Xrays scattered onsample, providing enough space to the focusing beam, and enabling sufficient absorption of stray Xrays from beam–air interactions along the pipes. In a 4π geometry, as the one proposed here, the cell holder and step motor could be mounted over a rail system, ideally placed along the polarization axis (as provided by standard undulators), where the is lower (Fig. 1). Due to the small focal distance of the MLLs, they should also be placed inside the TPC. The horizontally and vertically focusing MLLs, with slightly different focal lengths, would focus the incident beam to the same spot downstream, where the sample is located, following the socalled `nanoprobe' configuration described by Murray et al. (2019). A simplified sketch of a possible setup is shown in Fig. 7.
2.4. Image formation in the TPC
The parameters used for computing the TPC response rely largely on the experience accumulated during the NEXT R&D program. We consider a voltage of −8.5 kV at the cathode and 3 kV across the E_{drift} = 110 V cm^{−1} and E_{el} = 6 kV cm^{−1} in the drift and regions, respectively. The gas consists of Xe/CH_{4} admixed at 0.4% in volume in order to achieve a 40fold reduction in cluster size compared with operation in pure xenon [Fig. 3 (bottom)]. The plane will be optically coupled to a SiPM matrix, at the same pitch, forming a pixelated readout. The optical coupling may be typically done with the help of a layer of ITO (indium–tin oxide) and TPB (tetraphenyl butadiene) deposited on an acrylic plate, following Monrabal et al. (2018). This ensures wavelength shifting to the visible band, where SiPMs are usually more sensitive. The number of SiPM photoelectrons per incoming ionization electron, n_{phe}, that is the single most important figure of merit for an ELTPC, can be computed from the layout in Fig. 4 (bottom), after considering: an Y = 250 photons e^{−1} cm^{−1} at E_{el} = 6 kV cm^{−1} (GonzálezDíaz et al., 2019), a TPB wavelengthshifting efficiency WLSE_{TPB} = 0.4 (Benson et al., 2018), a solid angle coverage at the SiPM plane of Ω_{SiPM} = 0.3 and a SiPM QE_{SiPM} = 0.4. Finally, according to measurements by Henriques (2019), the presence of 0.4% CH_{4} reduces the scintillation probability by P_{scin} = 0.5, giving, for a h = 5 mmthick structure,
structure, with the anode sitting at ground, a situation that corresponds to fields aroundSince the energy needed to create an electron–ion pair in xenon is W_{I} = 22 eV, each 30–64 keV Xray interaction will give rise to a luminous signal worth 4000–9000 photoelectrons (phe), spanning over 4–8 pixels, hence well above the SiPM noise. The energy resolution (FWHM) is obtained from Henriques (2019) as
with σ_{G}/G being the width of the singlephoton distribution (around 0.1 for a typical SiPM) and F ≃ 0.17 the Fano factor of xenon. For comparison, a value compatible with = 5.5% was measured for acrylic hole multipliers by GonzálezDíaz et al. (2019). In present simulations, the contribution of the energy resolution has been included as a Gaussian smearing in the TPC response.
Finally, the time response function of the SiPM is included as a Gaussian with a 7 ns width, convoluted with the transit time of the electrons through the ΔT_{EL} = 0.36 µs, being both much smaller in any case than the typical temporal spread of the clusters (dominated by diffusion). The is taken to be ΔT_{s} = 0.5 µs as in Monrabal et al. (2018), and a matrix of 1800 10 mmpitch SiPMs is assumed for the readout. Images are formed after applying a 10 phethreshold to all SiPMs.
structureA fully processed TPC image for one time slice (ΔT_{s} = 0.5 µs), obtained at a beam rate of r = 3.7 × 10^{10} photons s^{−1} for a photon energy ɛ = 64 keV, is shown in Fig. 8. The main clusters have been marked with crosses, by resorting to `Monte Carlo truth', i.e. they represent the barycenter of each primary ionization cluster in Geant4. The beam has been assumed to be continuous, polarized along the xaxis, impinging on a 5 µm water cube surrounded by air, with a 33 nm DNA cubic feature placed at its centre. The Geant4 simulations are performed at fixed time, and the Xray interaction times are subsequently distributed uniformly within the dwell time corresponding to each position of the scan. It must be noted that interactions taking place at about the same time may be recorded at different times depending on the zposition of each interaction (and vice versa, clusters originating at different interaction times may eventually be reconstructed in the same time slice). This scrambling (unusual under typical TPC operation) renders every time slice equivalent for the purpose of counting. In principle, the absolute time and z position can be disambiguated from the size of the cluster, using the diffusion relation in equation (4), thus allowing photonbyphoton reconstruction in time, space and energy. A demonstration of the strong correlation between zposition and cluster width, for 30 keV Xray interactions, can be found in GonzálezDíaz et al. (2015) for instance.
The design parameters used in this subsection are compiled in Tables 2–5 of Appendix B.
3. Photoncounting capabilities
3.1. Ideal counting limit
The attenuation in the structural materials, rescatters, characteristic emission, as well as the detector inefficiency, are unavoidable limiting factors for counting. These intrinsic limitations can be conveniently evaluated from the signaltonoise ratio (S/N), defined from the relative spread in the number of ionization clusters per scan step (see Fig. 2), as obtained by the Monte Carlo method (n_{MC}),
Figure 9 shows the deterioration of the S/N for 64 keV photons, as the realism of the detector increases. It has been normalized to the relative spread in the number of photons scattered onsample per scan step, , so that it equals 1 for a perfect detector (see Appendix A),
The figure also shows S/N^{*} in `calorimetric mode', with the counting performed by simply integrating the total collected light per scan step (ɛ_{tot}), instead of photonbyphoton. S/N^{*} is defined in that case, equivalently, as S/N^{*} = . The values obtained are just slightly below the ones expected considering detector inefficiency alone (see Appendix A),
therefore suggesting a small contribution from rescatters in the materials or other secondary processes.
3.2. Real counting
Given the nature of the detector data (Fig. 8), consisting of voxelized ionization clusters grouped forming ellipsoidal shapes, generally separable, and of similar size, we select the Kmeans clustering method (MacQueen, 1967) to perform cluster counting. The counting algorithm has been implemented as follows:
(i) The `countable' clusters are first identified timeslice by timeslice using Monte Carlo truth information, as those producing a signal above a certain ɛ_{th}) in that slice. The is chosen to be much lower than the typical cluster energies. In this manner, only small clusters are left out of the counting process when most of their energy is collected in adjacent timeslices from which charge has spread out due to diffusion, and where they will be properly counted once the algorithm is applied there.
((ii) A weighted inertia (I) distribution is formed, as conventionally done in Kmeans, and a threshold (δI_{th}) is set to the variation of the inertia with the number of clusters counted by the algorithm (n) (Fig. 10). The threshold is optimized for each beam rate condition. We concentrate on beam rates for which the average efficiency and purity of the cluster identification in 2D slides is larger than 80%, as the ones illustratively depicted in Fig. 11. The and purity can be used as evaluation criteria for cluster assignment quality. is defined as the number of correctly assigned clusters, n_{matched}, divided by the total number of true (or MC) clusters, n_{MC}, while purity is defined as the number of correctly assigned clusters divided by the clusters counted by the algorithm n. Thus, bad clustering has purity (and efficiency) values close to 0, while a perfect clustering has a purity (and efficiency) of 1,
The Kmeans optimization parameters have been chosen to simultaneously maximize the while achieving n ≃ n_{MC}, therefore ε_{counting} ≃ p_{counting}.
Figure 12 (top) shows the performance of the counting algorithm, presenting the average number of clusters counted per 2D slice as a function of beam rate, with ɛ_{th} and δI_{th} optimized for each case as described above (green line). Red lines indicate the predictions outside the optimized case, that illustrate the consistent loss of linearity as the beam rate increases. Figure 12 (bottom) shows the relative spread in the number of counted clusters σ_{n}/n, and comparison with Monte Carlo truth. These results can be qualitatively understood if recalling that, by construction, the threshold inertia is strongly correlated with the average number of clusters and its size. Therefore, a simple Kmeans algorithm will inevitably bias the number of counted clusters to match its expectation on I, if no further considerations are made. Therefore, once δI_{th} has been adjusted to a certain beam rate, there will be systematic overcounting for lower beam rates, and undercounting for higher ones, as reflected by Fig. 12 (top). In present conditions, a secondorder polynomial is sufficient to capture this departure from proportionality introduced by the algorithm. A similar (although subtler) effect takes place for the cluster distributions obtained slicebyslice, where this systematic overcounting–undercounting effect makes the cluster distribution marginally (although systematically) narrower, as seen in Fig. 12 (bottom). As a consequence, the directly related magnitude S/N^{*} [equations (8) and (9)] is not deteriorated by the counting algorithm. On the other hand, proportionality is lost, and its impact needs to be addressed, depending on the application. The particular case of SCXM is scrutinized in the next section.
Finally, the photoncounting efficiency [equation (11)] can be assessed through Fig. 13 (top), where it is displayed as a function of the beam rate on target. It can be seen how, for the case of 30 and 64 keV photons, its value exceeds 85% for rates up to 10^{11} photons s^{−1} and 0.5 × 10^{11} photons s^{−1}, respectively. At these high beam rates, counting capability suffers from event pileup while, at low beam rates, it is limited by the presence of lowenergy deposits (corresponding to Xray interactions for which most of the energy is collected in adjacent slices). It must be recalled, at this point, that a complete reconstruction requires combining 2D timeslices as the ones studied here, in order to unambiguously identify clusters in 3D. Given that each cluster extends over 4–6 slices due to diffusion, and clusters are highly uncorrelated, a 3D well above 90% can be anticipated in the above conditions.
4. Projections for SCXM
We propose the characterization of the ELTPC technology in light of its performance as a cellular microscope, through the study of the smallest resolvable DNAfeature (size d) as a function of the scan time (ΔT_{scan}). Justification of the following derivations can be found in Appendix A, starting with
Here R equals 5 under the Rose criterion and the ratedependent coefficient C_{l} < 1 depends on the deviation of the counting algorithm from the proportional response, its expression being given in Appendix A. Other magnitudes have already been defined. Since the smallest resolvable feature size (d^{†}) is ultimately determined by the dose imparted at it when structural damage arises [equation (3), Fig. 2], the necessary scan time to achieve such performance () can be readily obtained,
For a detector with finite efficiency, the value of d^{†} can be recalculated by simply accounting for the necessary increase in fluence (and hence in dose), as
that results in slightly deteriorated values compared with Fig. 2: d^{†} = 36 nm instead of d^{†} = 33 nm for ɛ = 64 keV, and d^{†} = 44 nm instead of d^{†} = 37 nm for ɛ = 30 keV.
The limiting scan time (i.e. above which structural damage will appear) can be hence assessed from the behaviour of equation (14) with beam rate, as shown in Fig. 13 (bottom). For 64 keV, the loss of linearity of the counting algorithm at high rates results in a turning point at 9.3 × 10^{10} photons s^{−1}, above which an increase in rate stops improving the ability to resolve an image. For 30 keV, due to the absence of characteristic emission, only about half of the clusters are produced and the optimum rate is found at a higher value, r = 1.6 × 10^{11}. The and purity in these conditions is in the range 82–84%.
It is now possible to evaluate equation (13) under different scenarios: (i) a relatively simple calorimetric mode (total energy is integrated), for which we assume a hard Xray beam rate typical of the new generation of synchrotron light sources as r = 10^{12} photons s^{−1}, and (ii) a ratelimited photonbyphoton counting scenario, for the optimum rates r = 9.3 × 10^{10} photons s^{−1} (64 keV) and r = 1.6 × 10^{11} photons s^{−1} (30 keV), obtained above. Values for C_{l}(r) are extracted from secondorder fits as discussed in Appendix A. The remaining parameters are common to both modes: S/N^{*} = 0.71, efficiency = 58.5% (64 keV), S/N^{*} = 0.63, = 40.0% (30 keV); finally we assume l = 5 µm, a = 5 mm, R = 5, with the mean free paths (λ) taken from Table 1. Results are summarized in Fig. 14. At 64 keV, the doselimited resolution d^{†} = 36 nm can be achieved in approximately 24 h while, at 30 keV, d^{†} = 44 nm is reached in just 8 h. In the absence of systematic effects, operation in calorimetric mode would bring the scan time down to ≤1 h in both cases, although abandoning any photonbyphoton counting capabilities.
5. Discussion
The results presented here illustrate the potential of the proposed technology for highenergy Xray detection (up to ∼60–70 keV) at highbrightness synchrotron light sources, in particular for cellular imaging. In deriving them, we have adopted some simplifications, that should be superseded in future work, and are analyzed here:
(i) Availability of photonbyphoton information. Cluster reconstruction with high efficiency and purity enables x, y, t + t_{drift} and ɛ determination, and arguably the interaction time t and z position can be obtained from the study of the cluster size, as it has been demonstrated for 30 keV Xrays at nearatmospheric pressure before (GonzálezDíaz et al., 2015). This can help at removing backgrounds not accounted for, as well as any undesired systematic effect (beam or detector related). Since this technique provides a parallaxfree measurement, the concept may be extended to other applications, e.g. Xray crystallography. The presence of characteristic emission from xenon will unavoidably create confusion, so if unambiguous correspondence between the ionization cluster and the parent Xray is needed, one must consider operation at ∼30 keV.
(ii) Data processing and realism. Performing photonbyphoton counting at a rate nearing 5 × 10^{7} photons s^{−1} over the detector (≡ 10^{11} photons s^{−1} over the sample), as proposed here, is a computer intensive task, that will require high transfer rates too. Despite the relatively high the pixel occupancy is about 20% only, in the most extreme conditions considered, so zerosuppression of data before streaming is necessary. In that case, if assuming about 10 bits per 3D energyvoxel, a pixel multiplicity of 4 (Δ_{x,y}_{xray} = 16 mm, pixel size = 10 mm), and about four time bins per photon (ΔT_{xray} = 1.35 µs, ΔT_{s} = 0.5 µs), a transfer rate of about 8 Gb s^{−1} can be inferred. This is in the order of the typical data rates of Gb s^{−1} produced in tomography experiments, for which gigabitfast readout systems have been developed (Mokso et al., 2017).
Optimizing the counting algorithm and its speed will need to be accomplished, ultimately, with real data. To this aim, the availability of parallel processing (for groups of timeslices, for instance) as well as the possibility of simultaneous operation in calorimetric mode are desirable features. In stable detector and beam conditions, a calorimetric measurement will suffice for counting. This will be studied in the near future through a dedicated experiment.
(iii) Simplicity and compactness. The detector geometry proposed here has been conceived as a multipurpose permanent station. A portable device, however, could simply consist of a cubic 25 cm × 25 cm × 25 cm vessel that may be positioned, e.g. on top of the sample (at a distance of about ∼5 cm). The geometry would thus have an overall efficiency around 30% for 64 keV photons. For SCXM, and given that S/N^{*} ≃ as shown in this work, a loss of efficiency can be almost fully compensated by means of the corresponding increase in beam rate, at the price of a deteriorated value for the dose limited resolution d^{†}. In this case, a value corresponding to d^{†} = 41 nm could be achieved in 12 h, for our test study.
(iv) Feasibility. The proposed technology comes from the realm of highenergy physics, with an inherent operational complexity. On the one hand, the necessary high voltage and purity levels have been demonstrated in NEXTDEMO (Álvarez et al., 2013), that operates at 10 bar, conditions for which the technical specifications are much harsher than for the proposed detector (×10 higher operating voltage, ×100 less O_{2} contamination). On the other hand, adapting the SiPM readout seems to present some additional difficulties, related to design, prototyping and testing, needed to produce a rugged and stable readout structure, besides the need to develop waveform processing algorithms as well as a customized data acquisition system. While it is possible to build on the existing NEXT experience, it seems just timely to consider an alternative that largely simplifies the above aspects, by resorting to ultrafast (1.6 ns resolution) hitbased TimePix cameras (e.g. Amsterdam Scientific Instruments, 2019; Nomerotski, 2019). The camera would be coupled to a suitable VUV lens so a larger scintillating area can be fully imaged in small sensor, allowing 256 × 256 pixel readout at 80 Mhits s^{−1}, more than sufficient for this application. Largevolume optical TPCs have been read out with this concept already in ARIADNE (Lowe et al., 2020). The vessel would just house, in such a case, the acrylic hole multiplier and cathode mesh, together with the power leads; it would be filled with the xenon mixture at atmospheric pressure and interfaced to the outside with a VUVgrade viewport. This would compromise partly the ability to disentangle clusters by using time information, as well as energy information, since only the time over threshold would be stored and not the temporal shape of each cluster, or its energy. On the other hand, it would enhance the spatial information by a factor of 30 relative to the SiPM matrix proposed here (the hole pitch of the acrylic hole multiplier should be reduced accordingly). Indeed, TimePix cameras are regularly used nowadays for photon and ioncounting applications (Hirvonen et al., 2017; FisherLevine et al., 2018), but have not been applied to Xray counting yet, to the best of our knowledge. The counting and signal processing algorithms could be in this way directly ported, given the similarity with the images taken in those applications. The readiness of such an approach, aiming at immediate implementation, represents an attractive and compelling avenue.
The imaging criterion and study case chosen in this work are inspired by VillanuevaPerez et al. (2018), where a doselimited resolution of 34 nm was obtained for SCXM, compared with around 75 nm for CDI. A typical biomolecule feature was chosen, embedded in a 5 µm cell placed in vacuum. The present study shows that a 36 nm DNA feature can be resolved in similar conditions even after accounting for the presence of beamshielding, air, photon transport through a realistic detector, including the detector response in detail, and finally implementing photoncounting through a Kmeans algorithm.
6. Conclusions and outlook
We introduce a new 4πtechnology (ELTPC) designed for detecting ∼60 keV Xray photons at rates up to 5 × 10^{7} photons s^{−1} over the detector (10^{11} photons s^{−1} over the sample), with an overall (including geometrical acceptance) around 60%. At these rates, photonbyphoton counting can be achieved at an efficiency and purity above 80%, and plausibly well above 90% after straightforward improvements on the counting algorithm employed in this work. The technology has been repurposed from its original goal in particle physics (the experimental measurement of ββ0ν decay) and, with a number of simplifications, it has been optimally adapted to the task of Compton Xray microscopy in upcoming light sources. The proposed detector can be used either as a permanent station or a portable device. In the latter case, it is possible to combine Compton detection with other modalities, e.g. Xray diffraction measurements in the forward direction and perpendicular to the beam, where Compton scattering is minimal.
Concentrating on 5 µm cells as our test case, we estimate that, under the Rose imaging criterion, and assuming the dose fractionation theorem, 36 nm DNA features may be resolved in 24 h by using a permanent station, and 41 nm in 12 h with a portable device. Alternatively, the scan time could be brought down to less than 1 h by resorting to the calorimetric mode, although the photonbyphoton counting capability would need to be abandoned. Our analysis includes detailed Geant4 transport, a realistic detector response and a simplified 2Dcounting algorithm based on Kmeans. Thus, the obtained rate capability (and scan time) should be understood as lower (upper) limits to the actual capabilities when using more refined 3Dalgorithms, including constraints in energy and cluster size.
Although substantially below the nominal photoncounting capabilities of solidstate pixelated detectors, we believe that a number of applications could benefit from the proposed development, targeting specifically at the newly available fourthgeneration synchrotron light sources, capable of providing highbrightness hard Xrays. Indeed, previous conceptual studies point to about a factor of two increase in
for SCXM compared with CDI, in similar conditions to ours. The present simulation work just comes to support the fact that a complete 3D scan would be realizable in about 24 h time, under realistic assumptions on the experimental setup, detector response and counting algorithms.APPENDIX A
Relation between resolution and scan time
A1. Proportional (ideal) case
We start from the imaging criterion, applied to an arbitrary position of the step motor within a cellscan,
where R = 5 corresponds to the Rose condition. N_{f} is the number of scattered photons from a water medium with a `toberesolved' feature inside it, and N_{0} contains only water, instead [see Fig. 2 (top)]. This equation can be reexpressed as
that, under the assumption N_{f} ≳ N_{0}, and defining the signaltonoise ratio as S/N ≃ can be rewritten, in general, as
When considering n, standard deviation σ_{n}) and the distribution of scattered photons (mean N_{f} ≃ N_{0}, standard deviation ≃ ). If resorting to an unbiased counting algorithm, this relation will be proportional. In that case, the prefactors on the lefthandside of equation (19) remain, and any detectorrelated effect is contained in the quantity
it is understood that a relation can be established between the distribution of ionization clusters that are counted in the detector (meanAt fixed number of scattered photons (≃ N_{0}) the relative fluctuations in the number of counted clusters will increase due to efficiency losses, characteristic emission, and rescatters on the cell itself, air or structural materials, thereby resulting in a loss of signaltonoise. It is convenient to normalize this definition to the Poisson limit for a perfect detector,
and so the new quantity S/N^{*} is now defined between 0 and 1, with S/N = n/σ_{n} obtained, in the main document, from detailed simulations of the photon propagation through the experimental setup. Substitution of N_{f} and N_{0} by physical quantities in equation (19) yields
with d being the feature size, l the cell dimension, and λ_{f,w,a} the mean free paths in the feature, water and air, respectively, as defined in the text.
Now, we make use of the fact that N_{0} = , with r being the beam rate, ΔT_{step} a time step within the scan, and ΔT_{scan} the total time for a 2D scan: = . By replacing N_{0} in the previous equation we obtain
from which the time needed for a complete 2D scan can be expressed as
and, solving for d,
Expression (25) can be approximated under the simplifying assumption that S/N^{*} is mainly limited by Poisson statistics and by the efficiency of the detector (modelled through a simple binomial distribution), disregarding production of secondary particles or rescatters across structural materials, hence
from which it can be seen that detector efficiency and beam rate enter as a product in the denominator in formulas (24) and (25). Consequently, detector inefficiency increases the scan time linearly, as intuitively expected.
A2. Nonproportional case
We consider now the more realistic case where there is a nonproportional response of the counting algorithm. This is characterized, for the Kmeans algorithm implemented in the text, as a secondorder polynomial (Fig. 11),
By analogy, if the Kmeans parameters are optimized for a certain beam rate, r, the response to cell regions causing a different number of scattered photons N, relative to the wateronly case, will be
and a(r), b(r), c(r) are now ratedependent. Equation (19) should be rewritten, accordingly, as
and the relative variation in n becomes
that, for N_{f} ≃ N_{0}, can be reexpressed as
with C_{l}(r) = (b+2c)/(a+b+c). Hence, a loss of linearity during the counting process enters linearly in equation (19). The general expression for the resolvable feature size as a function of the beam rate is, finally, by analogy with equation (25),
that is the expression used in the main document, for the achievable resolution as a function of the scan time, under a given imaging criterion R. The detector response enters this final expression in three ways:
(i) Through the increased fluctuation in the number of detected clusters, relative to the ideal (Poisson) counting limit, characterized through the signaltonoise ratio, S/N^{*}.
(ii) The nonlinearity of the counting algorithm, C_{l}.
(iii) The assumed maximum operating rate, r, for which the product C_{l}^{ 2} r reaches a maximum, as for larger rates stops improving the ability to resolve an image.
APPENDIX B
ELTPC parameters
In Tables 2, 3, 4 and 5 we compile the main parameters used for the simulation of the TPC response, together with additional references when needed.



Footnotes
^{1}A nonzero value of F stems from the the intrinsic spread of primary ionization, as the partition of energy between excitations and ionizations changes event by event.
^{2}This unanticipated result, that might not look significant at first glance, results from a very subtle balance between the quenching of the xenon and the cooling of drifting electrons through inelastic collisions (Azevedo et al., 2018).
^{3}In the following we use σ_{x,y,z,t} to refer to the cluster width arising from diffusion after L = 50 cm drift.
Acknowledgements
We thank Ben Jones and David Nygren (University of Texas at Arlington), as well as our RD51 colleagues for stimulating discussions and encouragement, and specially to David José Fernández, Pablo Amedo, and Pablo Ameijeiras for discussions on the Kmeans method, and Damián García Castro for performing the Magboltz simulations.
Funding information
A. Saá Hernández is funded through the project ED431F 2017/10 (Xunta de Galicia) and D. GonzálezDíaz through the Ramon y Cajal program, contract RYC201518820. C. D. R. Azevedo is supported by Portuguese national funds (OE), through FCT  Fundação para a Ciência e a Tecnologia, IP, in the scope of the Law 57/2017, of 19 July.
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