A new imaging technology based on Compton X-ray scattering
Ángela Saá Hernández, Diego González-Díaz, Pablo Villanueva, Carlos Azevedo, Marcos Seoane
AA new imaging technology based on Compton X-ray scattering
Ángela Saá Hernández ∗ , Diego González-Díaz, Marcos Seoane Instituto Galego de Física de Altas Enerxías (IGFAE)Rúa de Xoaquín Díaz de Rábago, s/n, Campus Vida, 15782 Santiago de Compostela, Spain
Carlos Azevedo
I3N, Physics Department, University of AveiroCampus Universitário de Santiago, 3810-193 Aveiro, Portugal
Pablo Villanueva
Department of Physics, Lund UniversityP.O. Box 118, SE-22100 Lund, Sweden
Abstract
We describe a feasible implementation of a cellular microscope based on Compton X-ray scattering. The device, consistinglargely of a 25 cm-thick sensitive volume filled with xenon at atmospheric pressure, forms photoelectron images by resorting to theelectroluminescence produced in a custom multi-hole acrylic structure. Photon-by-photon counting can be achieved by processing theresulting images, taken in a continuous readout mode. The concept is amenable to permanent on-site π -coverage stations, but can bemade portable at an acceptable performance compromise, targeting a nearly π -coverage instead. Based on Geant4 simulations, anda realistic detector design and response, we show that photon rates up to around ph/s on-sample ( µ m water-equivalent cell)can be processed, limited by the spatial diffusion of the photoelectrons in the gas. Following the Rose criterion and assuming the dosepartitioning theorem, such a detector would allow obtaining 3d images of µ m unstained cells in their native environment in about20 h, with a resolution below 40 nm. I. I ntroduction
Recent work has shown that the use of incoherent (Comp-ton) scattering offers a plausible path, at fourth-generationsynchrotron Light Sources, towards obtaining 10’s of nm-accurate three-dimensional images of microscopic biologi-cal systems, before inducing structural damage. Notably,and despite its inelastic nature, the proposed ScatteringCompton X-ray Microscopy (SCXM) makes an optimal useof the number of scattered photons per unit of depositedenergy, contrary to coherent scattering at low energies,which is limited by photoelectric effect at the sample. Asa result, the back-to-back comparison performed in [1] re-vealed that a 34 nm biomolecular feature embedded in a5 µ m-cell may be resolved at a much reduced surface dose:three orders of magnitude less than the one needed forleading techniques currently used at Light Sources, suchas coherent diffraction imaging (CDI). Ultimately, whenbench-marking SCXM against CDI under the same imag-ing criterion, a factor of two improvement in the achievablespace resolution was obtained, due to a more favourablecell size ( l ) over feature size ( d ) scaling of the needed dose: ∗ Corresponding author: [email protected]
D ∼ l / d for SCXM compared to D ∼ l / d for CDI. Re-sults for the case study chosen in this work, illustratingthese observations, are presented in Fig. 1.Given that in SCXM virtually all interactions are used,a nearly 4 π -coverage is called for (Fig. 2), at an optimalenergy around 64 keV if aiming at DNA structures [1].This poses a formidable challenge for current detectiontechnologies. At lower X-ray energies, imaging based oncoherent scattering has benefited from the development ofultra-fast pixelated silicon detectors, capable of performingphoton-counting up to 10 counts/s/pixel. A nowadaystypical detection area is 20 ×
20 cm , sufficient for coveringthe coherent forward cone at a distance of about 1 m, atnear 100% quantum efficiency [2]. At higher energies, sili-con must be replaced by a semi-conductor with a higherstopping power to X-rays, e.g., CdTe. However, targeting ageometrical acceptance around 70% at 64 keV, while provid-ing enough space for the sample holder, pipes, shieldingand associated mechanics, would imply an imposing activearea for these type of detectors, well above 100 cm andpossibly up to 1000 cm . For comparison, PILATUS3 XCdTe, one of the latest high-energy X-ray detectors usedat synchrotron sources, has an active area of 30 cm [3].1 a r X i v : . [ phy s i c s . i n s - d e t ] J un new imaging technology based on Compton X-ray scattering l aa dd' case 0case fair (or helium)cellX-ray beam DNAscan
10 20 30 40 50 60 70 80 90 100 feature size [nm] r e q u i r e d d o s e [ G y ] max. tolerable dose = 30 keV, He = 64 keV, He = 30 keV, air = 64 keV, air Fig. 1.
Top: sketch of the case study chosen in this work,consisting of a cubic DNA feature of size d embedded ina cubic water cell ( l = µ m), surrounded by air/helium( a = d (cid:48) ) is roughly constant throughoutthe sample under study. Bottom: dose needed to resolvea DNA feature as a function of its size, for X-ray ener-gies of 30 keV and 64 keV, obtained respectively withGeant4 [11] (solid lines) and using NIST values [16] (dot-ted line), and the formulas in text. The maximum surfacedose at the feature before structural damage, estimatedfrom coherent scattering experiments in [15], appearsoverlaid. An ideal detector with a 100% photon coveragehas been assumed.Clearly, the availability of a 4 π /high energy X-ray detectorwould allow to exploit the potential of SCXM, if it can beimplemented in a practical way.This situation has motivated us to consider a deviceborrowed from particle physics: the electroluminescentxenon Time Projection Chamber (EL-TPC), and discuss itsperformance as an SCX-microscope. TPCs, introduced byD. Nygren in 1974 [4] are nowadays ubiquitous in particleand nuclear physics, chiefly used for reconstructing inter-actions at high track multiplicities [5], and/or when veryaccurate event reconstruction is needed [6, 7]. A recentreview on the TPC technology by one of us can be found in[8]. The main characteristics of the particular TPC-flavourproposed here can be summarized as: i) efficient to highenergy X-rays thanks to the use of xenon as the activemedium, ii) continuous readout mode with a time sam- (deg) d / d [ b / s r ] integrated over (scaled x10)
85 < < 95 (scaled x10)
Fig. 2.
Differential cross section for Compton-scatteredphotons on DNA (in barn per stereoradian), for a linearlypolarized X-ray beam of 64 keV as obtained with Geant4(histogram) and Hubbell parameterization (dashed lines),for different azimuthal regions: φ = [ − ] ◦ (green), φ = [ − ] ◦ (blue) and integrated over φ (red).pling around ∆ T s = µ s, iii) typical temporal extent ofan X-ray signal (at mid-chamber): ∆ T x − ray = µ s, iv)about 2000 readout pixels/pads, v) single-photon countingwith a Fano-limited energy resolution potentially down to2% FWHM for 60 keV X-rays, thanks to the electrolumi-nescence mode. Importantly, the main advantage of us-ing electroluminescence instead of conventional avalanchemultiplication is the suppression of ion space charge, tradi-tionally a shortcoming of TPCs under high rates.Our design is inspired by the proposal in [9], that hasbeen successfully adopted by the NEXT collaboration tomeasure neutrino-less double-beta decay [10]. However,compared to that, we propose three main simplificationshere: i) operation at atmospheric pressure, ii) removal ofthe photomultiplier-based energy-plane, and iii) introduc-tion of a compact all-in-one electroluminescence structure,purposely designed for counting. Armed with detailedGeant4 [11] simulations, we discuss first in section II themain ideas and working principles leading to the concep-tual design of the microscope; second, in section III, weintroduce the TPC response and propagate the ionizationclusters (stemming from the scattered X-rays) till the for-mation of 2d-images; finally, we discuss the counting per-formance, relying on custom algorithms. We present hencean assessment on the limits and scope of the proposedtechnology in section IV. II. TPC design i. Dose and intrinsic resolving power
The ability to resolve a feature of a given size embeddedin a medium can be studied through the schematic rep-resentation shown in Fig. 1-top, that corresponds to anarbitrary step within a 2d-scan for a specific sample-beam2 new imaging technology based on Compton X-ray scatteringorientation (perpendicular). Three main assumptions leadto this simplified picture: i) the dose fractionation theorem[12], based on which one can expect 3d reconstruction ca-pabilities at the same resolution (and for the same dose)than in a single 2d-scan: the total 3d-scan time is then sim-ply re-distributed among n orientations each at a fractionaldose D / n ; ii) the ability to obtain a focal spot, d (cid:48) , downto a size comparable to (or below) that of the feature tobe resolved, d , and iii) a depth of focus exceeding that ofthe sample under study, l . A possible technical solution tothe latter two problems was introduced in [1], targeting a10 µ m depth of focus at a 10 nm focal spot, thanks to thecombination of multi-layer Laue lenses [13] with a stackof negative refractive ones. Since that technique wouldenable any of the scenarios discussed hereafter, we adoptthe situation in Fig. 1-top as our benchmark case. Further,we use the Rose criterion [14] as the condition needed todiscern case f (feature embedded in the sample) from case0 (no feature), that reads in the Poisson limit as: | N f − N | (cid:113) σ N f + σ N = | N f − N | (cid:113) N f + N ≥ N being the number of scattered photons. Substi-tution of physical variables in eq. 1 leads directly to arequired fluence of: φ ≥ φ min = ( l − d ) · λ − w + d · λ − f + · a · λ − a d (cid:48) · d · ( λ − f − λ − w ) (2)and we assume d (cid:48) (cid:39) d . Here λ w , λ f , λ a are the Compton-scattering mean free paths of X-rays in water, DNA, andair (or helium), respectively (table 1), and dimensions aredefined in Fig. 1-top. Finally, we evaluate the surface dose that will be imparted at the feature in these conditions as: D = φ min · ε · N A M f · (cid:34) σ ph + (cid:90) d σ C d Ω · ( − + ε m e c ( − cos θ )) d Ω (cid:35) (3)where σ ph is the photoelectric cross section and d σ C / d Ω isthe differential cross section for Compton scattering, bothevaluated at the feature. M f is the feature molar mass, N A the Avogado number and ε the photon energy. Thedose inherits the approximate l / d behaviour displayed inequation (2). Table 1.
Mean free path for different materials at thestudied energies 30 and 64 keV, according to NIST.
Mean free path[cm] ε = 30keV ε = 64keV Material λ w λ f λ a µ m water-equivalent cell using Geant4,are shown as continuous lines. Results resorting to NISTvalues [16] and Hubbell parameterization for d σ C / d Ω [17]are displayed as dashed lines, highlighting the mutualconsistency in this simplified case. Clearly, SCXM canpotentially resolve 33 nm-size DNA features inside 5 µ mcells, and down to 26 nm if a stable He atmosphere aroundthe target can be provided.Using surface dose as a valid metric for inter-comparisonbetween SCXM and CDI is at the moment an open questionand will require experimental verification. In particular,eq. 3 assumes implicitly that the energy is released locally.However, a 10 keV photoelectron has a range of up to200 nm in water, while a 64 keV one can reach up to1.5 µ m. An approximate argument can be sketched basedon the fact that the average energy of a Compton electronfor 64 keV X-rays (in the range 0-14 keV) is similar to thatof a 10 keV photo-electron stemming from 10 keV X-rays,a typical case for CDI. Given that at 64 keV most (around70%) of the energy is released in Compton scatters, thesituation in terms of locality will largely resemble that ofcoherent scattering. Hence, compared to CDI, only about30% of the energy will be carried away from the interactionregion by the energetic 64 keV photoelectrons. On the otherhand, at 30 keV (the other energy considered in this study)the photoelectric effect contributes to 90% of the surfacedose, so one can expect a higher dose tolerance for SCXMthan the one estimated here.Naturally, shielding pipes, structural materials of thedetector, its efficiency, instrumental effects during the re-construction and the accuracy of the counting algorithmswill limit the achievable resolution, resulting in dose valueslarger than the ones in Fig. 1. These effects are discussedin the next sections. ii. Technical description of the TPC workingprinciple When X-rays of energies of the order of 10’s of keV interactin xenon gas at atmospheric pressure, the released photo-electron creates a cloud of secondary ionization ( O ( (cid:48) s ) electrons) with a typical (1 σ ) size of 0.25-1 mm (Fig. 3-top).If the X-ray energy is above that of the xenon K-shell, char-acteristic emission around 30-34 keV will ensue, in about70% of the cases. At these energies, X-ray interactions inxenon take place primarily through photoelectric effect,with just a small ( (cid:46) ab
50 100 150 200 250 300 350 400 electric field [V/cm] t e m po r a l [ s ] t r a n s v e r s e [ mm ] cluster size after charge collection pure xenonXe/CH (99.7/0.3) ab (99.6/0.4) Fig. 3.
Top(a): ionization distributions in xenon gas,stemming from X-rays interacting in an infinite volume.They are obtained after aligning each X-ray ionizationcloud by its barycenter, and projecting it over an arbi-trary axis. Calculations from Geant4 (green, orange) arecompared with the microscopic code DEGRAD devel-oped by S. Biagi [23]. Top(b): probability of characteristicX-ray emission in xenon for an incident photon energyof 30 keV (red) and 64 keV (blue), in Geant4. The K-shell(green) and L-shell (orange) lines, as tabulated in [24],are shown for comparison. Bottom(a): transverse size ofa point-like ionization cluster after drifting along 50 cm,obtained from Magboltz. Bottom(b): longitudinal size ofa point-like ionization cluster (in time units), in the sameconditions. Results for pure xenon and a fast ’counting’mixture based on Xe/CH are shown for comparison.The ionization clouds (hereafter ’clusters’) drift, due to the collection field of the TPC, towards the electrolumi-nescence/anode plane, as shown in Fig. 4-top, following adiffusion law as a function of the drift distance z : σ z ( x , y ) = D ∗ L ( T ) √ z (4)where D ∗ L and D ∗ T are the longitudinal and transverse dif-fusion coefficients, respectively. Simulations performedwith the electron transport code Magboltz [18, 19], indicatethat a small addition of CH will reduce the cluster sizewell below that in pure xenon, as required for photon-counting (Fig. 3-bottom). Recent work has demonstratedthat the electroluminescence signal is still copious in theseconditions [20]. For a collection field E c =
110 V/cm, thecluster’s longitudinal size can be kept at the σ z = σ t = µ s, while the transverse size willapproach σ x , y =
10 mm. The electron drift velocity wouldbe hence v d = σ z / σ t = µ s.The proposed detection concept is shown in Fig. 4-top,with Fig. 4-bottom displaying a close-up of the pixelatedreadout region, that relies on the recent developments onlarge-hole acrylic multipliers [21]. Provided sufficient fieldcollection can be achieved at the structure, as shown inFig. 4-bottom, the ionization clusters will enter a handfulof holes, creating a luminous signal in the correspondingsilicon photomultiplier (SiPM) situated right underneath,thus functioning as a pixelated readout. In summary: i)X-rays that Compton-scatter at the sample interact withthe xenon gas and give rise to clusters of characteristicsize somewhere in the range 1-10 mm- σ , depending onthe distance to the electroluminescence plane; ii) giventhe relatively large X-ray mean free path of around 20 cmin xenon at 1 bar, one anticipates a sparse distribution ofclusters, that can be conveniently recorded with 10 mm-size pixels/pads, on a readout area of around 2000 cm ( N pix = ∆ x , y | x − ray = √ · σ x , y =
16 mm, an average mul-tiplicity M of around 4 per cluster may be assumed ifresorting to 10 mm ×
10 mm pixels/pads. The tempo-ral spread, on the other hand, can be approximated by: ∆ T x − ray = √ · σ z / v d = µ s. Taking as a referencean interaction probability of P int = × − (5 µ m water-equivalent cell, 10 mm of air), a 70% detection efficiency (cid:101) ,and an m =
20% pixel occupancy, this configuration yieldsa plausible estimate of the achievable counting rate as: r max = (cid:101) P int m · N pix M ∆ T x − ray = × s − (5)compatible a priori with the beam rates foreseen at next-generation Light Sources [25]. However, in order to have a This unanticipated result, that might not look significant at firstglance, results from a very subtle balance between the quenching of thexenon triplet state and the cooling of drifting electrons through inelasticcollisions [22]. E drift 𝛥 t= 𝛥 z/v d 𝛾 phe phe phe t E [kV/cm]10mm TPB coatingITO coatingacrylic plateSiPMs
Fig. 4.
Top: schematic representation of the workingprinciple of the EL-TPC. Photons scattered at the sam-ple reach the xenon gas, creating ionization clusters thatdrift, while diffusing, towards the anode plane, wherethey induce electroluminescence. Bottom: close-up ofthe electroluminescence region, based on the recentlyintroduced acrylic-based electroluminescence multipli-ers, developed in collaboration between IGFAE and theCERN-RD51 workshops [21].realistic estimate of the actual counting performance it isimperative to understand which level of occupancy/pile-up can be really tolerated by the detector, before the photon-counting performance deteriorates above the Poisson-limitor proportionallity of response is lost. We address thisproblem specifically in section III. iii. Geometry optimization with Geant4
The suitability of the TPC technology for SCXM dependsprimarily on the ability to detect ∼
60 keV photons withina realistic gas volume, in the absence of pressurization.This can be anticipated, given that the mean free path of60 keV X-rays in xenon is 20 cm. Therefore, the most natu-ral 4 π -geometry adapting to this case is a hollow cylinderwith a characteristic scale of around half a meter. On the other hand, the geometrical acceptance is a function ofarctan ( R i / L ) , with L being the length and R i the innerradius of the cylinder. In order to place the sample holder,step motor, pipes and associated mechanics, we leave an R i = R o - R i ),that is the difference between the outer and inner TPCradii, becomes the main factor for the detector efficiency,as shown in Fig. 5. We discuss two photon energies: 30and 64 keV. The latter represents the theoretical optimumfor SCXM in terms of dose, while the former, sitting justbelow the K -shell energy of xenon, is more convenient forcounting due to the absence of characteristic X-ray emis-sion inside the chamber. The mean free path is similar forthe two energies, therefore no obvious advantage (or disad-vantage) can be appreciated in terms of detector efficiency.
10 20 30 40 50
Xe thickness [cm] E ff i c i e n c y ( % ) = 30 keV, L=25cm = 30 keV, L=50cm = 30 keV, L=100cm = 64 keV, L=25cm = 64 keV, L=50cm = 64 keV, L=100cm Fig. 5.
Efficiency as a function of the thickness of thexenon cylinder ( R i - R o ) for different lengths, at ener-gies of 30 and 64 keV. The dotted line indicates thebenchmark geometry considered in text, for a length L =
50 cm.We consider a realistic (and realizable) geometry, opt-ing for an inner cylinder shell made out of 0.5 mm-thickaluminum walls, with 2 mm HDPE (high density polyethy-lene), 50 µ m kapton and 15 µ m copper, sufficient for mak-ing the field cage of the chamber, that is needed to min-imize fringe fields (inset in Fig. 6). The HDPE cylindercan be custom-made and the kapton-copper laminates arecommercially available and can be adhered to it by thermalbonding, bolting, or even epoxied, for instance. The exter-nal cylinder shell may well have a different design, but wehave kept it symmetric for simplicity. We consider in thefollowing a configuration that enables a good compromisein terms of size and flexibility: L =
50 cm and R o =
25 cm.Additional 10 cm will be needed, axially, for instrumentingthe readout plane and taking the signal cables out of thechamber, and another 10 cm on the cathode side, for pro-viding sufficient isolation with respect to the vessel, giventhat the voltage difference will near 10 kV. Although thoseregions are not discussed here in detail, and have been5 new imaging technology based on Compton X-ray scatteringreplaced by simple covers, the reader is referred to [10] forpossible arrangements. With these choices, the geometryconsidered in simulations is shown in Fig. 6, having aweight below 10 kg.The necessary structural material of the walls and thepresence of air reduce the efficiency from 62.8% to 58.5%(64 keV) and 64.5% to 40.0% (30 keV). The beam enters theexperimental setup from the vacuum pipes (not included inthe figure) into two shielding cones (made of stainless steeland covered with lead) and from there into the sample re-gion. Our case study is that of a 33 nm DNA feature insidea 5 µ m cell, and 5 mm air to and from the shielding cones.The conical geometry is conceived not to crop the angularacceptance of the X-rays scattered on-sample, providingenough space to the focusing beam, and enabling sufficientabsorption of stray X-rays from beam-air interactions alongthe pipes. In a 4 π geometry as the one proposed here, thecell holder and step motor should be placed at 90 degrees,ideally along the polarization axis, where the photon fluxis negligible.
25 cm50 cm Lead shieldingSteel shieldingAluminum (0.5 mm)High densitypolyethylene (2 mm)Kapton (50 µ m)Copper (15 µ m)xy zA BC Fig. 6.
A) TPC geometry in Geant4, aimed at providingnearly 4 π -coverage for SCXM. B) detail of the regionfaced by X-rays when entering the detector, that includesthe vessel and field cage. C) detail of the sample regionand the shielding cones. iv. Image formation in the TPC The parameters used for computing the TPC responserely largely on the experience accumulated during theNEXT R&D program. We consider a voltage of -8.5 kVat the cathode and 3 kV across the electroluminescencestructure, with the anode sitting at ground, a situation thatcorresponds to fields around E c =
110 V/cm and E el = admixed at 0.4%in volume in order to achieve a 40-fold reduction in cluster size compared to operation in pure xenon (Fig. 3-bottom).The electroluminescence plane will be optically coupledto a SiPM matrix, at the same pitch, forming a pixelatedreadout. The optical coupling may be done with the helpof a layer of ITO (indium-tin oxide) and TPB (tetraphenylbutadiene) deposited on an acrylic plate, following [10].This ensures wavelength shifting to the visible band, whereSiPMs are usually more sensitive. The number of SiPM-photoelectrons per incoming ionization electron, n , thatis the single most important figure of merit for an EL-TPC, can be computed from the layout in Fig. 4-bottom,after considering: an optical yield Y =
250 ph/e/cm at E el = W LSE
TPB = Ω SiPM = QE SiPM = reduces the scintillationprobability by P scin = h = n = Y · h · W LSE
TPB · Ω SiPM · QE SiPM · P scin = W I =
22 eV, each 30-60 keV X-ray interaction willgive raise to a luminous signal worth 4000-8000 photoelec-trons (phe), spanning over 4-8 pixels, hence well above theSiPM noise. The expected energy resolution (FWHM) canbe approximated by: R ( ε =
60 keV ) (cid:39) (cid:118)(cid:117)(cid:117)(cid:116) F + n (cid:32) + σ G G (cid:33)(cid:114) W I ε = σ G / G being the width of the single-photon distribu-tion (around 0.1 for a typical SiPM) and F (cid:39) R ( ε =
60 keV ) =
5% was measured in [21]. These fluctua-tions in the detected light are correspondingly included inthe TPC response.Finally, the time response function of the SiPM is in-cluded as a Gaussian with a 7 ns width, convoluted withthe transit time of the electrons through the electrolumines-cence structure ∆ T EL = µ s , being both much smallerin any case than the typical temporal spread of the clusters(dominated by diffusion). The sampling time is taken to be ∆ T s = µ s as in [10], and a matrix of 1800 10 mm-pitchSiPMs is assumed for the readout. Images are formed afterapplying a 10 phe threshold to all SiPMs.A fully processed TPC image for one time slice( ∆ T s = µ s), obtained at a beam rate of r = × s − , is shown in Fig. 7. The main clusters have beenmarked with crosses, by resorting to ’Monte Carlo truth’,i.e., they represent the barycenter of each primary pho-toelectron in Geant4. The beam has been assumed to becontinuous, polarized along the x -axis, impinging on a5 µ m water cube surrounded by air, with a 33 nm DNA6 new imaging technology based on Compton X-ray scattering
200 100 0 100 200x [mm]2001000100200 y [ mm ] S i P M s i g n a l ( p h e ) Fig. 7.
A typical TPC image reconstructed from theSiPM signals (in phe), as recorded in one time-slice( ∆ T s = µ s), for a beam rate of r = × s − .The crosses show the clusters’ centroids, obtained from’MC-truth’ information.cubic feature placed at its center. The Geant4 simulationsare performed at fixed time, and the X-ray interaction timesare subsequently distributed uniformly within the dwelltime corresponding to each position of the scan. It mustbe noted that clusters coming from different z -positionsbut originating at different interaction times, may eventu-ally be reconstructed in the same time slice (and viceversa,interactions taking place at about the same time may berecorded at different times depending on the z -position ofeach interaction). This scrambling (unusual under typicalTPC operation) renders every time slice as equivalent forthe purpose of counting. In principle, the absolute timeand z position can be disambiguated from the size of thecluster, using the diffusion relation in eq. 4, thus allowingphoton-by-photon reconstruction in time, space, and en-ergy. A demonstration of the strong correlation between z -position and cluster width, for 30 keV X-ray interactions,can be found in [27] for instance.The design parameters used in this subsection are com-piled in tables 1-4 of the appendix B. III. P hoton counting capabilities
As mentioned, the attenuation in the structural materials,re-scatters, characteristic emission, as well as the detectorinefficiency, are unavoidable limiting factors for counting.These intrinsic limitations can be conveniently evaluatedfrom the signal to noise ratio, defined as the inverse of therelative spread in the number of clusters: S / N = N cl / σ N cl .In simulation, where each photoelectron can be tagged, wedefine N cl ≡ N phe .To illustrate the effect, Fig. 8 shows S / N for 64 keVphotons as the realism of the simulation increases, fromleft to right. It has been normalized to the relative spread in Case 0 Case 1 Case 2 Case 30.700.750.800.850.900.951.00 s i g n a l - t o - n o i s e r a t i o - point-like cell- 100% detection efficiency - realistic cell- xenon volume - realistic cell- xenon volume- TPC vessel - realistic cell- xenon volume- TPC vessel- air, beampipes Counting modeCalorimetry mode
Fig. 8.
Intrinsic counting performance (using MonteCarlo truth information) for 64 keV X-ray photons, char-acterized by the signal to noise ratio (relative to case 0).Counting (green) and calorimetry (red) modes are dis-played as a function of the realism of the simulations.the number of photons scattered on-sample, (cid:112) N cl ,0 , (case0): S / N ∗ ≡ ( N cl ,0 ) − · S / N (8)Overlaid, the S / N in calorimetric mode is also shown, withthe counting performed by integrating the total collectedlight, instead of photon-by-photon. S / N is defined in thatcase, equivalently, as: S / N ∗ = ( ε tot / σ ε tot ) / ( ε tot / σ ε tot ) | .The values obtained are very close to the ones expectedfrom the additional contribution of the binomial fluctu-ations originated from the detector inefficiency (see ap-pendix): S / N ∗ (cid:39) (cid:101) √ (cid:101) − (cid:101) (9)suggesting a small contribution from re-scatters in thematerials.As long as the counting algorithm employed does notincrease the spread in the number of clusters or causesa strong loss of proportionality, S / N ∗ = Due to technical reasons, a beam rate corresponding to N cl ,0 (cid:39) N cl ,0 isimmaterial in this calculation. ε th in that slice, with the thresh-old chosen to be much lower than the typical photoelectronenergies ( ε th =1-2.5 keV); the assumption is that, for those,most of the energy will be collected in a neighbour time-slice to which the charge has spread due to diffusion, andwhere they will be properly counted once the algorithmis applied there; ii) a weighted inertia is then defined, asconventionally done in K-means, and a threshold is set( I th = 10 ), optimized for a certain operating condition (Fig.9). We concentrated for this optimization on beam ratesfor which the average efficiency and purity of the clusteridentification in 2d slides is ultimately above 85%, as theones shown in Fig. 10. Once the K-means parameters areoptimized for a certain beam rate, it is possible to studythe counting performance as a function of it. threshold inertiainertia differencesmoothing filter Fig. 9.
The K-means cluster-counting algorithm evalu-ates the partition of N observations (photoelectrons invoxels in our case) in K clusters, so as to minimize the inertia , defined as the sum of the squared distances ofthe observations to their closest cluster center. In the plot:convergence of K-means for a beam rate of 10 ph/s.A Savitzkyâ ˘A¸SGolay filter is applied for the purpose ofsmoothing the data.Fig. 11 shows the performance of K-means and a com-parison with MC truth, as a function of the beam rate,when optimized for 7.5x10
64 keV ph/s on-sample. Itis noticeable that K-means tends to slightly over-count forcluster values lower than the average value for which it hasbeen trained, and under-counts for higher ones, resultingin a distribution slightly narrower than the real one. As aresult, a small deviation from the proportionality can beseen (top figure), resulting in saturation in the number ofclusters per slice for very high beam rates.
IV. D iscussion
The purpose of this work is to illustrate the viability of anew technology for the detection of high energy X-raysat synchrotron Light Sources, where solid state detectors y [ mm ]
200 100 0 100 200x [mm]2001000100200 y [ mm ]
200 100 0 100 200x [mm] 0255075100125150175200 S i P M s i g n a l ( p h e ) Fig. 10.
Cluster counting performance for typical ∆ T s = µ s time-slices, for different energies ( ε )and beam rates ( r ). Crosses indicate the cluster cen-troids from MC and circles are the clusters found byK-means. The average counting-efficiency and purityover 100 time-slices are given below in brackets. Top left: ε =64 keV and r =3.7x10 ph/s (efficiency = 88.6%, purity= 90.7%). Top right: ε =64 keV and r =7.5x10 ph/s (effi-ciency = 85.8%, purity = 86.7%). Bottom left: ε =30 keVand r =6.5x10 ph/s (efficiency = 90.8%, purity = 86.2%).Bottom right: ε =30 keV and r =1.3x10 ph/s (efficiency= 86.5%, purity = 85.1%). For ε =30 keV only about halfof the clusters are produced, which enables measuringat higher beam rates than ε =64 keV, at comparable effi-ciency and purity.are almost universally adopted, but suffer inevitably frompractical limitations when aiming at 4 π -coverage. We haveadopted approximations, that might be superseded in fu-ture work, and are scrutinized here:1.
2d vs 3d counting : a complete reconstruction requirescombining 2d time-slices as the ones studied here,in order to unambiguously identify photoelectrons.Given that each cluster extends over 4-6 slices due todiffusion, and the counting efficiency is above 85% perslice, one would easily reach 100% if the counting ef-ficiencies were independent slice by slice, that clearlycan not be the case. However, given that the posi-tions of the clusters’ centroids are largely independent,they will overlap for few time-slices only, allowing2-3 independent centroid estimates, that should raisethe efficiency to 98-99%. Purity, on the other hand,being limited by low-energy clusters, will be greatlyincreased when the photoelectron energy is recon-structed in 3d. These facts should allow to iterativelyimprove the counting accuracy to the point where im-purity and inefficiency levels will plausibly be %-levelor less.8 new imaging technology based on Compton X-ray scattering beam rate (ph/s)203040506070 c l u s t e r s p e r s li c e MC truthK-means0.4 0.6 0.8 1.0 1.2 1.4beam rate (ph/s) 1e110.000.050.100.150.200.250.30 r e l a t i v e s p r e a d Fig. 11.
Counting performance, characterized throughthe average number of clusters counted per 2d time-slice(top) and relative spread (bottom), as a function of thebeam rate. The 1/ √ r expectation (dashed) is shown forcomparison. The K-means parameters have been opti-mized for r =7.5x10 ph/s.2. Availability of photon-by-photon information : cluster re-construction with high efficiency and purity enables x , y , t + t dri f t and ε determination, and arguably theinteraction time t and z position can be obtained fromthe study of the cluster size, as has been demonstratedfor 30 keV X-rays at near-atmospheric pressure before[27]. This can help at removing backgrounds not ac-counted for as well as any undesired systematic effect(beam or detector related). Since this technique pro-vides a parallax-free measurement, the concept maybe extended to other applications, e.g., X-ray crystal-lography. The presence of characteristic emission fromxenon will unavoidably create confusion, so if unam-biguous correspondence between the photoelectronand the parent X-ray is needed, one must consideroperation at (cid:46)
30 keV.3.
Data processing and realism : photon-by-photon countingin 3d through processing 2d-images at a cluster ratenearing 35 MHz (for the conditions discussed here) isa computer intensive task. Achieving this with suffi-cient speed and accuracy will require the optimizationof the counting algorithm, something that will pre-sumably need to be accomplished, ultimately, withreal data. To this aim, both the availability of parallelprocessing as well as the possibility of operation incalorimetry mode are desirable features.To summarize our results, we study the scan time needed to obtain a certain space resolution (appendix): d = (cid:32) R l ( l λ − w + a λ − a )( λ − f − λ − w ) S / N ∗ ,2 · r · (cid:101) · ∆ T scan (cid:33) (10)We consider different scenarios: i) a relatively simplecalorimetry mode, for which we assume a beam rate typ-ical of next generation Light Sources as r = ph/s,and ii) a rate-limited photon-by-photon counting scenario,at r = × ph/s (64 keV) and r = × ph/s(30 keV), according to the results obtained in the previ-ous section. The remaining parameters are common toboth modes: S / N ∗ = (cid:101) = S / N ∗ = (cid:101) = l = µ m, a = R =
5, with the mean free paths ( λ )taken from table 1. The dose-limited resolution is slightlydeteriorated compared to Fig. 1, given that the neededfluence depends on the efficiency as (appendix): φ → φ (cid:48) = (cid:101) − (cid:101) (cid:101) × φ (11)A compilation of these results is shown in Fig. 12. At64 keV, a dose-limited resolution of 38.5 nm can be achievedin approximately 20 h, and 48.5 nm in 10 h at 30 keV. In theabsence of systematic effects, the possibility of operationin calorimetry mode would bring the scan time down toaround 1 h in both cases. scanning time [hours] f e a t u r e s i z e [ n m ] dose limit = 64 keVCalorimetry Counting10 scanning time [hours] dose limit = 30 keV Fig. 12.
Resolution achievable with a 64 keV photonbeam (left) and a 30 keV photon beam (right) as a func-tion of the scan time for a cell of 5 µ m (green line). Thered line shows the limit in which a calorimetric measure-ment is performed and photon-by-photon counting isabandoned. The horizontal line shows the dose-limitedresolution in each case, prior to inducing structural dam-age.The detector geometry proposed here has been conceivedas a multi-purpose permanent station. A portable devicefocused purely on SCXM, on the other hand, could simplyconsist of a cubic 25cm × × ∼ S / N ∗ → √ (cid:101) (validfor low efficiencies, eq. 9 in appendix), a dose-limitedresolution of 45.5 nm could be achieved in 40 h.A further possibility could be considered, by resorting toultra-fast (1.6 ns resolution) TimePix-based cameras (e.g.,[29], [30]) with suitable VUV-optics, allowing 256 × V. C onclusions and outlook
We introduce a new 4 π technology conceived for detecting ∼
60 keV X-ray photons at high rates and with high effi-ciency, and that could be optimally applied to ComptonX-ray microscopy in upcoming Light Sources. It can beimplemented either as a permanent facility or a portabledevice. We concentrate on 5 µ m cells as our test case, forwhich we estimate that, under a Rose imaging criterion,and assuming the dose fractionation theorem, 38.5 nmDNA features may be resolved in 20 h by using the perma-nent station and 45.5 nm in 40 h with the portable device.Our analysis includes detailed Geant4 transport, a realisticdetector response and a simplified 2d-counting algorithmbased on K-means. Thus, we understand that the obtainedrate capability (and scan time) should be understood aslower (upper) limits to the actual capabilities when us-ing more refined 3d-algorithms, including constraints inenergy and cluster size. F unding I nformation ASH is funded through project ED431F 2017/10 (Xunta deGalicia) and DGD through the Ramon y Cajal program,contract RYC-2015-18820. A cknowledgments We thank Ben Jones and David Nygren (University of Texasat Arlington), as well as our RD51 colleagues for stimulat-ing discussions and encouragement, and specially to DavidJosé Fernández and Pablo Amedo for discussions on theK-means method.
A. R elation between resolution and scantime
We start from the imaging criterion: | N f − N | (cid:113) σ N f + σ N = R (12)where R = f is thenumber of scattered photons from a water medium witha ’to-be-resolved’ feature inside it, and N contains waterinstead (Fig. 1-top in the main document). This equationcan be re-expressed as: | N f − N | (cid:114) N f (cid:16) σ Nf N f (cid:17) + N (cid:16) σ N N (cid:17) = R (13)that, under the assumption N f (cid:38) N , and S / N ≡ N f / σ N f (cid:39) N / σ N can be rewritten as:1 √ N f − N N × S / N = R (14)It is convenient to use the definition S / N ∗ = √ N o − · S / N as in text (again, we use N f (cid:39) N ). Substitution of N f and N o by physical quantities yields:1 √ d ( λ − f − λ − w ) l λ − w + a λ − a × S / N ∗ × √ N o = R (15)We make use of the fact that N = r · ∆ T step · ( l λ − w + a λ − a ) · (cid:101) , with r being the beam rate, ∆ T step a time stepwithin the scan, and ∆ T scan the total time for a 2d scan: ∆ T scan = (cid:16) ld (cid:17) ∆ T step . After substitution in previous equa-tion we obtain:1 √ d ( λ − f − λ − w ) l ( l λ − w + a λ − a ) × S / N ∗ × (cid:112) r · ∆ T scan · (cid:101) = R (16)from which the time needed for a complete 2d scan can beexpressed as: ∆ T scan = R l d ( l λ − w + a λ − a )( λ − f − λ − w ) S / N ∗ ,2 · r · (cid:101) (17)10 new imaging technology based on Compton X-ray scatteringand, solving for d : d = (cid:32) R l ( l λ − w + a λ − a )( λ − f − λ − w ) S / N ∗ ,2 · r · (cid:101) · ∆ T scan (cid:33) (18)that is the expression used in text, for the achievable resolu-tion as a function of the scan time, under a given imagingcriterion R . Expression 18 can be readily assessed if assum-ing that S / N ∗ is mainly limited by Poisson statistics andby the efficiency of the detector, yielding: S / N ∗ = √ N o N σ N = √ N o N o (cid:101) (cid:112) (cid:101) N o + (cid:101) · ( − (cid:101) ) · N o == (cid:101) √ (cid:101) − (cid:101) (19)So, in the limit of low efficiencies eq. 18 becomes: d = (cid:32) R l ( l λ − w + a λ − a )( λ − f − λ − w ) r · (cid:101) · ∆ T scan (cid:33) (20)underlining the importance of the detector efficiency, (cid:101) ,compared to the rate capability, r .Last, the necessary increase in fluence (hence in dose) tosatisfy Rose criterion in case of an inefficient detector, canbe approximated, from inspection of eq. 15 and eq. 19, as: φ → φ (cid:48) = (cid:101) − (cid:101) (cid:101) × φ (21) D → D (cid:48) = (cid:101) − (cid:101) (cid:101) × D (22) B. EL-TPC parameters
Here we compile the main parameters used for the sim-ulation of the TPC response, together with additional ref-erences when needed.
Table 2.
Parameters of the TPC R i R o
25 cm vessel outer radius L
50 cm vessel length
Table 3.
Main gas parameters (xenon + 0.4% CH ) in the drift/collection region E c
110 V/cm collection field V cat -8.5 kV cathode voltage F W I
22 eV energy to create an e − -ionpair [9] D ∗ T √ cm transverse diffusion coeffi-cient [19] D ∗ L √ cm longitudinal diffusion coef-ficient [19] v d µ s drift velocity [19]in the electroluminescence (EL) region E EL V gate -3 kV voltage at FAT-GEM en-trance (‘gate’) v d , EL µ s drift velocity [19] Table 4.
Parameters of the electroluminescent structure r h t p h
10 mm hole-to-hole pitch m opt
250 ph/e/cm optical gain [21] P scin Table 5.
Parameters of the readout p si
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