Effect of an electric field on liquid helium scintillation produced by fast electrons
N. S. Phan, V. Cianciolo, S. M. Clayton, S. A. Currie, R. Dipert, T. M. Ito, S. W. T. MacDonald, C. M. O'Shaughnessy, J. C. Ramsey, G. M. Seidel, E. Smith, E. Tang, Z. Tang, W. Yao
LLA-UR-20-23382
Effect of an electric field on liquid helium scintillation produced by fast electrons
N. S. Phan, ∗ V. Cianciolo, S. M. Clayton, S. A. Currie, R. Dipert, T. M. Ito, † S. W. T. MacDonald, C. M. O’Shaughnessy, J. C. Ramsey, G. M. Seidel, ‡ E. Smith, E. Tang, Z. Tang, and W. Yao Los Alamos National Laboratory, Los Alamos, New Mexico 87545, USA Oak Ridge National Laboratory, Oak Ridge, Tennessee 37831 USA Department of Physics, Arizona State University, Tempe, Arizona 85287, USA Department of Physics, Brown University, Providence, Rhode Island 02912, USA (Dated: May 8, 2020)The dependence on applied electric field (0 - 40 kV/cm) of the scintillation light produced byfast electrons and α particles stopped in liquid helium in the temperature range of 0.44 K to 3.12K is reported. For both types of particles, the reduction in the intensity of the scintillation signaldue to the applied field exhibits an apparent temperature dependence. Using an approximatesolution of the Debye-Smoluchowski equation, we show that the apparent temperature dependencefor electrons can be explained by the time required for geminate-pairs to recombine relative to thedetector signal integration time. This finding indicates that the spatial distribution of secondaryelectrons with respect to their geminate partners possesses a heavy, non-Gaussian tail at largerseparations, and has a dependence on the energy of the primary ionization electron. We discuss thepotential application of this result to pulse shape analysis for particle detection and discrimination. PACS numbers: Valid PACS appear here
I. INTRODUCTION
Scintillation of liquid helium (LHe) in response to thepassage of charged particles was discovered in the late1950s [1, 2]. Since then extensive studies have been car-ried out to illuminate the behavior of ions and neutralsin this unique substance [3–13]. More recently, there hasbeen renewed interest in studying liquid helium scintil-lation because of the potential application of LHe as aparticle detector and/or a target material in which toconduct nuclear, particle, and astroparticle physics ex-periments [14–22]. These experiments include solar neu-trino detection [23–25], a search for the permanent elec-tric dipole moment of the neutron [26–28], measurementof the free neutron lifetime [29], and detection of lightdark matter particles [30–33].These wide-ranging applications are motivated by oneor more of the following unique properties of LHe: (1)LHe can be made with very high purity. Apart from He, the only solute of any significance in liquid He ishydrogen, which has a solubility of 10 − at 1 K [34]. (2)Superfluid helium provides multiple signal channels, in-cluding electric charge, prompt scintillation, delayed scin-tillation, and elementary excitations, allowing for par-ticle identification. (4) The low mass of He providesrelatively good kinematic matching to search for GeVscale dark matter particles. (5) Superfluid helium canbe used to produce ultracold neutrons via downscatter-ing [35]. (6) LHe is a good electrical insulator [36]. (7) ∗ [email protected] † [email protected] ‡ george [email protected] Here the term “liquid helium” refers to either liquid He or liquidhelium with the natural isotropic abundance.
Dissolved spin-polarized He atoms can serve as a cohab-iting magnetometer [26]. (8) Dissolved He atoms allowneutron detection via the reaction He( n , p ) H, whose re-action products produce scintillation light in LHe [26].The passage of a charged particle in LHe deposits en-ergy into the medium by ionizing and exciting heliumatoms. Ionization creates electrons and ions, which thenthermalize with the LHe. The electron subsequentlyforms a “bubble” in the liquid, pushing away surround-ing helium atoms as a consequence of Pauli exclusion.The He + ion, on the other hand, forms a “snowball”attracting surrounding helium atoms. The bubbles andsnowballs recombine to form excited helium molecules(excimers). The excited atoms also form excimers byattracting nearby helium atoms. These excimers areformed in singlet and triplet states. The lowest singlet-state molecule radiatively decays in less than 10 ns to the(unbound) ground state, emitting an ∼
16 eV (80 nm)extreme-ultraviolet (EUV) photon and generating theprompt component of the LHe scintillation. The tripletstate molecule, on the other hand, has a lifetime of ∼
13 sin LHe [17]. In a high-excitation-density environment,the triplet-state excimers can undergo the Penning ion-ization processHe ∗ + He ∗ → + + e − , (1)or He ∗ + He ∗ → +2 + e − . (2)If a singlet excimer is formed as a result of Penning ion-ization, it produces the delayed scintillation component(sometimes referred to as “afterpulses”).The ionization density, and as a consequence, thecharge distribution about the particle track, depends onthe type of charged particle. For example, a 5.5 MeV α a r X i v : . [ phy s i c s . i n s - d e t ] M a y particle, such as those from an Am source, has a rangeof ∼ ρ = 0 .
145 g/cm ) [37].As a result, it produces dense, interpenetrating columnsof positive and negative charges. The radius of thecolumns was estimated to be ∼
60 nm [21]. The protonand triton from the He( n , p ) H reaction, which share akinetic energy of 760 keV, produce similar columns ofcharges, but with a lower density [21].On the other hand, a 364 keV electron, such as thosefrom
Sn, has a range of ∼ W value, defined as the average en-ergy loss by the incident particle per ion pair formed, of43 eV [39], the average separation of ionization eventsis ∼
840 nm, whereas the average separation betweenthe electron bubble and the helium snowball from anelectron-ion pair after they thermalized is ∼
40 nm [22].As a result, the thermalized ions from electron tracks aremost likely to recombine with their partners, a situationreferred to as geminate recombination.In both cases, if an electric field is applied, the re-combination process is suppressed since some fraction ofthe charges that would have otherwise recombined arepulled apart, resulting in a reduced scintillation yield forboth the prompt and delayed components. The separatedcharges are collected at the electrodes used to apply theelectric field. Importantly, only the component of scin-tillation light that results from recombination is affectedby the electric field. The component that results fromexcited atoms is left unaffected.To describe recombination under an applied electricfield, Jaffe’s columnar theory of recombination [40, 41]is most applicable to the high ionization density case,such as α particles and the protons and tritons from the He( n , p ) H reaction. On the other hand, Onsager devel-oped a theory to describe geminate recombination [42],taking into consideration the applied external field andthe Coulomb attraction between the two charges, as wellas thermal diffusion. The length scale of most rele-vance in this analysis is the so called Onsager radius, R = e / (4 π(cid:15)k B T ), which is the separation distance be-tween the two charges where the Coulomb potential iscomparable to the thermal energy. In LHe, this radiusis larger than 3.7 µ m, therefore thermal diffusion can beignored for moderate or higher fields [22].In applications of LHe to particle detection, an electricfield is often applied either to collect charge as one of thesignal channels or to satisfy other experimental require-ments. Therefore, it is of general interest to understandthe light/charge response of LHe as a function of thestrength of the applied electric field. We have measuredthe effect of an electric field on LHe scintillation producedby fast electrons using monoenergetic electrons with a ki-netic energy of 364 keV from a Sn for electric fields upto 40 kV/cm for helium temperatures between 0.44 Kand 3.12 K at a pressure of 600 Torr. This work wasconducted as part of the effort to develop an experimentto search for the permanent electric dipole moment ofthe neutron [26–28]. Previous works related to this effort as well as that for the measurement of the neutron life-time are found in Ref. [16–19]. Our group has previouslymeasured the effect of an electric field on LHe scintilla-tion produced by α -particles [21]. We have also reportedon the field dependence of the ionization current fromelectrons in LHe and the resulting determination of thecharge thermalization distribution [22]. Guo et al. [20]have reported on a measurement of the effect of an elec-tric field on LHe scintillation produced by electrons upto an electric field of 5 kV/cm at a single temperature of1.5 K. The work presented in this paper significantly ex-pands on both the electric field and temperature ranges.As described below, our data not only provide importantinformation for using LHe for particle detection in a widerange of physics experiments but also give new insightsinto the process of geminate charge recombination and itstime and temperature dependence. Furthermore, it maybe possible to apply these findings to particle detectionand identification through pulse shape analysis.This paper is organized as follows. Section II describesthe experimental apparatus and methods. Section IIIpresents the data and the details of the analysis methodemployed. Section IV discusses the results and their in-terpretation. II. EXPERIMENTAL APPARATUS ANDPROCEDUREA. Apparatus
The Medium-Scale High Voltage (MSHV) Test Appa-ratus [36] was used to perform the experiment presentedin this paper. The MSHV Test Apparatus is a cryo-genic apparatus designed and constructed to study elec-trical breakdown in LHe at temperatures as low as 0.4 Kfor pressures between the saturated vapor pressure and600 Torr. In this apparatus, the 6-liter Central Volume(CV), which can accommodate a pair of electrodes aslarge as 12 cm in diameter, is cooled down to 0.4 K us-ing a He refrigerator. A potential of up to ±
50 kVcan be applied to each of the electrodes. For this ex-periment, we replaced the MSHV electrode system withthe assembly depicted in Fig. 1. This assembly consistsof (1) a high voltage electrode 3.18 cm in diameter, onwhich
Sn and
Am radioactive sources were electro-plated, (2) an electropolished wire mesh that serves asa ground electrode, (3) a cylindrical light guide made ofpoly(methyl methacrylate) (PMMA), 2.54 cm in diam-eter and 2.54 cm in length, whose end facing the wiremesh was coated with vacuum evaporated tetraphenylbutadiene (TPB), which converts the 80 nm EUV lightto 400 nm visible light, and (4) a G10 structure to holdthese components together. As shown in Fig. 1, this as-sembly was mounted on one of the ports of the MSHVCV. The 400 nm light from the TPB-coated end of thelight guide is guided to the other end, through a sapphireviewport, and finally to a Hamamatsu R7725 photomuli-
PMT connections PMT HV feedline CV He pot HV electrode Ground ring Light guide Electroplated sources Wire mesh Sapphire window PMT TPB coating
FIG. 1: A diagram of the MSHV Test Apparatus (left) and an enlarged view of the light detection system (right).plier tube (PMT) for detection. This PMT is a modifiedversion with a Pt (platinum) underlay on the photocath-ode, which was shown to function at temperatures as lowas 2 K [21, 43].The PMT was thermally anchored to the 4 K shieldof the MSHV system [36]. As was done in Refs. [21] and[43], the base circuit for the PMT adopted the split de-sign, where the voltage dividing resistor chain, which isthermally anchored to the 4 K heat shield, was separatedwith a cryogenic ‘ribbon cable’ from the charge storingcapacitors, located directly on the PMT. The HV to biasthe PMT was supplied using a cryogenic HV coaxial ca-ble.Previous work has shown that the quantum efficiencyof the type of PMT used in this experiment decreasedby about 10% from room temperature down to 77 K,but became stable below this point [43]. Such an effectwould impact any comparison made between measure-ments taken far apart in time (many hours to a day)when the photocathode could potentially be at differenttemperatures, and hence result in different quantum effi-ciencies between measurements. To monitor the temper-ature of the PMT during the experiment, a temperaturesensor was attached to the base of the PMT. This partwas <
10 K when the first set of data were acquired.We chose
Sn as our electron source since it pro-vides monoenergetic electrons, giving a higher sensitiv-ity to the electric field effect than was possible with anelectron source with a continuous energy spectrum [20].The
Am source served as a calibration source andwas co-electrodeposited with the
Sn in a 6.35 mmdiameter spot at the center of the high voltage elec-trode. The
Sn and
Am sources had activities thatcorresponded to emission rates of 850 s − and 195 s − for 364 −
391 keV electrons and 5 . − .
544 MeV α -particles, respectively, into the liquid at the time of theexperiment; Sn has a half-life of 115 days whereas thehalf-life of
Am is 432.2 years.The gap between the wire mesh ground electrode andthe high-voltage (HV) electrode was 3.8 mm. This issmaller than the range of 364 keV electrons, which is ∼ ∼
62% of the emitted 364 keVelectrons ranged out in LHe. Those electrons that do notrange out in the liquid in the gap hit the PMMA surfacecoated with TPB or are backscattered toward the elec-trode. Due to the high penetrating power, <
1% of theelectron’s energy is deposited in the TPB layer, and sothe number of photons produced by these electrons di-rectly hitting the surface is negligibly small [38].One of the HV feedlines of the MSHV system [36] wasused to supply an electrical potential of up to 15 kV to theHV electrode. From the HV feedthrough on the CV tothe HV electrode, a HV feedline made of a polytetrafluo-roethylene (PTFE) insulated metal wire was used. Whenthe supplied HV was 15 kV, the electric field in the gapwas 40 kV/cm, and the highest field on the wire meshwas 90 kV/cm.FIG. 2: A sample digitized waveform. The red trianglesmark the locations of the detected peaks.
B. Data acquisition system
The signal from the PMT was sent to an ORTEC 474timing filter amplifier set with an 100 ns integration timeand 100 ns differentiation time. One of the outputs fromthe timing filter amplifier was sent to a linear amplifierand discriminator. The other output was transferred di-rectly to a DDC10 100 MHz waveform digitizer [44]. Thediscriminator, whose threshold level can be adjusted, wasused to trigger the digitizer. The digitized waveformswere continuously transferred to a DAQ computer andsaved to disk as the data were being acquired.The data for α -particles and electrons were acquiredsimultaneously with the trigger threshold set to a levelthat corresponded to ∼ µ s waveform was captured.Each waveform consisted of 2048 samples which were cap-tured at rate of 10 ns per sample. The trigger was offsetby 2.1 µ s from the start of the waveform time windowso that there was 2.1 µ s of pre-trigger data and 18.37 µ sof post-trigger data. Figure 2 shows a sample digitizedwaveform.It was found after the data had already been acquiredthat input reflection at the linear amplifier used in thedata acquisition system caused a reduction in the am-plitudes of the digitized signals. To correct for this, a pulse generator was used to measure the reduction as afunction of the amplitude of the input signal. We foundthat only the large signals produced by α scintillationwere affected, and the amplitudes of these signals werecorrected using the calibration measurements made withthe pulse generator. III. DATA AND ANALYSISA. Range of the measurements
Data were acquired for six different temperatures,ranging from 0.44 K to 3.12 K, at a pressure of approxi-mately 600 Torr rather than at saturated vapor pressure(SVP). Measurements were made at two temperaturesabove the lambda transition and four below it. For a setof measurements at a fixed temperature, the potentialdifference between the electrode and the wire mesh thatacts as a ground plane was ramped up from 0 to 15 kV.At 0.44 K, measurements were made with both polaritieson the high voltage electrode. Table I summarizes the pa-rameters for each of the datasets. The temperatures andpressures shown represent the average of the start andend values for each dataset. Here, we are referring toa dataset as a complete series of measurements over therange of voltages – for both polarities, when applicable –at one temperature.Uncalibrated ruthenium oxide (ROX) sensors wereused to monitor the temperature of the experimental vol-ume. The ROX sensors have a stated accuracy of ± ±
75 mK at 2.0 K [45]. The largest tem-perature variation (2.43 - 2.27 K), one which was welloutside the stated uncertainty of the ROX sensors wasexhibited by dataset V, and was the result of difficultyin stabilizing the temperature over the duration of thesemeasurements. With the exception of the 3.12 K dataset,where the pressure varied between 648 −
605 Torr betweenthe start and end of the measurement cycle, the varia-tion in pressure for all other datasets was only a few Torrduring the data acquisition of that dataset.In total, the measurements consisted of 43 subsets ofdata, and for each subset at a particular temperature,pressure, and electrode voltage setting, 2 × eventswere acquired. Of these, approximately 30% were Am α -particles [5485.7 keV (85.2%), 5443.0 keV (12.80%),TABLE I: Dataset and conditions Dataset T [K] P [Torr] E [kV/cm] ρ [g/cm ]I 0.44 607 (-40, 40) 0.1466II 0.84 611 (0, 40) 0.1466III 1.15 600 (0, 40) 0.1466IV 1.65 597 (0, 40) 0.1468V 2.35 601 (0, 40) 0.1472VI 3.12 627 (0, 40) 0.1418 Sn conversion elec-trons [363.76 keV (28.2%), 387.46 keV (5.48%), 391.0 keV(1.245%)]. Here the paired values in square bracketsrepresent the energies of the particles and their corre-sponding branching ratios, restricted to cases where thebranching ratio is at least one percent. The average en-ergy, ε , for the α -particles and electrons were 5446 keVand 368 keV, respectively. An event trigger was catego-rized as either an α -particle or an electron by the numberof photoelectrons in the prompt pulse. The total acqui-sition time for each data subset was approximately 10minutes, and each subset was calibrated with the singlephotoelectron distribution acquired in the same subset. B. Waveform analysis
A peak detection algorithm is applied to each digitizedwaveform. The algorithm detects the amplitude and lo-cation of each peak in the waveform. Examples, markedby red triangles, are shown in Fig. 2. Peaks are classi-fied based on their time locations as one of the follow-ing: a pre-trigger peak, a trigger peak (prompt pulse),an afterpulse. The afterpulses are generated by singlephotoelectrons, and their distribution is used to deter-mine the PMT gain. For this purpose, only afterpulsesafter 6 µ s are used for fitting the afterpulse ADC spec-trum. Furthermore, to prevent distortion to the singlephotoelectron spectrum from overlapping afterpulses, apulse time separation analysis cut is made to remove af-terpulses separated by less than 800 ns. However, theseanalysis cuts are not used in analyzing the time spectrumof the afterpulses, results of which will be presented in aforthcoming paper. C. Spectrum fitting
1. Afterpulse spectrum
Single photoelectron (SPE) pulses in the afterpulse re-gion of both the α and electron triggered waveforms areused to characterize the PMT gain, providing for the con-version between the measured ADC channel to numberof photoelectrons. The SPE spectrum is fitted with thePMT response function proposed by Bellamy et al. [46].There are 7 free parameters in the fitting function: Q , σ , Q , σ , µ , w , and α . The parameters Q and σ char-acterize the mean and width of the pedestal distribution. Q , σ , and µ characterize the PMT gain, distributionwidth, and source intensity, respectively. The final twoparameters, w and α describe the discreet backgrounddistribution. Once Q and σ are determined from a fitof the pedestal distribution, their values are fixed in thefit of the SPE distribution. Shown in Fig. 3a is a fit of theSPE distribution for one subset of the data. The energyresolution of the SPE response ( σ /Q ) obtained fromthe fit is ∼
32% and is consistent with the value expected for the type of PMT used in this experiment. The fittingprocedure is performed to determine the PMT gain foreach data subset. α prompt scintillation spectrum The α ADC spectrum of the prompt scintillation signalis fitted with an analytic peak-shape function proposedby Bortels & Collaers [47] to fit α spectra in Si detectors.The function consists of the convolution of a Gaussianwith the weighted sum of two left sided exponential func-tions to model the low energy tail in the spectrum and isgiven by f ( x ) = A (cid:40) − ητ exp (cid:18) x − µτ + σ τ (cid:19) × erfc (cid:20) √ (cid:18) x − µσ + στ (cid:19)(cid:21) + ητ exp (cid:18) x − µτ + σ τ (cid:19) × erfc (cid:20) √ (cid:18) x − µσ + στ (cid:19)(cid:21)(cid:41) . (3)In Eq. 3, µ and σ are the mean and standard deviationof the Gaussian component, τ and τ are the param-eters of the two exponential functions, η is a weighingfactor, and A is the overall normalization. A third expo-nential can also be included to further improve the fit insome instances, but we find that two terms are sufficientfor a good fit to our data and choose not to include theadditional term in our analysis. In principle, the α spec-trum for Am is fitted with multiple peaks and Eq. 3is written as a sum over the number of peaks in the fit.However, we choose to fit the spectrum with only onepeak because the secondary peaks in the spectrum arevery close in energy to the primary peak and also havemuch smaller branching ratios. The number of photo-electrons in the peak, N P E , is equal to µ/Q , where Q is the ADC gain determined from a fit of the afterpulseADC spectrum. Figure 3b shows a fit to the α promptscintillation spectrum of one data subset using Eq. 3.
3. Electron prompt scintillation spectrum
The
Sn electron prompt scintillation spectrum is fit-ted by a function composed of the sum of three Gaus-sians, each representing one of the energies of the con-version electrons, and an exponential function. The fit C oun t s (a) Afterpulse spectrum C oun t s (b) α spectrum C oun t s (c) Electron spectrum FIG. 3: (a) A sample fit of the afterpulse ADC spectrum for one subset of the data. The fit returns the PMT gain, Q , which is the ADC channel corresponding to one photoelectron. The left edge corresponds to an analysisthreshold for identifying pulses above the noise level. (b) A fit of the α ADC spectrum for one subset of the datausing the fit function given by Eq. 3. (c) A fit of the electron ADC spectrum for one subset of the data using thefitting function given by Eq. 4.FIG. 4: Simulated spectrum of the scintillation signalproduced by 364 keV electrons arriving at the PMT inthe experimental setup. The contributions to the totalspectrum (solid-filled gray) are from back-scatteredelectrons that range out in the electrode (dashed red),electrons that range out in the PMMA light guide(dotted purple), and electrons that range out in LHe(solid blue). Detector efficiency, resolution, and noiseare not included in the simulation.function is given by f ( x ) = N (cid:40) a exp (cid:34) − (cid:18) x − µσ (cid:19) (cid:35) + a exp (cid:34) − (cid:18) x − b µσ (cid:19) (cid:35) + a exp (cid:34) − (cid:18) x − b µσ (cid:19) (cid:35)(cid:41) + A exp ( − λx ) . (4) There are five free parameters in the fit. Two of the pa-rameters are the mean, µ , and width, σ , of the Gaussianfunction for the 364 keV peak. The mean values of theother two peaks are expressed as the mean of the firstpeak weighted by the energy of these two peaks relativeto the first. The widths are taken to be the same for allthree peaks, and the amplitude of each peak is weightedby its branching ratio. N is the overall normalization forthe Gaussian component. Two other parameters, A and λ characterize the exponential component.A fit of the electron prompt scintillation spectrum ofone subset of the data is shown in Fig. 3c, where the leftedge is due to an analysis threshold. The events that donot range out in the gap between electrodes form a lowenergy tail that extends below this threshold as shown inthe simulated spectrum in Fig. 4, which is obtained withGEANT4 [48] and the PENELOPE Low Energy Elec-tromagnetic Physics Model [49, 50]. In the simulation,the geometry of the light detection setup (Fig. 1) is mod-eled in GEANT4. Electrons with an energy of 364 keVare emitted from the surface of the electrode isotropicallyinto the liquid, generating EUV photons, some of whichstrike the TPB surface. These photons are then wave-length shifted by the TPB, and a fraction of them arecaptured and transmitted down the light guide, throughthe sapphire window, and finally end up at the PMTwhere they are counted. There are two components thatcontribute to the low energy tail, one from electrons thatrange out in the PMMA light guide( ∼ ∼ ∼ FIG. 5: PMT gain from fits of the afterpulse spectra forall 43 datasets acquired during this experiment. Thesize of the statistical error bars is smaller than the datapoints.
D. Results
1. PMT gain
The PMT gain for all subsets of the data acquired inthe experiment is shown in Fig. 5. The time order of thedata is as follows: 3.12 K (VI), 2.35 K (V), 0.44 K (I),0.84 K (II), 1.15 K (III), and 1.65 K (IV). For the 0.44 Kdata, the order of the voltage settings is the following:0 kV, 3 kV, 6 kV, 9 kV, 12 kV, 15 kV, 1 kV, 2 kV,-3 kV, -6 kV, -9 kV, -12 kV, -15 kV. We observed atrend of decreasing PMT gain with time, highlightingthe importance of individually calibrating each subset ofthe data with the afterpulse distribution from the samesubset.
2. Mean number of prompt photoelectrons
The mean number of photoelectrons in the prompt sig-nal (first 100 ns), N P E , is defined as the fitted ADCchannel peak value of the α (electron) spectrum dividedby the PMT gain, Q , measured at the same temperatureand field. The mean number of prompt photoelectronsas a function of electric field for all temperature datasetsis shown in Figures 6a and 6b.For the zero-field α data in this work, an approximately8% decrease in the detected prompt scintillation yieldbetween 2.35 K and 0.44 K is observed (Fig. 7a). Incomparison, the data from Ito et al. [21] which were takenat SVP show about a 9% reduction over approximatelythe same temperature range. Density effects are a likelyexplanation for this small difference. For more discussion,see Sec. IV A.For the zero-field electron data, we observe an increasein the detected prompt scintillation signal with decreas- ing temperature as shown in Fig. 7b. The trend exhibitedby these data is the inverse of the trend observed for α -particles (Fig. 7a). In comparison, the results for electronscintillation from Kane et al. [11] are strikingly differ-ent. They observe a relatively flat yield for temperaturesabove the lambda transition and a similarly flat yield fortemperatures below it. However, the low temperatureyield is reduced by about 5% relative to their high tem-perature yield and a very steep, almost discontinuity-likechange around the lambda transition is observed, form-ing a step-like function. We discuss in detail the originof the observed temperature dependence in Sec. IV B.Our data and our understanding of the phenomenon arenot consistent with the sharp discontinuity observed byKane et al. . Their observation could be the result of theirparticular methodology and experimental setup.
3. Normalized prompt scintillation yield
For analyzing the effect of the applied field on the scin-tillation signal, it is more convenient to examine the zero-field normalized scintillation yield, y , which is defined as y ( E, T ) ≡ N P E ( T, E ) N P E ( T, E = 0) . (5)Here, N P E is mean number of detected prompt pho-toelectrons at a given temperature T and electric field E and N P E ( T, E = 0) is the number of photoelectronsdetected at the same temperature T and zero field. InFig. 8a and 8b, the zero-field normalized detected promptscintillation yield as a function of the applied electric fieldis shown. The normalized α prompt scintillation yieldexhibits an interesting, and perhaps surprising, temper-ature dependence of the yield reduction with field. Thisfeature was not observed by Ito et al. [21] over the tem-perature range of 0.2 K to 1.1 K. Consistent with theirobservation is the absence of a temperature dependencein our data below 1.15 K. The yield reduction that weobserve between 0 and 40 kV/cm at 0.44 K is about 11%,which is in good agreement with the results from Ito etal. . Small differences are attributable to the uncertaintyin the electric field in both experiments. For this ex-periment, the estimated uncertainty in the gap spannedby the the high voltage electrode and the ground grid,and hence the electric field, is ∼ ∼
600 Torrwhile those from Ref. [21] are taken at SVP, and den-sity effects may play a role. The electron light yield andits field dependence will be discussed in more detail inSection IV B.
4. Effect of voltage polarity on electrons
Given that most of the ∼
364 keV electrons from the ra-diation source used in this experiment will transverse the -5 0 5 10 15 20 25 30 35 40 45140150160170180190200210 (a) Detected prompt N PE for α -particles (b) Detected prompt N PE for electrons FIG. 6: The absolute mean number of detected PEs in the prompt signal (first 100 ns) for α -particles (a) andelectrons (b) as a function of the applied electric field. (a) Zero field detected prompt N PE for α -particles (b) Zero field detected prompt N PE for electrons FIG. 7: The absolute mean number of detected PEs in the prompt signal (first 100 ns) at zero field for (a) Am α -particles and (b) 364 keV electrons from Sn.entire high field region between the electrodes, the effectof the electric field on their energy and trajectories mustbe considered. For this purpose, measurements were ob-tained for a potential difference between the electrodeand wire mesh of -15 kV to +15 kV. These measure-ments are only made at T = 0 .
44 K, and the normalizedscintillation yield of the prompt signal at this tempera-ture for the two electrode polarities is shown in Fig. 9(top plot). Note that the horizontal axis represents theabsolute value of the electric field.With a range of ∼ Sn conversion electronsin LHe and a high field region gap size of ∼ ∼
1% at the highestfield measured. Also notice that the normalized detectedlight yield for the negative polarity data is higher thanthat for the positive polarity data over the entire rangeof electric fields. This is consistent with the expectationthat the electron energy is slightly boosted by the fieldwhen the voltage on the source plated electrode is nega-tive. α/β ratio The co-electrodeposition of the α and conversion elec-tron sources in our experiment allows for simultaneousmeasurements of the scintillation yield produced by these (a) Normalized detected prompt yield y ( E ) for α -particles (b) Normalized detected prompt yield y ( E ) for electrons FIG. 8: The normalized detected scintillation yield of the prompt signal (first 100 ns) as a function of the appliedelectric field at six different temperatures for (a) α -particles and (b) electrons. The temperature dependence seen inthe electron data is due to the signal integration time and is discussed further in Sec. IV B. FIG. 9: Top: The normalized scintillation yield in theprompt signal as a function of electric field for the twovoltage polarities at T = 0.44 K. Bottom: The halfdifference in the yield between the polaritymeasurements.two sources. From this, a determination of the α/β ratio,or “quenching factor”, η , is quite straightforward. Thisquantity is the ratio of the number of prompt photo-electrons from α excitation per unit energy deposition toprompt photoelectrons from electron excitation per unitenergy deposition and is defined as η ≡ R α R β , (6)with R α ( β ) = N P Es,α ( β ) ε α ( β ) × ω α ( β ) . (7)The parameters ε and ω are the primary particle en-ergy and the scintillation photon geometric acceptance factor, respectively. The latter is the fraction of emit-ted EUV photons generated by the corresponding par-ticle that strike the TPB surface, which is determinedthrough a GEANT4 simulation using the PENELOPELow Energy Electromagnetic Physics Model. For every α decay, N γ,α primary photons are generated and emit-ted isotropically, and of those, the number that strikethe TPB surface on top of the light guide, N TPB ,α , iscounted. A histogram of N TPB ,α is accumulated and fit-ted with a Gaussian. The fitted mean value is dividedby the number of primary photons to obtain the photongeometric acceptance factor, ω α . The same procedure iscarried out for electrons from Sn to obtain the elec-tron geometric acceptance factor, ω β , but the fit functionused for the electrons is given by Eq. 4. The simulationis carried out for each liquid density listed in Tab. I. For α -particles, the average geometric acceptance factor is0 . ± .
001 with negligible changes with liquid den-sity. The factor for electrons is 0 . ± .
002 between0 . − .
15 K, 0 . ± .
002 between 1 . − .
35 K, and0 . ± .
003 at 3.12 K. All stated errors are statistical.We note that the estimated uncertainty in the positionsof the components of the light collection system is ∼ ∼ η , as a function of the temper-ature at zero field. The error bars are statistical and arederived from the errors in the fits of the spectral peaks.The ratio has a striking temperature dependence, andthis dependence is not merely a density effect becausesimulations shows that the geometric acceptance factorchanges very little over the temperature range in the ex-periment for both particle types. This effect is tied tothe temperature dependence of the zero-field scintillation0 FIG. 10: The α/β ratio as a function of temperaturecorrected for geometric acceptance. Note that theprompt signals from both particles are integrated overthe first 100 ns and the apparent temperaturedependence of the ratio is a result of this. Refer toSec. IV B for more details.yield as shown in Figs. 7a and 7b. The reason behind thisis discussed in more detail in Sec. IV.Several previous investigations of the α/β ratio havebeen made. One of the earliest measurements was madeby Miller[12], who measured a value of 0.182 (3% stateduncertainty). The two sources used in his experimentwere
Am and Co, and the wavelength shifters werePOPOP and DPS. Presumably, the measurement wasmade at ∼ α/β ratio in the literatureinclude those from Adams [14] and Adams et al. [15]. Inthe former work, it is stated that 35% of an electron’senergy goes into scintillation light while for an α particleit is only 10%. This implies an α/β ratio of 0.29. How-ever, in Ref. [15], the fraction of energy that goes intoscintillation is stated to be 24% and 10% for the electronand α , respectively. The ratio implied by these valuesis 0.42, and this value appears to be consistent with ourlowest temperature result of 0.45.McKinsey et al. [19] has also measured the ratio, statinga value of 0 . ± .
10. The α and electron sources used intheir experiment were Am and
Sn, respectively, andthe wavelength shifter used was TPB doped polystyreneat 40% concentration; thus the setup of their experimentis very similar to the one used in this work. Within thestated uncertainty, their result is consistent with our new results.
IV. DISCUSSIONA. Temperature and electric field dependence of α prompt scintillation The temperature dependence of the prompt scintilla-tion yield at zero electric field for α -particles plotted inFig. 7a is consistent with what was reported in Ref. [21].A similar temperature dependence, that is, a reductionin scintillation yield with decreasing temperature, wasobserved in the past [4–10]. As pointed out in Ref. [21],however, the results from these past experiments cannotbe directly compared to the results reported here and inRef. [21]. The electronics integration time for the scin-tillation pulse was ∼ µ s in the previous experiments,whereas it was ∼
100 ns in the experiments reported hereand in Ref. [21]. The scintillation pulse in the past ex-periments are likely to have included part of what we call“afterpulses”, which have their own temperature depen-dence [21].In a series of papers, Hereford and collaborators (seeRef. [10] and references therein) described a model forscintillation light production. They attributed LHe scin-tillation to radiative destruction of some metastablestates due to interactions with some collision partners.The temperature dependence of the scintillation yieldcomes from the temperature dependent diffusion con-stant of these species in LHe, which affects the rate ofexpansion of the column that contains these species.However, this picture is incompatible with our currentunderstanding of the phenomenon. Here, the promptscintillation is due to radiative decay of excited singletspecies (excimers and atoms). Furthermore, the temper-ature inside the column created by the passage of an α particle is about 2 K irrespective of the temperature ofthe bulk LHe when it is below ∼ α -induced scintillation shown in Fig. 8a. As discussed inRef. [21], in the presence of an electric field, the fractionof ions that recombine depends on a single parameter f = √ π(cid:15) bE/ ( N e ), where b is the Gaussian width ofthe charge column and N is the number of charges perunit length along the track. At lower temperatures, b tends to expand faster, but an increase in b has the sameeffect as increasing the electric field, E . Thus, at a givenfield, its effect is magnified when b increases.1 B. Temperature and electric field dependence ofelectron prompt scintillation
At the highest field measured, the electron data showa reduction in the prompt scintillation yield of >
1. Model of scintillation yield vs electric field
Consider the following model for the dependence of theprompt scintillation yield on electric field. Let a be thenumber of electrons and positive ions that recombine assinglets, b the number that recombine as triplets, and x the ratio of the number of singlet excitations to the to-tal number of ionizations. Then the prompt scintillationyield as a function of field is Y ( E ) = a − aa + b I ( E ) + x ( a + b ) , (8)with I ( E ) being proportional to the ionization current.The first term corresponds to the prompt scintillationyield due to excimers in the absence of an electric field.The second term represents the reduction due to theelectric field. The third term represents the contribu-tion from excited atoms. Normalizing the current to thevalue at E = ∞ , I ( ∞ ) = a + b , the normalized current,i.e. the fraction of charges that escape recombination atthe given electric field, is i ( E ) = I ( E ) a + b . (9)The prompt scintillation normalized to the value at E =0, namely, Y (0) = a + x ( a + b ) , (10)results in the normalized prompt scintillation as y ( E ) = 1 − i ( E )1 + x (1 + b/a ) = 1 − f s i ( E ) , (11)where f s is the fraction of the prompt scintillation lightfrom ionization.Sato et al. [51] calculated the ratio of the number ofdirect excitations to ionizations in helium to be 0 .
45 forelectron recoils. Among the excited atoms, 83% are inthe spin-singlet state and the remaining 17% are in thespin-triplet states[51]. For the excimers that form onrecombination, experiments indicate that approximately 50% are in excited spin-singlet states and 50% are in spin-triplet states [14]. With x = 0 .
37 and b/a = 1 .
0, Eq. 11reduces to y ( E ) (cid:39) − . i ( E ) . (12)Thus, the zero-field normalized electron scintillation yieldhas a very simple dependence on the ionization current,and measurements of the latter have been made by Seidel et al. [22]. However, there is some uncertainty in theprefactor, f s = 0 .
57, in front of the ionization currentterm in Eq. 12. This is primarily due to the uncertaintyin the parameter b/a . Conservatively, we can take theuncertainty of this parameter to be 50%, and this wouldcorrespond to an estimated lower and upper bound on theprefactor of f s = 0 .
52 and f s = 0 .
64, respectively. Hence,the sensitivity of Eq. 12 to deviations of the parameter b/a from 1.0 appears to be quite modest. In the following,we will perform calculations using all three values of f s to gauge the robustness of the model to the imperfectknowledge of its parameters, and whenever not specified,the value of f s is taken to be 0.57.Utilizing i ( E ) from Seidel et al. [22], a comparison ofthe model prediction with data is shown in Fig. 11. Thereis fair agreement between the f s = 0 .
57 curve and thelow temperature data ( < f s . Additionally, eventhough the model appears to have better agreement withour low temperature data, there still exists a discrepancy,particularly in the low field region (below ∼ FIG. 11: The normalized prompt light yield as afunction of the applied electric field for 364 keVelectrons from
Sn. The curves are the expected yieldfrom Eq. 12 with the ionization current from Seidel etal. [22] for different values of f s . Refer to text for moredetails about the apparent temperature dependence.2
2. Zero-field temperature dependence
It is clear that the model is unable to account for theobserved temperature dependence shown in Fig. 11, andit is possible that this is merely a consequence of an in-adequate model. But from a different perspective, thereasonable agreement between model and data even withthe use of an independently obtained ionization currentmeasurement and numerical values of model parametersindicates there is at least some merit to this model. Let usthen suppose that there is an alternative explanation forthe discrepancy and that our model has some veracity. Ifsuch is the case, a clue to the origin of the temperaturedependence appears to lie in the zero-field scintillationdata given that the normalization is performed with re-spect to them. That an effect emergent at zero field couldalso appear in the finite field data is not surprising. Theabsolute scintillation yield shown in Fig. 6b is suggestiveof this very possibility. It shows that for fields greaterthan approximately 15 kV/cm, the absolute yield for dif-ferent temperature datasets converge to approximatelythe same value. This is indicative of an effect that ismanifested at zero or low fields but become greatly di-minished, or possibly vanishes entirely, at higher fields.If we consider the prompt signal at zero field as a func-tion of temperature as shown in Fig. 7b, it is immediatelyapparent that the amount of detected light is much higherfor the data measured at temperatures below the lambdatransition. For instance, the detected yield at 3.12 K isapproximately 12% lower than that measured at 0.44 K.Perhaps, the behavior is merely a result of a liquid den-sity temperature dependence? However, a more carefulexamination reveals that a change in the liquid densitycannot be the primary reason for the observed behav-ior. Since our measurements are acquired with the liquidunder a pressure of ∼
600 Torr, the difference in densitybetween 0.44 K and 3.12 K is only about 3%, and thereare very small density differences for temperatures in therange of 0.44 K to 2.35 K (Tab. I). Therefore, densityeffects alone cannot explain the observed temperaturedependence of the zero-field data, particularly the <
3. Ion mobility and recombination time
Of the many properties of LHe that are known tochange with temperature, the mobilities of ions are ofparticular interest because they affect the recombina-tion process. As shown in Fig. 12 the zero-field mobil-ity of both the positive and negative ions changes ratherrapidly below the lambda transition [52]. The changehere mirrors what is observed in the absolute scintilla-tion yield of Fig. 7b. It is therefore conceivable that aneffect arising from a temperature dependent ion mobil-ity may explain the zero-field temperature dependence ofthe electron prompt scintillation yield.The mechanism by which the temperature dependent -2 FIG. 12: The zero-field mobility of positive andnegative ions in LHe as a function of temperature fromDonnelly & Barenghi [52].mobility would affect the detected prompt light yield hasto due with the finite recombination time for the thermal-ized ion pairs. In this experiment, the integration timefor the prompt pulse is set to 100 ns, so that scintillationlight emitted by excimers formed through recombinationafter this time will not be accumulated in the prompt sig-nal. Qualitatively, the trend of the ion mobilities shownin Fig. 12 is not inconsistent with the observed lowerlight yield detected at higher temperatures; the mobili-ties are much smaller at these temperatures, so that theions will take longer to recombine as compared to whenthe temperature is below the lambda transition, resultingin a reduced detected light yield for a given finite signalintegration time window.To illustrate this point more quantitatively, considerthe simple case of a single pair of ions separated by dis-tance r . Once they have thermalized, the pair will drifttowards one another due to their mutual Coulomb at-traction. Under the assumption of a field-independentmobility and ballistic ion motion, an estimate of the re-combination time, τ r , is given by τ r = 4 π(cid:15) (cid:15) r qµ r = 4 π(cid:15) q ( µ + + µ − ) r , (13)where µ = µ + + µ − is the combined mobility, q theelectric charge, and (cid:15) = (cid:15) (cid:15) r is the permittivity of LHe.Williams [53] obtained the same estimate of the recom-bination time and also applied the Nernst-Einstein rela-tion, D = µk B T /q , to relate the mobility to the diffusioncoefficient.From Seidel et al. [22], the typical separation for ion-pairs produced by an electron is ∼
40 nm with 10% ofthe ion-pairs having an initial separation greater than100 nm. Considering that the difference in the zero-fieldlight yield between 0.44 K and 3.12 K is ∼ r = 100 nm is τ r ≈
38 ns3for T = 3 .
12 K but is < T = 3 .
12 K is of the same orderas the signal integration time. Certainly, the assumptionof ballistic motion is not entirely accurate because diffu-sion is present. But in the more accurate description thatincludes the effects of diffusion, the true recombinationtime is longer than the estimate obtained from Eq. 13.In fact, Ludwig [54] has shown that Eq. 13 overestimatesthe recombination rate (i.e. underestimates the recombi-nation time). Thus, this estimate can be thought of asrepresenting somewhat of a lower bound on the recombi-nation time.An upper bound on recombination time can be ob-tained by considering the case of diffusion dominated mo-tion. In such a case, the average time for a displacement, r , is τ r,D = r D + + D − ) , (14)with D + and D − being the coefficients of diffusion forthe positive and negative ions, respectively [55]. For r = 100 nm and T = 3 .
12 K, the diffusion dominatedrecombination time is τ r,D ≈ T λ ; the pos-itive ion does not become trapped in a vortex ring un-less T (cid:46) .
65 K, and a negative ion will bind only upto T ∼ . T = 1 .
12 K,the transit time for a negative ion to transverse somegiven distance is about a factor of four higher when itis trapped versus when it is free [56]. But at this tem-perature, the estimated recombination time is < r = 100 nm, suggesting that the effect of vortex rings ismostly negligible to the present discussion.A more rigorous treatment of the time-dependentproblem of geminate recombination, that of isolatedpairs of ions undergoing diffusive motion in their mutualCoulomb field, must start with the Debye-Smoluchowskiequation, ∂ρ ( (cid:126)r, t ) ∂t = ∇ D · (cid:20) ∇ ρ + ρk B T ∇ U ( r ) (cid:21) , (15)where ρ ( (cid:126)r, t ) is the probability density of one ion relativeto its partner at time t , D = D + + D − is the diffusioncoefficient, T is the absolute temperature, and U is theinteraction energy for an isolated pair of ions [57].The well-known Onsager theory of geminate recombi-nation [42] corresponds to the solution of this equationin the limit of t → ∞ . As such, the solution is onlyapplicable in the steady-state situation and does not in-clude any time dependence of the recombination process. Nonetheless, approximate time-dependent solutions havebeen obtained by several authors [58–66], and the fullanalytic solution was obtained by Hong & Noolandi forboth the special and general cases of without and with anexternally applied electric field [67, 68]. Unfortunately,their full analytic solutions are found in Laplace trans-form space and as a consequence are immensely compli-cated and difficult to evaluate numerically, so that ap-plication to the analysis of experimental data is not en-tirely straightforward. In the discussion that follows, anapproximate solution in the time domain developed byGreen et al. [66], which is applicable to the situation inwhich the initial ion pair separation distance is signifi-cantly smaller than the Onsager radius and the mediumhas a low permittivity, is used for the analysis of ourdata.
4. Recombination survival probability -10 -9 -8 -7 -6 -5 FIG. 13: The recombination survival probability as afunction of time for two different values of r at T = 1 . T = 3 . etal. [66].When discussing recombination, the quantity of partic-ular interest is the survival probability, Ω( r, t, T ), whichis the fraction of particles that have not recombined inthe system at time t , temperature T , and initial sep-aration r . Shown in Fig. 13 is the calculated survivalprobability using the Green et al. approximate solution[66] to the Debye-Smoluchowski equation. The probabil-ity is calculated for two different initial ion separationsof 100 nm and 150 nm at a temperature of 1.0 K and3.0 K by making use of the mobility data from Donnelly& Barenghi [52]. The results indicate that the survivalprobability at 3.0 K approaches 1.0 for an initial separa-tion of r = 150 nm with t = 100 ns, the signal integra-tion time in the experiment. At 1.0 K, the survival proba-bility is zero for both initial separations when t = 100 ns.4Notably, the survival probability changes rather rapidly,falling from one to zero within a single decade of time.This behavior is implied by the power-law dependence ofthe initial separation in the recombination time from ourearlier estimate from Eq. 13.The implication of this is that any geminate pairs withan initial separation greater than ∼
150 nm will not con-tribute to the prompt scintillation signal for the chosenintegration time at 3.0 K but will contribute at 1.0 K.The degree to which the recombination of geminate pairscontributes to the signal will depend on the distributionof their initial separation.
5. Thermalization distribution
A determination of the temperature dependence of thelight yield requires knowledge of the thermalization dis-tribution. The initial separation of an ion-pair is alsotypically referred to as the thermalization length, so wewill adopt this latter terminology in the discussion thatfollows. The secondary electrons produced in the wakeof an ionization track will have a distribution of thermal-ization lengths relative to their geminate partners, N ( r ),and in general the thermalization length distribution isalso a function of temperature. The survival probabilityaveraged over this distribution is given byΩ( t, T ) = (cid:90) ∞ Ω( r, t, T ) N ( r )4 πr dr. (16)It then follows that the normalized detected light yieldas a function of temperature is y ( t, T, E = 0) = f r (cid:2) − Ω( t, T ) (cid:3) + f ex , (17)where f r and f ex are the fraction of scintillation due torecombination and excitation at zero field, respectively.Implicit in Eq. 17 is the assumption that any time depen-dence in the excitation component of the signal is negligi-ble compared to the experimental integration time. Thisis supported by experimental observation that the singletexcimers radiatively decay within 10 ns of the initial ion-ization event [18], a significantly shorter timescale thanthe signal integration time of our experiment.As discussed below, N ( r ) can be determined from theionization current data, or equivalently the scintillationyield dependence on the electric field. We will determine N ( r ) from our scintillation data. However, fitting thelight yield for the two highest temperature datasets toderive the distribution is not possible because the pres-ence of the recombination effect on the zero-field signaldirectly affects the field dependence. For this reason, wewill use the 0.44 K data to derive the distribution.The thermalization distribution is determined by usingthe analytic method developed by Seidel et al. [22]. Theionization current, i ( E ), is determined from the normal-ized detected light yield, y ( E ), from Eq. 12. Then, N(r) is obtained from the relation N ( r ) = 4 π / (cid:15) / E / e / ddE (cid:18) i + E didE (cid:19) (18)which applies to motion in a viscous medium [22].For fitting the detected light yield, the use of a splinepolynomial model is not preferred because it would re-strict the fit to the range of the data. Furthermore, thesparseness of our data points makes such a fit model un-reliable. Instead, a more reliable fit can be obtained byspecifying a functional form. For this, we choose thesimple analytic expression proposed by Boag [69] andrediscovered by Thomas & Imel [70] in their efforts todetermine the fraction of charges escaping initial recom-bination.We note that there are possible objections to using thismodel given its lack of physical justification. However,its experimental success and general applicability to bothcolumnar and cluster recombination show its usefulness,even if only as an empirical model. But it is also im-portant to note that although the model’s use provides astraightforward and convenient means for obtaining thesought after thermalization distribution, our use is lim-ited to only its functional form and does not represent ameans to obtain a determination of any physical param-eters. FIG. 14: Fits of the normalized current derived fromthe scintillation measurements and the model in Eq. 12for the T = 0 .
44 K data with a function of the formproposed by Refs. [69] and [70].The fit of the normalized current for the T = 0 .
44 Kdataset using the chosen functional form is shown inFig. 14. The goodness of fit is indicated by the degree offreedom (DOF) adjusted R value. We note that the datapoints used in these fits are derived from the midpointsspanned by the positive and negative polarity datasets.For the two field data points ( ∼ f s , and the best fit is obtained for f s = 0 . -6 -5 -4 FIG. 15: The thermalization distribution derived fromscintillation measurements in this work for differentvalues of f s and the distribution from Seidel et al. [22].In Fig. 15, the distributions obtained from the fit ofthe T = 0 .
44 K dataset with different f s values alongwith the distribution derived from the ionization currentmeasurements of Seidel et al. [22] are shown. The de-rived distributions are similar regardless of the value of f s , and so the shape of the distribution appears to berobust against uncertainty in the parameter b/a of themodel. The important feature to highlight is the presenceof a much heavier tail in the distributions derived fromthe scintillation measurements in this work. In particu-lar, the fraction of charges with separation greater than100(150) nm is 25%(15%) for the distribution ( f s = 0 . et al. [22]. At separations greaterthan 100 nm, our distribution has approximately an r − dependence. The probability density function for a ther-malization length r is N ( r ) r and thus has a r − depen-dence. Interestingly, in a Levy walk, the distribution ofstep sizes is f ( x ) ∼ | x | − (1+ α ) with 0 < α < et al. [22]. Infact, this result is expected if the field dependence of theelectron scintillation is well described by the model pro-posed in Section IV B 1. The reason for this expectationis that the curves from Fig. 11 are suggestive of a heaviertailed thermalization distribution; the data show a largeryield reduction at low fields than what the model predictswith the ionization current from Seidel et al. [22]. How-ever, it is important to note that this does not necessarilysuggest that the distribution from their work is in error.Rather, it is likely that their distribution is not applicableto the scintillation data from this work due to differencesin experimental setups. In particular, the energies of the electron sources used in the two experiments are quitedifferent (mean energy of 17 keV for Ni vs 364 keV for
Sn ), and so the distribution of separation distances ofthe thermalized charges need not necessarily be the samebecause the initial energy and spatial distribution of thesecondary electrons are not necessarily identical. We willexpand upon this point in more detail in Section IV C.Utilizing the model derived thermalization distribu-tions, the predicted light yield as a function of tempera-ture at zero field as given by Eq. 17 is shown in Fig. 16.The distributions are obtained from a fit of the scintilla-tion data at 0.44 K. The fraction of prompt scintillationlight due to recombination and excitation is taken to be f s and 1 − f s , respectively. Corrections for density effectson the thermalization distribution are included in the cal-culations. This is done by letting r → r ( ρ /ρ T ), where ρ is the density at 0.44 K and ρ T is the density at tempera-ture T . Additionally, the data points incorporate changesto the photon geometric acceptance onto the TPB wave-length shifting coating due to density/temperature ef-fects, and these are accounted for with simulations. FIG. 16: The normalized scintillation yield as afunction of temperature normalized to the yield at0.44 K (black triangles), and the yield obtained fromcalculations of the recombination probability for thethermalization distributions derived from model fitswith different values of f s .The calculated temperature dependence of the zero-field yield reproduces the general trend of the data. Un-certainties in the value of the prefactor f s does not appearto significantly alter the behavior of the yield with tem-perature. The lack of complete agreement between calcu-lations and data may be due to the presence of other ef-fects besides finite recombination time. Perhaps, anothertemperature dependent effect influencing the scintillationyield exists, one which is not accounted for by our model.Moreover, it must also be mentioned that in the discus-sion thus far we have assumed a field-independent mo-bility. However, experiments show that this assumptionis only valid for fields up to a few 100 V/cm, beyondwhich point the mobility decreases. When the ion-pair6separation is less than ∼
100 nm, the field strength is > ∼
60 m/s),above which the mobility reduces with increasing fieldstrength [3]. But when the drift velocity is at the Lan-dau limit, the time to transverse the final 100 nm is onlya few ns, so this effect appears negligible when comparedto the signal integration time. At higher temperatures,a decrease in mobility at high fields may still play a sig-nificant role.
C. Energy dependence of thermalizationdistribution
The spatial distribution of thermalized secondary elec-trons with respect to their geminate partners depends onthe initial energy distribution of the electrons and the en-ergy loss processes they undergo in the medium. For in-stance, a 10 eV electron is estimated to require on the or-der of 10 collisions before being thermalized in a sphereof approximately 100 nm from its positive ion partner[22]. In principle, knowledge of the initial energy distri-bution and all the energy loss processes allows for thedetermination of the spatial distribution of the thermal-ized electrons. However, in practice, that determinationis not so straightforward. By restricting our attention toonly the energy distribution of secondary electrons, theproblem is made much more tractable while still provid-ing useful information about the energy dependence ofthe thermalization distribution. -15 -10 -5 FIG. 17: The single differential ionization cross-sectionfor primary electron energies of ε = 17 and 364 keV inhelium calculated using the model proposed by Kimand Rudd [72].To start, let us consider the single differential ioniza-tion cross-section for electron impact, dσ/dW , as a func-tion of the secondary electron energy, W , in helium. Thisis plotted for two different primary electron energies inFig. 17 [72]. The formulas used for this calculation arevalid for relativistic primary electron energies and are an extension of the binary-encounter-dipole model pro-posed by Kim and Rudd [73]. The lower energy curverepresents the energy distribution of secondary electronsfrom the Ni source (mean primary electron energy of17 keV) used in the ionization current measurements ofSeidel et al. [22], while the 364 keV curve represents thedistribution of the
Sn source used in the scintillationmeasurements from this work.Interestingly, the shape of the energy distribution ofsecondary electrons is nearly identical for the two pri-mary electron energies being considered as shown inFig. 17. The main difference is that the lower primary en-ergy distribution is shifted upwards towards higher cross-sections relative to the other distribution. Therefore, thiswould seem to imply that the thermalization distribu-tion is independent of the energy of the primary electron.However, even though the shape of the energy distribu-tion of secondaries is similar, the relative shift betweenthe two is key to understanding the energy dependenceof the thermalization length distribution which is ulti-mately related to the difference in the ionization densityand the mean separation distance between ion-pairs.Before discussing the importance of the ionization den-sity, let us further inspect the shape of the energy distri-bution by considering the fraction of the secondary elec-trons having energies in the range of 10 eV to 19.8 eV.The estimated thermalization length for an electron withan energy of 10 eV is 100 nm [22], and 19.8 eV corre-sponds to the energy of the first excitation level in he-lium, below which electrons can only lose energy throughthe inefficient process of scattering with helium atoms.To estimate the fraction of secondary electrons that hasa thermalization length greater than 100 nm, we inte-grate the distribution in Fig. 17 from W = 10 eV to W = 19 . ∼
21% and20% of secondary electrons being thermalized at separa-tions >
100 nm for a primary electron energy of 364 keVand 17 keV, respectively. These values represent only alower bound because very high energy secondary elec-trons can further ionize, producing tertiary electrons,some of which will have energies in the range of 10 to19.8 eV. Interestingly, the fraction obtained from this es-timate is similar to the value (25%) obtained from thedistribution derived from the model fit of our scintilla-tion data, but at the same time is significantly higherthan the value (10%) obtained from the distribution bySeidel et al. [22].The difference between the thermalization distributionderived from the scintillation data in this work and thatobtained from the ionization current measurements ofSeidel et al. [22] is likely the result of the energy of theelectron source used in the experiment. The electronsfrom the Ni electron emitter used in Seidel et al. [22]have an end point energy of 66 keV and a mean energy17 keV. Taking the value of 17 keV as the characteristicelectron energy of the Ni source and a W value of 43 eVin helium, the electron average range is ∼ . × − mm7in LHe and the average separation distance between ad-jacent ion-pairs, ¯ x , is ∼
100 nm. By comparison, the364 keV electrons from the
Sn conversion source usedin this work have an average range of ∼ ∼
840 nm. Considering that the thermalization lengthfor an electron is a few 10s to 100s of nm, there is a non-negligible chance for an electron, once thermalized, tobecome paired/matched with a new positive ion partner,one in which it did not originate from, when the averageseparation between ion-pairs is much smaller than thethermalization length. This shifts the distribution N ( r )to shorter r values. On the other hand, when the en-ergy of the electron is such that the average separationbetween ion-pairs is much greater than the average ther-malization length, the exchange of ion partners does notoccur. 𝑥𝑥 𝑥𝑥 𝑟𝑟⃗ 𝑟𝑟⃗ 𝑟𝑟⃗ (0,0,0) ( 𝑥𝑥 , 0,0) ( −𝑥𝑥 , 0,0) 𝑟𝑟⃗ ′ FIG. 18: A diagram showing the model setup used inthe simulation of the thermalization distribution and itsdependence on the primary electron energy. Red-dashedcircles represent the ion locations and blue circles theassociated electrons.We further explore such an effect on the thermalizationdistribution through Monte Carlo simulations. Considerthree electron-ion pairs as shown in Fig. 18. The firstpair is located at the origin while the second and thirdpairs are located at distances x and x from the firstpair, respectively. The pairs are arbitrarily chosen tolie on the x-axis, and the x ’s, which are the separationsbetween ion-pairs, are sampled from an exponential dis-tribution. The mean of this distribution, λ , is dependenton the energy of the primary electron. The appropriatevalue for λ can be set in the simulation for an arbitraryprimary electron energy. For the case of a 17 keV elec-tron in LHe, λ ∼
100 nm. At the location of each ion-pair, a random isotropic vector direction, ˆ r , is chosen.The length of the vector is sampled from the scintillationdata derived distribution, N ( r ) r . These two randomlygenerated quantities represent the direction and thermal-ization length of the electron. The distance between thepositive ion at the origin and the three thermalized elec-trons are then calculated, and the minimum distance, r (cid:48) ,is determined. The process is repeated many times toaccumulate a distribution of r (cid:48) . -6 -5 -4 FIG. 19: The simulated thermalization distributions for17 keV (blue dash-dotted) and 80 keV (purple dashed)primary electrons and the distributions from Seidel etal. [22] (red-dotted ) and this work ( f s = 0 .
57) (blacksolid). The red dotted and black solid vertical linesegments at r = 4 . × − cm and r = 1 . × − cmdenote, respectively, the reach of the measurement ofthe corresponding experiment. Refer to text for furtherdetails.The results of the simulations are shown in Fig. 19along with the thermalization distribution from Seidel etal. [22] and the one derived from the scintillation mea-surements in this work with f s = 0 .
57. There is relativelygood agreement between the simulated distribution andthe one from Seidel et al. [22] from r ≈ × − cmto r ≈ . × − cm, with the latter separation corre-sponding to the highest field measurement in their ex-periment. Here, r is related to the applied field by r = ( e/ π(cid:15) E ) / . The small vertical separation be-tween the two in this region is likely due to a normal-ization difference, whereas the wider separation at largethermalization distances is most probably the result ofthe simplistic nature of the simulations. One such sim-plification is the use of a single primary electron energy,which does not describe the full emission spectrum of the Ni source which has an end point energy at 66 keV. Asa result, the high electron energy component is missingfrom the simulation, and this can be seen as the steepertail at large thermalization distances in Fig. 19. Furtherhighlighting this point is the simulated distribution for80 keV electrons, which shows an enhancement of thetail with primary electron energy.We note that these rudimentary simulations are onlymeant to illustrate some qualitative features of the ther-malization distribution and its energy dependence. Thecomplexity necessary to accurately simulate the distri-bution for an arbitrary electron energy is considerableand well beyond the scope of this work. But in prin-ciple, once the thermalization distribution is known, itis a straightforward excercise to determine the scintilla-tion and ionization current yields as a function of applied8electric field.Figure 20 shows the ionization current yields deducedfrom our scintillation measurements for f s = 0 .
57 alongwith those from Ref. [22]. Just as the thermalizationdistribution has an energy dependence, the same depen-dence is present in the ionization yield. This has impor-tant implications for particle detection in that the mea-sured ionization yield at one energy cannot be assumedto be applicable to the entire range of the measurementsin the experiment. For example, with an applied fieldof 1 kV/cm, approximately 10% of charges are extractedfrom the ionization produced by 364 keV electron, butthis decreases to ∼
2% when the mean energy is 17 keV.At even lower energies, the yield should further decreaseand approach that for heavier ionizing particles. How-ever, a determination of the precise level of reductionbecomes untenable with the present analysis because theconditions for geminate recombination are no longer metwhen the charge density is sufficiently high, as is the casefor electron energies of a few keV.The ionization yields are shown to converge when theelectric field increases. The characteristic field strengthfor this is approximately the strength required to pullapart a pair of ions separated by the mean length of thethermalization distribution, above which the contribu-tion to the total ionization yield from the tail of the distri-bution becomes less significant. This behavior is perhapsthe reason for the improvement in particle discriminationobserved for noble liquid detectors with increasing driftfield in the detection volume. -2 -1 -3 -2 -1 FIG. 20: The ionization current as a function of appliedelectric field. The solid black line is the ionizationcurrent deduced from the scintillation measurements inthis work and the red-dotted curve is the measurementsof Ref. [22].
D. Prompt pulse shape and particle identification
The existence of a temperature dependent recombina-tion time in LHe may provide for a potential applicationin particle detection and identification. As we have dis-cussed, this dependence results from the thermalizationdistribution of the ion-pairs produced in the wake of anionizing particle and the influence of a temperature de-pendent mobility on the recombination of the ions. As itis known that the ionization density produced is relatedto the properties of the charged particle, the effect is pos-sibly very different for minimally ionizing particles (e.g.fast electrons) as compared to heavy ionizing particles(e.g. α particles and heavier nuclei). For instance, therecombination time for α -particles is estimated to occuron a timescale of only ∼ τ d .This implies that regardless of how quickly recombina-tion proceeds, the singlet state lifetime will represent anirreducible time resolution for any measurement.In practical applications, the experimental quantity ofinterest for particle detection and identification is therate at which the signal is generated rather than the sur-vival probability of charges, a quantity that has thus farbeen the focus of our discussion. The signal shape is de-fined by the probability of emission of scintillation lightwithin a time interval t and t + dt and temperature T .We denote this probability as S ( t, T ) dt with S ( t, T ) = (cid:90) ∞ R ( t, t − t (cid:48) ) F ( t − t (cid:48) , T ) dt (cid:48) (19)9 -20 0 20 40 60 8000.20.40.60.81 (a) 0.1 ns detector resolution -20 0 20 40 60 8000.20.40.60.81 (b) 2 ns detector resolution FIG. 21: Calculations of the expected prompt scintillation signal pulse shape for electron ionization in LHe fortemperature of 0.4 K and 4.0 K and for detector resolutions of 0.1 ns (a) and 2 ns (b). The singlet-state decay timeconstant is taken to be 5 ns.where F ( t, T ) = f r ∂∂t (cid:2) − Ω( t, T ) (cid:3) ∗ H ( t ) + f ex H ( t ) . (20)Here, f r and f ex are the fraction of the prompt scintilla-tion signal due to recombination and excitation, respec-tively, and H ( t ) = e − t/τ d /τ d is the rate of scintillationemission from singlet state decays. The component ofthe prompt signal due to recombination is the convolu-tion of the rate of recombination, ∂ (1 − Ω( t )) /∂t and therate of decay of the singlets, H ( t ), that are created bythe recombination process. The second term in Eq. 20is the contribution to the signal from the radiative decayof singlet state excimers produced from neutral excitedatoms in the initial ionization process. Both terms areconvolved with the function, R ( t, t (cid:48) ), which representsthe detector response at time t to an input at time t (cid:48) ,to produce the measured signal. The contribution frominstrument noise is neglected.Measurements from Ref. [16] show a scintillation pulseshape decay time constant of < α particleinduced scintillation. However, if α recombination hap-pens on a very fast timescale as suggested by [21], thenthe pulse shape will be comparable to that for electronsat 0.4 K and will be primarily determined by the singletstate lifetime. If this presumption holds true, then a LHedetector operated around a temperature of 4 K will beable to observe a difference in the prompt pulse shapeof the scintillation induced by α versus that from a fastelectron.Measurements of the prompt scintillation pulse shapegenerated by α -particles and 1 MeV electrons in LHeat 4.2 K and 1.8 K made by Habicht [16] do show thatthere is, indeed, a difference between the pulses gener-ated by these two types of particles at 4.2 K. This dif-ference, however, appears to vanish at 1.8 K. Altogether,their observations are consistent with our discussion ofthe temperature dependence of recombination.There are several important aspects that must be con-sidered in implementing this idea for particle identifica-tion. First, the energy of the electron needs to be suf-ficiently high that the particle is truly minimally ioniz-ing and the description of geminate recombination ap-plies. More precisely, the thermalization distributionmust have a sufficiently heavy tail so that the effectsof recombination are more easily detected. Moreover,to accurately characterize the scintillation pulse shape, asufficient number of photons needs to be detected, andprocesses that can further distort the time profile of thephotons must be keep to a minimum. One such processis the conversion of the EUV scintillation light to longerwavelengths by wavelength shifting fluors such as TPB.The choice of wavelength shifter must then be guided bythe need for both high conversion efficiency and fast de-0cay time, but if possible, the removal of the wavelengthshifter from the light detection process is most preferred.Another process that must also be suppressed is multi-ple scattering of photons as it would further degrade thetiming resolution of the measurement.Missing from the discussion thus far is the presence ofan applied electric field which is commonly utilized inparticle detectors employing liquid and gaseous scintilla-tors as the detection medium. This allows for a secondarysignal (S2), in addition to the primary scintillation sig-nal (S1), to be observed by extracting the ions onto acharge readout. The purpose of this is to make use ofthe parameter S /S α particleis collected [21]. Whenever the application requires largescale detectors, the voltages needed on electrodes are im-mensely high and may not be achievable in practice. Ifthe scintillation pulse shape (S1) can solely provide forthe necessary discrimination power, then this potentialproblem may be circumvented, and detector design andoperation are greatly simplified. If in the case the elec-tric field is necessary for the application, the pulse shapeof S1 may be used in conjunction with S /S S α particles. However, this should notbe entirely ruled out without definitive experimental ev-idence. The zero-field prompt scintillation yield data inSection III D 2 hint at the possibility that the tempera-ture dependent recombination effect for α -particles mayin fact be the inverse of electrons, though this is merelyspeculation and requires further study for clarification.Future measurements of the temperature and field depen-dence of the prompt scintillation pulse shape producedby the stopping of α -particles and electrons in LHe will help answer this question.Finally, considering that free electrons also form bub-ble or localized states in liquid neon [77–81] as it doesin LHe, the effects of recombination time may be alsoapplicable to particle discrimination in liquid neon de-tectors. Previous studies of particle discrimination inliquid neon [82, 83] have been made but in the context ofmeasuring the ratio of the prompt to delayed scintillationcomponents. The latter is analogous to the afterpulsesobserved in LHe scintillation. Whether discriminationcan be obtained through only the shape of the promptpulse and what operational parameters (pressure, tem-perature, field, etc.) will maximize it are open questionsworth considering. Answers to these, however, are left tofuture work. V. CONCLUSION
The prompt scintillation signal generated by the pas-sage of ∼
364 keV electrons in LHe at 0.44 K exhibits a ∼
42 % reduction at a field of 40 kV/cm. We show thatthe apparent temperature dependence of this reductionfor electrons can be explained by an effect due to finiteion recombination time and signal integration time. Tothe best of our knowledge, this is the first time that theeffects of recombination time have been used to explainsuch an observation in noble liquids. The observation ofthis effect indicates the existence of a heavy-tailed distri-bution of thermalization distances produced by electronsas has been suggested by the work of Seidel et al. [22].Furthermore, this thermalization distribution appears tohave a dependence on the energy of the primary electron,with higher energy electrons producing a heavier-taileddistribution. A potential application of the recombina-tion time effect is the use of pulse shape analysis for par-ticle identification and discrimination in particle/nuclearphysics experiments.
ACKNOWLEDGMENTS
The authors greatly appreciate the help provided bythe following individuals and organizations: E. Bond(LANL, C-NR) for electroplating the electrode with ra-dioactive sources, M. Febbraro (ORNL) for coating thelightguide with TPB, and LANSCE Facility Operationsfor providing support for the experimental activities.This work was supported by the US Department of En-ergy, Office of Science, Office of Nuclear Physics throughcontract number 89233218CNA00000 (LANL) under pro-posal number LANLEEDM, contract number DE-AC05-00OR22725 (ORNL), and contract number de-sc0019309(ASU). [1] E. H. Thorndike and W. J. Shlaer, Rev. Sci. Instrum. ,838 (1959). [2] H. Fleishman, H. Einbinder, and C. S. Wu, Rev. Sci.Instrum. , 1173 (1961). [3] L. Mayer and F. Rief, Phys. Rev. , 279 (1958): Phys.Rev. Lett. , 1 (1960): Phys. Rev. , 1164 (1960).[4] F. E. Moss and F. L. Hereford, Phys. Rev. Lett. , 63(1963).[5] F. L. Hereford and F. E. Moss, Phys. Rev. , 204(1966).[6] M. R. Fischbach, H. A. Roberts, and F. L. Hereford,Phys. Rev. Lett. , 462 (1969).[7] R. J. Manning, F. J. Agee, Jr., J. S. Vinson, andF. L. Hereford, Phys. Rev. A , 2377 (1971).[8] H. A. Roberts, W. D. Lee, and F. L.Hereford, Phys. Rev.A , 2380 (1971).[9] H. A. Roberts and F. L. Hereford, Phys. Lett. A , 395(1972).[10] H. A. Roberts and F. L. Hereford, Phys. Rev. A , 284(1973).[11] J. R. Kane, R. T. Siegel, and A. Suzuki, Phys. Lett. ,256 (1963).[12] T. G. Miller, Nucl. Instrum. Methods , 239 (1965).[13] P. B. Dunscombe, in Liquid Scintillation Counting ,edited by A. Dyer (Heyden and Son Ltd., London, 1971),Vol. 1, Chap. 4, pp. 3742.[14] J. S. Adams, Energy Deposition by Electrons in Super-fluid Helium, Doctoral dissertation, Brown University,2001.[15] J. S. Adams, Y. H. Kim, R. E. Lanou, H. J. Maris, andG. M. Seidel, Journal of Low Temperature Physics ,1121 (1998).[16] K. Habicht, Szintillationen in fl¨ussigem Helium ein De-tektor f¨ur ultrakalte Neutronen, Doctoral dissertation,Technischen Universit¨at Berlin, 1998.[17] D. N. McKinsey, C. R. Brome, J. S. Butterworth,S. N. Dzhosyuk, P. R. Huffman, C. E. H. Mattoni,J. M. Doyle, R. Golub, and K. Habicht, Phys. Rev. A , 200 (1999).[18] D. N. McKinsey et al. , Phys. Rev. A , 062716 (2003).[19] D. N. McKinsey et al. , Nucl. Instrum. Methods Res.,Sect. A , 475 (2004).[20] W. Guo, M. Dunfault, S. B. Cahn, J. A. Nikkel, Y. Shin,and D. N. McKinsey, J. Inst. , P01002 (2012).[21] T. M. Ito, S. M. Clayton, J. Ramsey, M. Karcz, C.-Y. Liu,J. C. Long, T. G. Reddy, and G. M. Seidel, Phys. Rev.A , 042718 (2012).[22] G. M. Seidel, T. M. Ito, A. Ghosh, and B. Sethumadha-van, Phys. Rev. C , 025808 (2014).[23] R. E. Lanou, H. J. Maris, and G. M. Seidel, Phys. Rev.Lett. , 2498 (1987).[24] R. E. Lanou, H. J. Maris and G. M. Seidel, in Dark Mat-ter , Proc. XXIII Ren. de Moriond, ed. J. Adouze and J.Tran Thanh Van, p. 79, Editions Frontieres (1988).[25] Y. H. Huang, R. E. Lanou, H. J. Maris, G. M. Seidel,B. Sethumadhavan, and W. Yao, Astropart. Phys. , 1(2008)[26] R. Golub and S. K. Lamoreaux, Phys. Rep. , 1 (1994).[27] T. M. Ito, J. Phys. Conf. Ser. , 012037 (2007).[28] M. W. Ahmed et al. (SNS nEDM Collaboration), JINST14 P11017 (2019).[29] P. R. Huffman et al. , Nature
62 (2000).[30] W. Guo and D. N. McKinsey, Phys. Rev. D , 115001(2013).[31] T. M. Ito and G. M. Seidel, Phys. Rev. C , 025805(2013).[32] H. J. Maris, G. M. Seidel, and D. Stein, Phys. Rev. Lett. , 181303 (2017). [33] S. A. Hertel, A. Biekert, J. Lin, V. Velan, and D. N. McK-insey Phys. Rev. D , 092007 (2019).[34] C. Jewell and P. V. E. McClintock, Cryogenics et al. , Rev. Sci. Instrum. , 045113 (2016).[37] ASTAR, https://physics.nist.gov/PhysRefData/Star/Text/ASTAR.html [38] ESTAR, https://physics.nist.gov/PhysRefData/Star/Text/ESTAR.html [39] W. P. Jesse and J. Sadauskis, Phys. Rev. , 1668 (1955);N. Ishida, J. Kikuchi, and T. Doke, Jpn. J. Appl. Phys. , 1465 (1992).[40] G. Jaffe, Ann. Phys. , 303 (1913).[41] H. A. Kramers, Physica , 665 (1952).[42] L. Onsager, Phys. Rev. 54, 554 (1938).[43] H.-O. Meyer, Nucl. Intrum. Methods Phys. Res., Sect. A , 437 (2010).[44] SkuTek, [45] Lakeshore, [46] E. H. Bellamy et al. , Nucl. Intrum. Methods Phys. Res.,Sect. A , 468 (1994).[47] G. Bortels and P. Collaers, Appl. Radiat. Isot. Vo1. 38,No. 10, 831-837 (1987).[48] S. Agostinelli, J. Allison, K.A. Amako, J. Apostolakis, H.Araujo, P. Arce, M. Asai, D. Axen, S. Banerjee, G. Bar-rand, F. Behner, GEANT4 - a simulation toolkit, Nucl.Instrum. Methods, Sect. A , 250 (2003).[49] J. Baro, J. Sempau, J. M. Fernndez-Varea, and F. Salvat,PENELOPE: an algorithm for Monte Carlo simulation ofthe penetration and energy loss of electrons and positronsin matter, Nucl. Instrum. Methods Phys. Res., Sect. B
31 (1995).[50] F. Salvat, J. M. Fernandez-Varea, J. Sempau, PENE-LOPE - A code system for Monte Carlo simulationof electron andphoton transport, NEA-OECD, Paris(2003).[51] S. Sato, K. Kowari, and S. Ohno, Bull. Chem. Soc. Jpn. , 2174 (1974).[52] R. J. Donnelly and C. F. Barenghi, J. Phys. Chem. Ref.Data , 1217 (1998).[53] F. Williams, J. Am. Chem. Soc. , 3954 (1964).[54] P. K. Ludwig, J. Chem. Phys. , 1787 (1969).[55] G. Jaff´e, Phys. Rev. , 968 (1940).[56] C. Gamota, J. Phys. Colloques , C3-39 (1970).[57] S. A. Rice, in Diffusion-limited Reactions, Chapter 3 Re-actions between Ions in Solution , edited by C. H. Bam-ford, C. F. H. Tipper, and R. G. Compton (Elsevier Sci-ence Publishers, Amsterdam, 1985).[58] E. W. Montroll, J. Chem. Phys. , 202 (1945).[59] A. Mozumder, J. Chem. Phys. , 1659 (1968).[60] G. C. Abell, A. Mozumder, and J. L. Magee, J. Chem.Phys. , 5422 (1972).[61] S. A. Rice, P. R. Butler, M. J. Pilling, and J. K. Baird,J. Chem. Phys. , 4001 (1979).[62] J. B. Pedersen and P. Sibani, J. Chem. Phys. , 5368(1981).[63] J. Y. Parlange, R. D. Braddock, D. Lockington, andG. Sander, J. Chem. Phys. , 4171 (1984). [66] N. J. B. Green, M. J. Pilling, and P. Clifford, Molec.Phys. , 1085 (1989).[67] K. M. Hong and J. Noolandi, J. Chem. Phys. , 5163(1978).[68] K. M. Hong and J. Noolandi, J. Chem. Phys. , 5026(1978).[69] J. W. Boag, Br. J. Radiol. , 601 (1950).[70] J. Thomas and D. A. Imel, Phys. Rev. A , 614 (1987).[71] O. C. Ibe, Elements of Random Walk and Diffusion Pro-cesses (John Wiley & Sons, Inc., Hoboken, NJ, 2010).[72] Y. -K. Kim, J. P. Santos, and F. Parente, Phys. Rev. A , 052710 (2000).[73] Y. -K. Kim and M. E. Rudd, Phys. Rev. A , 3954(1994).[74] G. F. Knoll, Radiation Detection and Measurement (Wi-ley, New York, 2010), 4th ed. [75] A. V. Benderskii, R. Zadoyan, N. Schwentner, andV. A. Apkarianb, J. Chem. Phys. , 1542 (1999).[76] P. Hill, Phys. Rev. A , 5006 (1989).[77] C. G. Kuper, Phys. Rev. , 1007 (1961).[78] J. Jortner, N. R. Kestner, S. A. Rice, and M. H. Cohen,J. Chem. Phys. , 2614 (1965).[79] B. E. Springett, M. H. Cohen, and J. Jortner, Phys. Rev. , 183 (1967).[80] L. Bruschi, G. Mazzi, and M. Santini, Phys.Rev.Lett. ,1504 (1972).[81] R. J. Loveland, P. G. L. Comber, and W. E. Spear, Phys.Lett. A , 225 (1972).[82] J. A. Nikkel, R. Hasty, W. H. Lippincott, and D. N. McK-insey, Astropart. Phys. , 161 (2008).[83] W. H. Lippincott, K. J. Coakley, D. Gastler, E. Kearns,D. N. McKinsey, and J. A. Nikkel, Phys. Rev. C86