Meter-scale spark X-ray spectrumstatistics
B. E. Carlson, N. Østgaard, P. Kochkin, Ø. Grondahl, R. Nisi, K. Weber, Z. Scherrer, K. LeCaptain
JJournal of Geophysical Research: Atmospheres
Meter-scale spark X-ray spectrum statistics
B. E. Carlson , N. Østgaard , P. Kochkin , Ø. Grondahl , R. Nisi , K. Weber ,Z. Scherrer , and K. LeCaptain Department of Physics, Carthage College, Kenosha, Wisconsin, USA, Birkeland Center for Space Science, University ofBergen, Bergen, Norway, Technische Universiteit Eindhoven, Eindhoven, Netherlands
Abstract
X-ray emission by sparks implies bremsstrahlung from a population of energetic electrons, butthe details of this process remain a mystery. We present detailed statistical analysis of X-ray spectra detectedby multiple detectors during sparks produced by 1 MV negative high-voltage pulses with 1 μ s risetime. Withover 900 shots, we statistically analyze the signals, assuming that the distribution of spark X-ray fluencebehaves as a power law and that the energy spectrum of X-rays detectable after traversing ∼ ± keV and the spark X-ray fluence power lawdistribution has index − . ± . and spans at least 3 orders of magnitude in fluence.
1. Introduction
Spark discharge is a complicated multiscale process. The physics involved ranges from subnanosecondsubmillimeter electron avalanches governed by atomic and molecular ionization and attachment crosssections to centimeter-scale propagating space charge waves (streamers) to microsecond-scale meter-longchannels of ionized gas described by nonequilibrium plasma and electrodynamic behavior. On a larger scale,spark behavior in nature ranges from 100 km lightning channels that last seconds and span conditions fromthe upper reaches of a thundercloud to the microsecond-scale processes by which such a channel attachesto the ground.In such a diverse range of behavior one can find many puzzling phenomena. The process by which a dischargeinitiates in the cloud is not understood nor is the process by which lightning channels (leaders) extend andbranch. The extension process sometimes proceeds as a series of microsecond-scale steps separated by 10 s ofmicroseconds of relative quiescence. More puzzling, the steps seem to be immediately preceded by formationof a hot conductive channel displaced from the end of the existing channel that rapidly connects with themain channel, though the mechanism by which this “space leader” forms is not understood. On a smaller scale,the transition from streamer to leader is the focus of much research as are the dynamics of the streamer itselfand the role of energetic particles in the process. See
Dwyer and Uman [2014] for a review of recent resultsand open questions.This role of energetic particles is especially interesting in the light of observations of X-ray production bylightning and the hypothesis that energetic particles produced by lightning contribute to the production ofterrestrial gamma ray flashes. X-rays as produced by lightning seem to occur coincident with stepwise exten-sions of the channel [
Howard et al. , 2010;
Dwyer et al. , 2011] and have energies around a few 100 keV [
Mooreet al. , 2001;
Dwyer et al. , 2003;
Dwyer , 2005b]. These X-rays imply the existence of a much larger populationof higher-energy electrons. While low-energy electrons encounter high dynamic friction, relativistic electronsencounter much lower friction and can “run away” to very high energies when driven by electric fields, afact first recognized by
Wilson [1925] and modeled in the context of electric fields near lightning channels in
Carlson [2009, chapters 5 and 6]. “Seed” energetic electrons necessary to initiate such a process can come fromenergetic background radiation (i.e., cosmic rays [
Carlson et al. , 2008]), low-energy electrons in local electricfields strong enough to overcome the maximum friction force [e.g.,
Moss et al. , 2006;
Li et al. , 2009;
Chanrionand Neubert , 2010], or by feedback from prior generations of energetic electrons [
Dwyer , 2003, 2007]. Regard-less, electric fields may drive avalanche growth of populations of such runaway electrons [
Gurevich et al. , 1992].
RESEARCH ARTICLE
Key Points: • We describe the statistics of X-rayproduction by sparks with a data setof over 900 sparks• The results are well described byroughly 85 keV mean X-ray energy• X-ray fluence is power law distributedand spans 3 orders of magnitudein fluence
Correspondence to:
B. E. Carlson,[email protected]
Citation:
Carlson, B. E., N. Østgaard, P. Kochkin,Ø. Grondahl, R. Nisi, K. Weber,Z. Scherrer, and K. LeCaptain(2015), Meter-scale spark X-rayspectrum statistics,
J. Geophys.Res. Atmos. , , 11,191–11,202,doi:10.1002/2015JD023849.Received 24 JUN 2015Accepted 19 OCT 2015Accepted article online 10 NOV 2015Published online 13 NOV 2015©2015. The Authors.This is an open access article under theterms of the Creative CommonsAttribution-NonCommercial-NoDerivsLicense, which permits use anddistribution in any medium, providedthe original work is properly cited, theuse is non-commercial and nomodifications or adaptations are made. CARLSON ET AL. SPARK ENERGY SPECTRA ,,
J. Geophys.Res. Atmos. , , 11,191–11,202,doi:10.1002/2015JD023849.Received 24 JUN 2015Accepted 19 OCT 2015Accepted article online 10 NOV 2015Published online 13 NOV 2015©2015. The Authors.This is an open access article under theterms of the Creative CommonsAttribution-NonCommercial-NoDerivsLicense, which permits use anddistribution in any medium, providedthe original work is properly cited, theuse is non-commercial and nomodifications or adaptations are made. CARLSON ET AL. SPARK ENERGY SPECTRA ,, ournal of Geophysical Research: Atmospheres Unfortunately, the relative importance of these processes for phenomena like X-ray production by lightningor terrestrial gamma ray flashes is not well understood.In the context of these puzzles, any additional information about the processes involved may potentially beuseful. While limited in scale, laboratory studies can shed light on the detailed dynamics of leader extension,and while the energy scales are necessarily smaller (of order 1 MV compared to the 10 s of megavolts of naturallightning), lab sparks even produce X-rays.X-ray production by meter-scale sparks has been observed on several occasions.
Dwyer [2005a] report thefirst detection of X-rays associated with such sparks in a study of 14 discharges of 1.5 MV, with total energydeposited in a detector up to MeV scale. Variation of signals among X-ray detectors with di ff erent attenuatorssuggest X-ray energies from 30 keV to 150 keV piling up to produce the MeV-scale observations. Rahman et al. [2008] report similar results with total energy deposited up to several MeV with 1 MV discharge.
Nguyen et al. [2008] (elaborated in
Nguyen [2012]) also report similar results for 0.88 – 1 MV sparks with a much larger dataset and with a variety of detectors, attenuators, and positions of detector.
Nguyen et al. [2008] show intensityvariations with high-voltage (HV) polarity and with distance between detector and spark gap, suggesting thatsuch intensity variations could result from X-ray production by positive streamers from metal structures nearthe detectors. This highlights the importance of positioning detectors at a su ffi cient distance from the gap. Dwyer et al. [2008] report a further study of 241 discharges of 1 MV with various polarities and gap distancesand push the maximum energy deposited in a single detector up to 50 MeV with varied attenuators providingevidence for individual photon energies exceeding 300 keV and statistical evidence for an average photonenergy at most 230 keV.
Dwyer et al. [2008] also report collimator experiments supporting the production ofX-rays by a di ff use source in the gap between the electrodes and that bursts of X-rays produced late in thedischarge come from elsewhere. March and Montanyà [2010] report that HV pulses with rapid risetime tendto produce more X-rays, and
March and Montanyà [2011] examines the e ff ect of electrode geometry. Kochkinet al. [2012],
Kochkin et al. [2014], and
Kochkin et al. [2015] add nanosecond-resolution photography as a use-ful tool, mapping the streamer clouds produced by positive [
Kochkin et al. , 2012] and negative [
Kochkin et al. ,2015] HV discharges. In such experiments the HV risetime is much longer than the time for streamers to prop-agate across the gap, and in experiments with negative high voltage the negative streamers tend to appearand grow in bursts [see, e.g.,
Kochkin et al. , 2014, Figure 6].
Kochkin et al. [2015] demonstrate for negative HVthat the X-rays tend to be emitted during these bursts of streamer development but are emitted on a muchshorter timescale than the burst, supporting the suggestion that the large and transient electric fields thatresult from interaction of negative and positive streamer fronts may play a role in energetic electron and thusX-ray production.
Kochkin et al. [2012, 2015] also describe that attenuator experiments they claim are consis-tent with 200 keV characteristic X-ray energy. These estimates are based simply on comparison of registrationrate (the fraction of sparks for which X-rays are observed) for various attenuators with the expected attenua-tion in number of photons incident on the detector, assuming all sparks emit the same number of photons andthat pileup does not a ff ect their observations, though they acknowledge that pileup has a significant e ff ect.Together, these studies provide a reasonably complete picture of X-ray emissions from sparks: free elec-trons are pushed to overcome friction in the high-field region ahead of a negative streamer (possibly brieflyenhanced by a nearby positive streamer), gain at least enough energy to run away in the lower fields sur-rounding the streamer, then undergo bremsstrahlung in the surrounding air to emit X-rays. These intensestreamer fields exist only as the discharge develops and are unlikely to exist during the discharge itself,though the intense electrodynamic environment during discharge may drive discharges elsewhere in the labthat also produce detectable X-rays. However, this picture remains incomplete in several aspects: beyond an“average” energy, the properties of the photon spectrum are unknown and di ffi cult to judge due to pileup,and the fact that some sparks produce copious X-rays while others produce no measurable X-rays is notaddressed. The present paper attempts to shed light on these topics by examining the energies measuredin X-ray detectors placed near a 1 m spark gap at the Technical University of Eindhoven during 921 shots asdescribed in section 2. These shots give us the opportunity to statistically analyze the distributions of sparkX-ray fluence, number of photons detected by multiple detectors, and photon energy. These distributions aredescribed in section 3 and constrained by comparison to observations in section 4. We conclude and discussthe implications of our results in section 5.CARLSON ET AL. SPARK ENERGY SPECTRA ,,
Kochkin et al. [2012, 2015] also describe that attenuator experiments they claim are consis-tent with 200 keV characteristic X-ray energy. These estimates are based simply on comparison of registrationrate (the fraction of sparks for which X-rays are observed) for various attenuators with the expected attenua-tion in number of photons incident on the detector, assuming all sparks emit the same number of photons andthat pileup does not a ff ect their observations, though they acknowledge that pileup has a significant e ff ect.Together, these studies provide a reasonably complete picture of X-ray emissions from sparks: free elec-trons are pushed to overcome friction in the high-field region ahead of a negative streamer (possibly brieflyenhanced by a nearby positive streamer), gain at least enough energy to run away in the lower fields sur-rounding the streamer, then undergo bremsstrahlung in the surrounding air to emit X-rays. These intensestreamer fields exist only as the discharge develops and are unlikely to exist during the discharge itself,though the intense electrodynamic environment during discharge may drive discharges elsewhere in the labthat also produce detectable X-rays. However, this picture remains incomplete in several aspects: beyond an“average” energy, the properties of the photon spectrum are unknown and di ffi cult to judge due to pileup,and the fact that some sparks produce copious X-rays while others produce no measurable X-rays is notaddressed. The present paper attempts to shed light on these topics by examining the energies measuredin X-ray detectors placed near a 1 m spark gap at the Technical University of Eindhoven during 921 shots asdescribed in section 2. These shots give us the opportunity to statistically analyze the distributions of sparkX-ray fluence, number of photons detected by multiple detectors, and photon energy. These distributions aredescribed in section 3 and constrained by comparison to observations in section 4. We conclude and discussthe implications of our results in section 5.CARLSON ET AL. SPARK ENERGY SPECTRA ,, ournal of Geophysical Research: Atmospheres Figure 1.
Sample records from a single spark. (top left) The voltage of the high-voltage electrode with respect to theground electrode. (bottom left) The magnitude of the current flowing to the high-voltage electrode. (right) Signals fromthe two LaBr detectors prior to calibration.
2. Experimental Setup
The experiment was conducted in the high-voltage lab at the Technical University of Eindhoven with a Haefly2 MV Marx generator configured to produce 1 MV pulses with 1 μ s risetime. All tests were carried out with a1 m point-point gap with negative HV polarity. The spark voltage, ground electrode current, and high-voltageelectrode current were recorded by Lecroy four-channel storage oscilloscopes configured to record 2 μ s ofdata at 10 GHz sample frequency when triggered by the Marx generator, as were signals direct from the out-put of photomultiplier tubes monitoring a variety of energetic radiation detectors. No photomultiplier pulseshaping electronics was used. See Kochkin et al. [2012] for a more detailed description of the setup.The radiation detectors used in the experiment included scintillating plastic optical fibers placed at a varietyof locations around the spark to monitor energetic electrons. Analysis of these data has been presented[
Ostgaard et al. , 2014] and will be published separately. Due to the need to run many shots with the scintillat-ing fiber detectors at various locations, a total of 950 shots were carried out from which we recorded usabledata for 921. During all shots, two additional X-ray detectors were running, providing a wealth of data aboutthe photon population. These X-ray data are the focus of this paper.These two detectors are each composed of a 1.5 inch long 1.5 inch diameter LaBr (Ce + ) scintillator monitoredby a photomultiplier tube (PMT). These detectors are placed next to each other (separation ∼
10 cm) in an elec-tromagnetic compatibility (EMC) cabinet to shield the detectors from the electromagnetic noise produced bythe spark. The location of the detectors relative to the spark corresponds to location H as shown in Figure 1of
Kochkin et al. [2015]. The scintillator material is separated from the spark by roughly 0.5 mm of aluminumin the scintillator housing and EMC cabinet wall and approximately 2 m of air.The signals as recorded by the oscilloscopes are in volts and here have been converted to MeV by use of Cs-137and Co-60 gamma ray emitters as sources of known energy. The lowest energy detectable by this setup is ∼
20 keV, comparable to the minimum photon energies transmitted through the 0.5 mm aluminum shielding,while the maximum depends on the settings of the oscilloscope. Signal from detector 1 clips at just above5 MeV, while detector 2 clips at just above 3.5 MeV.Sample data are shown in Figure 1. The high-voltage trace shown is very consistent from one spark to thenext, as is the high-voltage electrode current. The pulse-like features on the high-voltage electrode currentoccur when bursts of corona and streamer activity carry charge away from the electrode. See
Kochkin et al. [2014] for a detailed discussion of such processes as seen in high-speed camera imagery. The timing of theX-ray pulses is also quite consistent. In this experiment, X-rays are typically not seen during the high-currentphase of the discharge.The pulses seen in the oscilloscope traces indicate deposition of energy in the scintillator, but a single pulsefrom the PMTs may be produced by multiple X-ray photons entering the scintillator. These photons may arriveat slightly di ff erent times and produce a visibly altered pulse shape, but typically no time structure is visible.The photon spectrum is therefore not directly measurable, and pulse pileup must be treated statistically.CARLSON ET AL. SPARK ENERGY SPECTRA ,,
Kochkin et al. [2014] for a detailed discussion of such processes as seen in high-speed camera imagery. The timing of theX-ray pulses is also quite consistent. In this experiment, X-rays are typically not seen during the high-currentphase of the discharge.The pulses seen in the oscilloscope traces indicate deposition of energy in the scintillator, but a single pulsefrom the PMTs may be produced by multiple X-ray photons entering the scintillator. These photons may arriveat slightly di ff erent times and produce a visibly altered pulse shape, but typically no time structure is visible.The photon spectrum is therefore not directly measurable, and pulse pileup must be treated statistically.CARLSON ET AL. SPARK ENERGY SPECTRA ,, ournal of Geophysical Research: Atmospheres Figure 2.
Scatterplot of observed energy deposition events. Each pointrepresents a spark, where E was deposited in detector 1, while E wasdeposited in detector 2. The dashed lines show the 0 and saturationenergy levels. The dotted line shows E = E . At E = E = , 523 pointsoverlap, and 20 points overlap at the upper right, where both detectorswere saturated. To compile data for such statistics, wesearch through the data for pulses. Tosearch, we first smooth the data bylow-pass filtering, then identify pulsesas significant deviations from the back-ground. Once pulses are identified, wecollect a variety of data including pulsetime, height, integral, and duration. Pulseheight and integral are calibrated toenergy as described above. Pulse integralis a more robust measure of depositedenergy in case multiple pulses arrivenear simultaneously, so our analysis herereports energy as determined by pulseintegral. While pulse integral is somewhatresistant to saturation when oscilloscopesignals clip, the energy of a saturatedpulse cannot reliably be determined, sowe enforce a maximum energy for eachdetector corresponding to the energy ofthe largest unclipped pulse. In cases whenmultiple pulses are observed in a single detector, we add their energies together to better capture variationfrom one spark to the next, though note that the majority of sparks have zero or only one burst so this com-bination of bursts only a ff ects a small fraction of our data. The data set we consider here is therefore a setof pairs of numbers, each pair associated with a single spark and each number with the pair giving the totalenergy deposited in a detector during the given spark.A scatterplot showing the energies deposited in each detector in each shot is shown in Figure 2. The energydeposited varies widely from one spark to the next. Most sparks (57%) produce no detectable signals in eitherdetector, indicating less than 20 keV deposited, while approximately 3% of sparks saturate both detectors,indicating at least 3 – 5 MeV of energy deposited.The points clearly cluster in the lower left, but a significant number of points appear in the middle and upperregions of the plot. Attempting to explain this distribution most simply, one might assume all sparks areidentical and that all photon energies are equal and treat the observed distribution as solely due to Poissonstatistical fluctuations in the number of photons observed. In this case, the high number of points in the lowerleft (57% of events undetected) implies a Poisson mean of 0.562. With this mean, observation of events thatsaturated both detectors is then incredibly unlikely as deposition of more than 8 MeV (3 MeV in one and 5 MeVin the other) requires at least 40 photons for the mean energy 200 keV consistent with the results quotedabove. Observation of at least 40 photons from a Poisson distribution with mean 0.562 has a probabilityaround − , much less than the 3% of observed sparks that saturate both detectors.Adding the complication of a photon energy distribution is logical, but does not help enough, since themaximum photon energy must be less than 1 MeV as the photons will be much less energetic than the1 MV maximum spark voltage. Assuming the same Poisson mean as before and attempting to explainthe events that saturated both detectors as due to an extreme fluctuation in photon energy such that allphotons carried 1 MeV of energy means that we must have at least eight photons instead of at least 40, but thePoisson probability of observing at least eight photons given a mean of 0.562 is roughly − , still muchsmaller than the 3% observed.Clearly, there must be a significant variability in spark X-ray fluence at the location of the detectors from onespark to the next. This variability could come from intrinsic variability in spark X-ray luminosity or it couldcome from some variability in the geometry if X-ray emissions are not isotropic. Regardless, a full explanationof the distribution of points in Figure 2 must include the e ff ect of the distribution of spark X-ray fluence, thedistribution of numbers of photons detected by each detector (given the X-ray fluence), and the distributionof photon energies. These distributions and their properties are the focus of this paper.CARLSON ET AL. SPARK ENERGY SPECTRA ,,
Scatterplot of observed energy deposition events. Each pointrepresents a spark, where E was deposited in detector 1, while E wasdeposited in detector 2. The dashed lines show the 0 and saturationenergy levels. The dotted line shows E = E . At E = E = , 523 pointsoverlap, and 20 points overlap at the upper right, where both detectorswere saturated. To compile data for such statistics, wesearch through the data for pulses. Tosearch, we first smooth the data bylow-pass filtering, then identify pulsesas significant deviations from the back-ground. Once pulses are identified, wecollect a variety of data including pulsetime, height, integral, and duration. Pulseheight and integral are calibrated toenergy as described above. Pulse integralis a more robust measure of depositedenergy in case multiple pulses arrivenear simultaneously, so our analysis herereports energy as determined by pulseintegral. While pulse integral is somewhatresistant to saturation when oscilloscopesignals clip, the energy of a saturatedpulse cannot reliably be determined, sowe enforce a maximum energy for eachdetector corresponding to the energy ofthe largest unclipped pulse. In cases whenmultiple pulses are observed in a single detector, we add their energies together to better capture variationfrom one spark to the next, though note that the majority of sparks have zero or only one burst so this com-bination of bursts only a ff ects a small fraction of our data. The data set we consider here is therefore a setof pairs of numbers, each pair associated with a single spark and each number with the pair giving the totalenergy deposited in a detector during the given spark.A scatterplot showing the energies deposited in each detector in each shot is shown in Figure 2. The energydeposited varies widely from one spark to the next. Most sparks (57%) produce no detectable signals in eitherdetector, indicating less than 20 keV deposited, while approximately 3% of sparks saturate both detectors,indicating at least 3 – 5 MeV of energy deposited.The points clearly cluster in the lower left, but a significant number of points appear in the middle and upperregions of the plot. Attempting to explain this distribution most simply, one might assume all sparks areidentical and that all photon energies are equal and treat the observed distribution as solely due to Poissonstatistical fluctuations in the number of photons observed. In this case, the high number of points in the lowerleft (57% of events undetected) implies a Poisson mean of 0.562. With this mean, observation of events thatsaturated both detectors is then incredibly unlikely as deposition of more than 8 MeV (3 MeV in one and 5 MeVin the other) requires at least 40 photons for the mean energy 200 keV consistent with the results quotedabove. Observation of at least 40 photons from a Poisson distribution with mean 0.562 has a probabilityaround − , much less than the 3% of observed sparks that saturate both detectors.Adding the complication of a photon energy distribution is logical, but does not help enough, since themaximum photon energy must be less than 1 MeV as the photons will be much less energetic than the1 MV maximum spark voltage. Assuming the same Poisson mean as before and attempting to explainthe events that saturated both detectors as due to an extreme fluctuation in photon energy such that allphotons carried 1 MeV of energy means that we must have at least eight photons instead of at least 40, but thePoisson probability of observing at least eight photons given a mean of 0.562 is roughly − , still muchsmaller than the 3% observed.Clearly, there must be a significant variability in spark X-ray fluence at the location of the detectors from onespark to the next. This variability could come from intrinsic variability in spark X-ray luminosity or it couldcome from some variability in the geometry if X-ray emissions are not isotropic. Regardless, a full explanationof the distribution of points in Figure 2 must include the e ff ect of the distribution of spark X-ray fluence, thedistribution of numbers of photons detected by each detector (given the X-ray fluence), and the distributionof photon energies. These distributions and their properties are the focus of this paper.CARLSON ET AL. SPARK ENERGY SPECTRA ,, ournal of Geophysical Research: Atmospheres
3. Statistical Model
In attempting to build a statistical model of the distribution of points in Figure 2, we need to know the formof the distributions of physical properties relevant to the point locations. There are three main distributions atwork: the distribution of spark X-ray fluence, the distribution of numbers of photons incident on each detector(given spark fluence), and the distribution of photon energies.The distribution of spark fluence is determined in principle by the processes at work in electron acceleration.One spark has an intrinsically higher luminosity than another by having more energetic electrons, perhapsbecause random streamer branching events happened to lead to more negative streamers or perhaps fewerbut more intense or more rapidly growing negative streamers or perhaps more interactions between negativestreamers and positive streamers. Fluence variability may also result from anisotropy in the directionality ofX-ray emissions: perhaps all sparks emit many X-rays, but the emissions are not always beamed toward thedetectors. These processes may be chaotic or perhaps intrinsically random, but either way there will be somevariability in X-ray fluence. To represent this variability, we treat the X-ray fluence as the expected value ofthe number of photons hitting a detector, 𝜂 , and assume some probability distribution for the occurrence ofvarious values of 𝜂 . As there is yet no theoretical expectation for the form of this distribution, we must makesome assumption. Examining the clustering of points in the lower left of Figure 2, we expect lower fluencesto be more common than higher fluences, but the existence of points in the upper right suggests that the tailof the distribution to high fluences is quite strong. To represent distributions that follow this general trend ofdecreasing with a potentially long tail, we assume the distribution for 𝜂 is a power law with index 𝜆 : p 𝜆 ( 𝜂 ) = ( 𝜆 + ) 𝜂 𝜆 + − 𝜂 𝜆 + 𝜂 𝜆 (1)where 𝜂 min and 𝜂 max are lower and upper limits on 𝜂 necessary to ensure that the distribution is normalizablefor 𝜆 ≥ − and the leading constant ensures normalization. By examining the clustering in the lower left inFigure 2, we expect 𝜆 < , but the number of points saturating both detectors suggests that 𝜆 is not toonegative and that the distribution of 𝜂 is in some sense quite hard.Given the fluence of the spark (i.e., given 𝜂 ), the number of photons hitting a detector, N , should be Poissondistributed with mean 𝜂 . Normalized, p 𝜂 ( N ) = 𝜂 N e − 𝜂 N ! (2)This assumes that there is a very large number of photons produced by the spark, each with a very small prob-ability of hitting the detector, and that the two detectors have the same probability of catching each photon.This implicitly assumes that X-ray emissions from the spark are uniform on the scale of the ∼
10 cm separationof the detectors, but this is reasonable as the detectors are roughly 2 m from the spark and bremsstrahlungfrom electrons with only a few hundred keV of energy is not strongly beamed.Finally, given a number of X-rays hitting each detector, one can calculate the distribution of deposited energyby applying a distribution of X-ray energies. This unfortunately is unknown as we do not know the distribu-tion of electron energies responsible for the bremsstrahlung emissions [
Chanrion and Neubert , 2010]. Figure 2presents calculations of energy distributions of free electrons as produced in constant electric fields thatappear linear on a semi – log plot (inset) and that show the e ff ects of an exponential cuto ff up to the thresh-old of runaway electron behavior. These exponential distributions of electron energy are characteristic ofavalanche growth processes. The resulting photon distribution will share some of these characteristics, butin principle connecting the photon and electron distributions requires detailed treatment of bremsstrahlung,a topic beyond the scope of the current work. As such, we simply assume that the photon energy is alsoexponentially distributed with mean photon energy 𝜇 : p 𝜇 ( ) = e − ∕ 𝜇 𝜇 (3)where the division by 𝜇 ensures normalization for < < ∞ . Note that for simplicity we are assuming thatthe photon energy distribution does not depend on X-ray fluence. It is reasonable to expect that 𝜇 dependson 𝜂 somewhat, but including this dependence would require still more assumptions and additional freeCARLSON ET AL. SPARK ENERGY SPECTRA ,,
Chanrion and Neubert , 2010]. Figure 2presents calculations of energy distributions of free electrons as produced in constant electric fields thatappear linear on a semi – log plot (inset) and that show the e ff ects of an exponential cuto ff up to the thresh-old of runaway electron behavior. These exponential distributions of electron energy are characteristic ofavalanche growth processes. The resulting photon distribution will share some of these characteristics, butin principle connecting the photon and electron distributions requires detailed treatment of bremsstrahlung,a topic beyond the scope of the current work. As such, we simply assume that the photon energy is alsoexponentially distributed with mean photon energy 𝜇 : p 𝜇 ( ) = e − ∕ 𝜇 𝜇 (3)where the division by 𝜇 ensures normalization for < < ∞ . Note that for simplicity we are assuming thatthe photon energy distribution does not depend on X-ray fluence. It is reasonable to expect that 𝜇 dependson 𝜂 somewhat, but including this dependence would require still more assumptions and additional freeCARLSON ET AL. SPARK ENERGY SPECTRA ,, ournal of Geophysical Research: Atmospheres parameters. Assuming an exponential photon energy distribution is especially convenient since the distribu-tion of energies deposited in a detector E , a sum of the energies of N photons ( E = ∑ Ni = i ) will be distributedas the sum of N independent exponentially distributed random variables. The distribution of such a sum ofexponential random variables can be expressed in closed form and is known as the Erlang distribution: p N , 𝜇 ( E ) = E N − e − E ∕ 𝜇 𝜇 N ( N − )! (4)Note, however, that if N = , no photons strike the detector, the energy deposited is exactly zero, and p N = , 𝜇 ( E ) = 𝛿 ( E ) , a Dirac delta function, instead of the continuous distribution above.If the fluence of the spark ( 𝜂 ) and the number of X-rays hitting each detector ( N , N ) were known, the jointdistribution of energy depositions would simply be given by a product of two Erlang distributions. Unfortu-nately, the fluence of the spark and the number of X-rays hitting each detector are not known. As such, wemust compute the marginal distribution of observed energies by integrating (i.e., taking an weighted aver-age) over the possible values of the unknown quantities, weighted by the probability distributions associatedwith those quantities. For one detector, p 𝜆 , 𝜇 ( E ) = ∫ 𝜂 max 𝜂 min d 𝜂 p 𝜆 ( 𝜂 ) ∞ ∑ N = p 𝜂 ( N ) p N , 𝜇 ( E ) (5)and for the joint distribution of simultaneous observations with two detectors, p 𝜆 , 𝜇 ( E , E ) = ∫ 𝜂 max 𝜂 min d 𝜂 p 𝜆 ( 𝜂 ) ∞ ∑ N = ∞ ∑ N = p 𝜂 ( N ) p N , 𝜇 ( E ) p 𝜂 ( N ) p N , 𝜇 ( E ) (6)where the integral over 𝜂 covers the range of possible spark X-ray fluence and the sums over N or N and N cover the possible numbers of photons observed by each detector. The resulting joint probability distributionof E and E should be able to reproduce the observations in Figure 2.Note that this is a hybrid continuous-discrete probability distribution. There is a finite probability that bothdetectors observe exactly zero energy, a discrete probability represented by the N = N = term in thesums for which p N , 𝜇 and p N , 𝜇 become delta functions as described above. Likewise, the N = , N > , and N = , N > terms in the probability distribution contain a single delta function. Only the terms for whichboth N and N are greater than zero lack delta function divergence behavior. These delta functions make theprobability distribution di ffi cult to work with directly but are capable of representing the clustering of pointsat the origin and along the axes in Figure 2.These probability distributions form the basis of our model, and we seek to infer the parameters 𝜆 , 𝜇 , 𝜂 min , and 𝜂 max by comparison of our model to data.
4. Comparison to Data
Comparison of a probability distribution to data can be done in many ways, for example, theKolmogorov-Smirnov test or the Anderson-Darling test. Such tests work on the basis of the cumulativedistribution function (CDF) F , which for a probability distribution p ( x ) in one dimension is defined as F ( x ) = ∫ x −∞ p ( 𝜉 ) d 𝜉 . Since here we work in two dimensions ( E , E ), we need a two-dimensional analog of thecumulative distribution function. Here we follow Fasano and Franceschini [1987] in taking our set of data points { P i } = {( E i , E i )} and for each point, assigning four CDF-like values: F << dat i , F < ≥ dat i , F ≥ < dat i , and F ≥≥ dat i , each givingthe fraction of the observed data set in the corresponding quadrant relative to the data point in question. Forexample, F << dat i gives the fraction of data { P j } for which E j < E i and E j < E i .These { F … dat i } can be plotted if an order is assigned to the data points for use as an abscissa. Here it is convenientto order each set of F … dat i separately such that F … dat i is monotonically increasing, i.e., sort each set of F … dat i fromsmallest to largest and use its place in the resulting list as its abscissa. The order has no physical significanceand is used only for plotting. The reordered F … dat i are shown in Figure 3. For example, the grey ≥ , ≥ curve startsat 0.03 since 3% of sparks saturate both detectors: the data point that has the fewest data points with energiesgreater than or equal to its energy must be one of those doubly saturated events, and thus, 3% of the datasatisfies the ≥ , ≥ condition. Moving to the right along the ≥ , ≥ curve, larger CDF values correspond to dataCARLSON ET AL. SPARK ENERGY SPECTRA ,,
Comparison of a probability distribution to data can be done in many ways, for example, theKolmogorov-Smirnov test or the Anderson-Darling test. Such tests work on the basis of the cumulativedistribution function (CDF) F , which for a probability distribution p ( x ) in one dimension is defined as F ( x ) = ∫ x −∞ p ( 𝜉 ) d 𝜉 . Since here we work in two dimensions ( E , E ), we need a two-dimensional analog of thecumulative distribution function. Here we follow Fasano and Franceschini [1987] in taking our set of data points { P i } = {( E i , E i )} and for each point, assigning four CDF-like values: F << dat i , F < ≥ dat i , F ≥ < dat i , and F ≥≥ dat i , each givingthe fraction of the observed data set in the corresponding quadrant relative to the data point in question. Forexample, F << dat i gives the fraction of data { P j } for which E j < E i and E j < E i .These { F … dat i } can be plotted if an order is assigned to the data points for use as an abscissa. Here it is convenientto order each set of F … dat i separately such that F … dat i is monotonically increasing, i.e., sort each set of F … dat i fromsmallest to largest and use its place in the resulting list as its abscissa. The order has no physical significanceand is used only for plotting. The reordered F … dat i are shown in Figure 3. For example, the grey ≥ , ≥ curve startsat 0.03 since 3% of sparks saturate both detectors: the data point that has the fewest data points with energiesgreater than or equal to its energy must be one of those doubly saturated events, and thus, 3% of the datasatisfies the ≥ , ≥ condition. Moving to the right along the ≥ , ≥ curve, larger CDF values correspond to dataCARLSON ET AL. SPARK ENERGY SPECTRA ,, ournal of Geophysical Research: Atmospheres Figure 3.
Cumulative distribution functions as calculated for the data.On a given curve, each horizontal point represents a point in the (2-D)data set, for which the vertical coordinate represents the fraction of thedata set with energies related to the energies of the given data point bythe inequalities associated with the given curve (i.e., the F … dat i describedin the text). Each curve represents a di ff erent inequality, and the datahave been reordered for each curve to make the curve in questionmonotonically increasing. Two sets of curves are shown, thick curvesfrom the data and thin curves predicted given the best fit parametersdiscussed in the text, but the curves overlap so much any deviations aredi ffi cult to judge. points that are below and/or to the leftof the upper right corner, i.e., data pointsfor which larger fractions of the data sat-isfy the ≥ , ≥ condition. The large jumpfrom about 0.35 to 1.0 at CDF rank 400occurs when incrementally increasingthe fraction of the data set that satis-fies the ≥ , ≥ condition requires lookingat a point where E = E = , whichthus includes all other such points (57%of all data) as well as points for which E = or E = . The CDF thereforejumps up to 1 since all of the data has E ≥ and E ≥ . The direction of theinequality is related to the shape of thedistribution, with ≥≥ and << related tothe lower left/upper right balance andwidth while the < ≥ and ≥ < versionsrelated to upper left/lower right balanceand width. One can imagine other waysof constructing such CDFs, but this isa straightforward technique that as wewill see shortly is quite e ff ective.Such curves as in Figure 3 can be predicted on the basis of the probability density (equation (6)), assigning apredicted F … pred i to each data point by integrating the expected distributions over the region relevant to theinequalities in question. For example, F << pred i = ∫ E i −∞ d E ∫ E i −∞ d E p 𝜆 , 𝜇 ( E , E ) (7)Similar calculations can be made for predicted F ≥ < pred i , F < ≥ pred i , F ≥≥ pred i by changing the limits on the integrals to gofrom the point in question to + ∞ as appropriate. Regardless, the exponentials and powers in the integrandcan be manipulated to convert the integrals here and in equation (6) into upper and lower incomplete gammafunction evaluations (e.g., Γ ( k , x ) = ∫ ∞ x x k − e − x d x ), and the summations in equation (6) can be computednumerically and truncated without significant loss of accuracy.For a given 𝜆 , 𝜇 , 𝜂 min , and 𝜂 max , the result of this exercise is four sets of numbers ( { F << pred i } , { F ≥ < pred i } , { F < ≥ pred i } ,and { F ≥≥ pred i } ) that can be compared to the corresponding sets associated with the data ( { F << dat i } , { F ≥ < dat i } , { F < ≥ dat i } , and { F ≥≥ dat i } ). Treating these numbers as analogous to the cumulative distribution function, theKolmogorov-Smirnov test statistic would be the maximum deviation between any two corresponding F …… i .The Kolmogorov-Smirnov test is sometimes criticized as not very powerful [e.g., Razali and Wah , 2011], soinstead, we calculate an analog of the Anderson-Darling test statistic: S = ∑ i ( ( F << dat i − F << pred i ) F << pred i ( − F << pred i ) + ( F ≥ < dat i − F ≥ < pred i ) F ≥ < pred i ( − F ≥ < pred i ) + ( F < ≥ dat i − F < ≥ pred i ) F < ≥ pred i ( − F < ≥ pred i ) + ( F ≥≥ dat i − F ≥≥ pred i ) F ≥≥ pred i ( − F ≥≥ pred i ) ) (8)i.e., the squared deviation between the predicted CDF and the observed CDF, weighted as in theAnderson-Darling statistic, summed over all four CDF types (inequalities), and summed over all points in theobserved data set. Some numerical problems arise when both numerator and denominator are exactly zero,i.e., one of the F … pred i is exactly 0 or 1, as occurs, for example, for F << pred i for a point, where E i = E i = . Sincesuch points involve exact match between predicted and observed CDF and therefore should not contributeto the test statistic, we add a small factor (0.001) to the denominator of each term to ensure the divisionof zero by zero results in zero. This does not significantly a ff ect the other terms in the test statistic or theoverall results.We then fit our predicted distribution to the observed distribution — minimizing S — by varying the param-eters of our calculated CDF with the downhill simplex algorithm ( Nelder and Mead [1965] as implemented in
Johnson [2014]). The optimal values are 𝜆 = − . , 𝜇 = keV, 𝜂 min = . , and 𝜂 max = .CARLSON ET AL. SPARK ENERGY SPECTRA ,,
Johnson [2014]). The optimal values are 𝜆 = − . , 𝜇 = keV, 𝜂 min = . , and 𝜂 max = .CARLSON ET AL. SPARK ENERGY SPECTRA ,, ournal of Geophysical Research: Atmospheres Figure 4.
Cumulative distribution function residuals for the best fitresults as discussed in the text. A positive deviation indicates a largerdata CDF than computed from the best fit parameters. Deviations aretypically less than 1%.
Since our multidimensional CDF andmodified Anderson-Darling test statis-tic are so far removed from their origi-nal application, we only use the statisticin judging the quality of the fit in theoptimization process described aboveand make no attempt to apply thedistributions typically associated withthe Anderson-Darling test statistic toassess the uncertainty in our fit results.Instead, we judge uncertainty by boot-strap, repeating our fit process manytimes with alternative data sets pro-duced from our original data by sam-pling with replacement. The fit resultsare each approximately normally dis-tributed, with mean and standard devi-ation given as follows: 𝜆 = − . ± . , 𝜇 = ± keV, 𝜂 min = . ± . , and 𝜂 max = ± . Plotting these bootstrapped fit results, one param-eter versus another, shows very little correlation between results, with the exception of 𝜆 and 𝜂 min which arenegatively correlated: more negative 𝜆 is associated with higher 𝜂 min . This makes sense given that more neg-ative 𝜆 is associated with more events with low X-ray fluence, so raising the minimum fluence is necessary toretain the balance between events with and without detectable signal.The CDFs computed from the best fit parameters are also shown in Figure 3 as thin curves, but they overlapwith the data so much that no systematic deviations are evident. A residuals plot showing the di ff erencesbetween the two sets of curves is shown in Figure 4. Some deviations can be seen but are well within thetypical size of the fluctuations due to random distribution of points in the data set.As a sanity check, Monte Carlo simulations of data sets drawn from the distributions described above showno obvious deviations from the distribution of the data when plotted as in Figure 2. It is also worth notingthat since our fitting procedure captures the joint distribution, it also captures the distribution within eachdetector separately. In this paper, we have not attempted to compensate for saturation to construct a trueenergy distribution, but Kochkin et al. [2015], using essentially the same experimental setup as used here, dopresent such a spectrum [
Kochkin et al. , 2015, Figure 12] constructed by a sophisticated procedure of fittingpulse shapes to observed oscilloscope records. Though the analysis in
Kochkin et al. [2015] includes the dataset used here, the analysis procedures are completely di ff erent, and Kochkin et al. [2015] include an additional2000 sparks and thus represent an approximately independent analysis. The
Kochkin et al. [2015] results arepresented for energy deposition per burst not per spark, but the fact that relatively few sparks have multiplebursts means that it is still useful to directly compare the results from
Kochkin et al. [2015] to equation (5).Evaluation of equation (5) with our fit result values of 𝜆 , 𝜇 , 𝜂 min , and 𝜂 max , normalized and overlaid with thedata from Kochkin et al. [2015], is shown in Figure 5, demonstrating much better agreement than the singleexponential distribution used in
Kochkin et al. [2015].
Kochkin et al. [2015] inferred a 200 keV characteristicburst energy (which is asserted to be roughly equal to characteristic photon energy) but showed an extremelypoor fit at high burst energies that they attribute to pileup. Properly accounting for pileup as we do here withthe Poisson distribution of observed counts ( p 𝜂 ( N ) ) and for fluence variability with the assumption of a powerlaw ( p 𝜆 ( 𝜂 ) ) gives us a much better fit with a much lower mean photon energy (86 keV), suggesting that pileupand fluence variation is essential to consider in interpretation of such data.As a final comparison to data, it is useful to consider attenuator experiments. While we ran no attenuatorexperiments in the data set described above, Kochkin et al. [2015] describe such experiments, again withessentially the same setup as used here. They report results for lead attenuators of varying thickness, deter-mining the fraction of sparks that produce observable signals in a single detector by running 50 sparks witheach attenuator.The distributions determined above can easily be used to construct results for attenuated scenarios forcomparison with the observations in
Kochkin et al. [2015] by Monte Carlo simulation. First, we draw a randomCARLSON ET AL. SPARK ENERGY SPECTRA ,,
Kochkin et al. [2015] by Monte Carlo simulation. First, we draw a randomCARLSON ET AL. SPARK ENERGY SPECTRA ,, ournal of Geophysical Research: Atmospheres Figure 5.
Comparison of the predicted total energy deposition spectrumfrom this paper with observations from Figure 12 of
Kochkin et al. [2015].The red squares show the observed spectrum, while the black curvesshow the predictions based on our distributions. The thick curve is theprediction, while the thin curves represent the ± √ N standard deviationexpected for the spread in the data. spark fluence, then draw a random num-ber of photons from such a spark, thendraw that many random photon ener-gies from the photon energy spectrum,then attenuate those photons proba-bilistically by use of the mass attenua-tion coe ffi cients from Hubbell and Seltzer [2004], and then finally classify the eventas containing detected X-rays or notbased on whether or not any pho-tons made it through the lead shield.We repeat this procedure times,compute the overall fraction of eventswith detected X-rays, and repeat foreach attenuator thickness. Results areshown in Table 1. Comparison with datarequires an estimate of the uncertaintyin the data, here estimated by calcu-lating a 68% confidence interval basedon the binomial distribution. Sixty-eightpercent confidence corresponds to a ± 𝜎 error bar, and all but one of our predictions are consistent with the observations at this confidencelevel. The distributions determined in this paper are thus also consistent with attenuator experiments, thoughfurther attenuator observations would be useful to narrow the uncertainties in the observations.
5. Discussion
To summarize, we have studied observations of X-ray emission by sparks and attempted to understand thoseobservations by modeling the process as a combination of a distribution of spark fluence, Poisson statistics,and an X-ray energy spectrum. These distributions successfully reproduce not only the data used here but alsoindependent analysis of similar experiments on overall energy distribution and signal attenuation in
Kochkinet al. [2015]. The two main results are a well-defined photon mean energy and the distribution of overall sparkfluence.The ± keV mean photon energy determined above is consistent with most earlier estimates but some-what lower than some. We feel this mean energy is quite reasonable given that electrons accelerated in theenvironment of a 1 MV spark realistically must have at most a few hundred keV of energy and bremsstrahlungphotons produced must have a lower energy than that of the source electrons. As a crude estimate of electronenergy, assume bremsstrahlung produced by the relatively low energy electrons here follows approximatelythe d N ∕ dE 𝛾 ∝ ∕ E 𝛾 distribution seen for bremsstrahlung produced by high-energy electrons. This ∕ E 𝛾 distri-bution spans the range from the energy of the electron as a maximum to a minimum energy at most 20 keV.Using 20 keV as the minimum energy and requiring that the average photon energy is 86 keV require theelectron energy to be roughly 200 keV, while using 5 keV as the minimum energy gives nearly 400 keV as the Table 1.
Attenuator Experiments From
Kochkin et al. [2015] and Predictions Based on theDistributions Determined in This Paper a Thickness (mm) Observed Fraction (%) 68% Confidence Interval (%) Predicted Fraction (%)0.0 32 25–40 351.5 8 4–14 113.0 5 2.7–11 6.64.5 0 0–3.6 4.26.0 0 0–3.6 2.87.5 2 0.3–6.4 1.9 a The 68% confidence intervals correspond to ± 𝜎 error bars and are calculated based onbinomial statistics. CARLSON ET AL. SPARK ENERGY SPECTRA ,,
Kochkin et al. [2015] and Predictions Based on theDistributions Determined in This Paper a Thickness (mm) Observed Fraction (%) 68% Confidence Interval (%) Predicted Fraction (%)0.0 32 25–40 351.5 8 4–14 113.0 5 2.7–11 6.64.5 0 0–3.6 4.26.0 0 0–3.6 2.87.5 2 0.3–6.4 1.9 a The 68% confidence intervals correspond to ± 𝜎 error bars and are calculated based onbinomial statistics. CARLSON ET AL. SPARK ENERGY SPECTRA ,, ournal of Geophysical Research: Atmospheres electron energy. This 200 – 400 keV energy range is not inconsistent with the potential available for electronacceleration at the time of X-ray production as seen in Figure 1, but this is a crude estimate in need of refine-ment by modeling of electron energy distributions and the resulting bremsstrahlung before it can be takenvery seriously.The exponential X-ray spectrum motivated by the expectation of an at least approximately exponential dis-tribution in electron energies very closely reproduces the observed distributions, though is likely not theonly such distribution to do so, and as noted above we have assumed that fluence ( 𝜂 ) and mean energy areindependent which may or may not be true. While the average energy should be close to correct, the otherdistributions at work in this system complicate the analysis.The more important factor at work is the very broad distribution of observed spark X-ray fluence ( 𝜂 ), which,as mentioned earlier, we assume captures both intrinsic variability in spark luminosity and variability in thespatial distribution of X-ray emissions. The results obtained above, which the distribution of 𝜂 is very hard(power law index − Kochkin et al. , 2014, Figure 2], so the number or distributionof streamer production cannot directly explain the broad distribution of spark X-ray fluence. As discussed in
Kochkin et al. [2015], the very short duration and appearance of multiple bursts of X-ray emissions suggeststhat X-ray emission occurs during some fast transient process like streamer collision, but it is not clear howstreamer collision frequency would be distributed and whether such a distribution could reproduce the char-acteristics observed here. While we have not proved that X-ray fluence truly follows a power law, only thatour assumption of a power law is consistent with our data, it is perhaps not unreasonable that a power lawdistribution might arise here in the context of dielectric breakdown since power laws arise in studies of sys-tems that similarly approach breakdown and then collapse [
Bak et al. , 1987]. Regardless of the true form of thedistribution, we hope our analysis here is useful for evaluation of X-ray production mechanisms, which mustexplain the frequency of both dim and bright sparks in roughly the balance described here.As for the relative contributions of spark luminosity variation and nonuniform X-ray spatial distribution to theobserved fluence variability, some crude analysis is instructive. Assuming no intrinsic luminosity variability, theobserved fluence variability should correspond roughly to the degree of spatial variability in X-ray emissions.One can set an upper limit on the spatial variability present in X-ray emissions in an experiment such as thisby examining the directional distribution of bremsstrahlung from a unidirectional beam of 500 keV electrons.Such directional distributions can be found, for example, in
Tseng et al. [1979, Figure 5] and
Köhn and Ebert [2014, Figure 7b]. Both figures present plots of the intensity of emissions versus angle relative to the directionof the electron beam, with
Tseng et al. [1979] showing a factor of around 100 di ff erence from highest to lowestfluence, while Köhn and Ebert [2014] show a factor of around 300. That the best fit distribution constructedhere requires a factor of 𝜂 max ∕ 𝜂 min = ∕ . = from highest to lowest fluence suggests that even thisextremely aggressive assumption of unidirectional 500 keV electrons falls short of explaining the observedfluence variability, but this analysis is admittedly crude.A slightly less crude assessment can be made by Monte Carlo: first, draw a random beam direction uniformlyfrom the high-voltage electrode in a hemisphere facing the ground electrode, then determine the angle fromthe center of that beam to the location of the detector and evaluate the relative fluence by extracting thedistributions shown in Tseng et al. [1979] and
Köhn and Ebert [2014] for a relatively high energy photon. Thoughthis ignores the X-ray energy dependence of the bremsstrahlung angular distribution, it should reasonablyaccount for the degree of variability. Repeat this procedure many times, compiling a histogram estimating thefluence distribution, then scale these histograms such that they are normalized and adjust the relative fluenceto correspond to 𝜂 by shifting the relative fluence until the histograms correctly predict the fraction of eventsthat would fall at intensities too low to be detected. Histograms constructed in this manner are compared tothe best fit 𝜂 power law in Figure 6.The similarity of slopes is striking, especially for the Köhn and Ebert [2014] case; but it is clear that fluence vari-ability due to beaming cannot fully account for the observations. In the context of the 86 keV average photonenergy determined here, the maximum fluence of the bremsstrahlung-derived cases, 𝜂 ∼ , would resultin 800 keV deposited on average, far from the 3 – 5 MeV needed to saturate the detectors. Put another way,if we attempt to account for the large fraction of low-fluence events simply as emissions beamed away fromCARLSON ET AL. SPARK ENERGY SPECTRA ,,
Köhn and Ebert [2014] for a relatively high energy photon. Thoughthis ignores the X-ray energy dependence of the bremsstrahlung angular distribution, it should reasonablyaccount for the degree of variability. Repeat this procedure many times, compiling a histogram estimating thefluence distribution, then scale these histograms such that they are normalized and adjust the relative fluenceto correspond to 𝜂 by shifting the relative fluence until the histograms correctly predict the fraction of eventsthat would fall at intensities too low to be detected. Histograms constructed in this manner are compared tothe best fit 𝜂 power law in Figure 6.The similarity of slopes is striking, especially for the Köhn and Ebert [2014] case; but it is clear that fluence vari-ability due to beaming cannot fully account for the observations. In the context of the 86 keV average photonenergy determined here, the maximum fluence of the bremsstrahlung-derived cases, 𝜂 ∼ , would resultin 800 keV deposited on average, far from the 3 – 5 MeV needed to saturate the detectors. Put another way,if we attempt to account for the large fraction of low-fluence events simply as emissions beamed away fromCARLSON ET AL. SPARK ENERGY SPECTRA ,, ournal of Geophysical Research: Atmospheres Figure 6.
Predicted fluence distributions assuming only spatialvariability due to beaming of bremsstrahlung from unidirectional500 keV electrons. The curve marked “Tseng” is derived from Figure 5of
Tseng et al. [1979], “Koehn” is derived from Figure 7b of
Köhn andEbert [2014], and “Power” is the best fit power law determined here. the detectors, the maximum fluencewould then be too low to accountfor events that saturate the detectors.The scale of variability in the distri-butions derived from bremsstrahlung,even for the extreme case of unidirec-tional 500 keV electrons, is therefore toolow to fully account for the variabilityobserved, providing clear evidence forstrong spark-to-spark variability. Crudelycomparing the spread of the
Köhn andEbert [2014] curve to our power law inFigure 6, there is at least another orderof magnitude variability in fluence thatcannot be explained as spatial variabilityand that is a very conservative estimatesince in a more realistic scenario therewill be directional dispersion of elec-trons, lower energy electrons, and possi-bly multiple beams, all of which lead toreductions in the spatial variability dueto beaming alone, thus requiring evenmore spark-to-spark variation.We hope the results described above will provide a basis for comparison to results of other spark experimentsto determine if the distributions of mean photon energy and spark X-ray fluence depend on the experimentalsetup. We further hope that theoretical work can find a basis for such distributions to confirm or refute ourassumptions and that such theories can shed light on the process of runaway electron acceleration in sparksas suggested to be relevant to other electrical phenomena such as lightning or terrestrial gamma ray flashes.
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The data described in this paper areavailable from the authors on request([email protected]). We arevery grateful to Lex van Deursen forthe valuable discussions. This studywas supported by the EuropeanResearch Council under the EuropeanUnion’s Seventh FrameworkProgramme (FP7/2007-2013)/ERCgrant agreement 320839 and theResearch Council of Norway undercontracts 208028/F50, 216872/F50,and 223252/F50 (CoE) and by theCarthage Summer UndergraduateResearch Experience program.
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