Scintillation of Liquid Helium for Low-Energy Nuclear Recoils
SScintillation of Liquid Helium for Low-Energy Nuclear Recoils
T. M. Ito ∗ Los Alamos National Laboratory, Los Alamos, New Mexico 87545, USA
G. M. Seidel † Department of Physics, Brown University, Providence, Rhode Island, 02912, USA (Dated: October 4, 2018)The scintillation properties of liquid helium upon the recoil of a low energy helium atom arediscussed in the context of the possible use of this medium as a detector of dark matter. It 1sfound that the prompt scintillation yield in the range of recoil energies from a few keV to 100 keVis somewhat higher than that obtained by a linear extrapolation from the measured yield foran 5 MeV alpha particle. A comparison is made of both the scintillation yield and the chargeseparation by an electric field for nuclear recoils and for electrons stopped in helium.PACS numbers: 34.50.-s, 33.50.-j, 29.40.Mc
I. Introduction
The noble gas liquids have become a very attractivemedia for particle detection. Argon[1] and xenon[2, 3]are being used extensively in searches for dark matterparticles. Neon is being investigated[4, 5] for possibleapplication for detecting dark matter and neutrinos.And while helium has been proposed[6–8], as well, as atarget material for studies of neutrinos and dark matter,it has yet to be employed in such an application for anumber of reasons. But because of the possibility thatdark matter may consist of WIMPs of lower mass thanpreviously expected, in the range of 10 GeV or below,there is some reason to consider detectors that couldprovide better sensitivity in this low mass range. Inthat regard, it is natural to consider the advantages thatliquid helium might provide over the heavier liquefiednoble gases currently being used. To this end, this paperdiscusses the expected scintillation properties of liquidhelium in the energy region below 100 keV where nuclearrecoils resulting from elastic scattering of WIMPS wouldbe expected to occur.The recoil energy of a nucleus from which a WIMP iselastically scattered is E nr = 2 m x m n v ( m x + m n ) cos θ , (1)where m x and m n is the mass of the WIMP and recoilnucleus, respectively, v is the velocity of the WIMPwith respect to the detector, and θ is the angle of therecoil nucleus relative to the direction of the motion ofthe WIMP. For a WIMP of mass 10 GeV in the galactichalo with a velocity of 250 km/s with respect to thesolar system, the recoil energy of a helium nucleus isapproximately 10 keV for θ = 0. Since dark matter ∗ [email protected] † corresponding author; george [email protected] particles are expected to have a velocity distribution upto the escape velocity from the galaxy of 680 km/s, therecoil energy of a helium nucleus will extend up to the100 keV for 10 GeV WIMPs.The heavy liquefied noble gases, argon and xenon, arebeing used as dark matter detectors since they possessa number of desirable properties. As cryogenic liquidsthey can be made with very high purity, they are rea-sonably dense so self shielding of background radiationis possible, and they have high scintillation yields beingtransparent to their own emission. Furthermore, anelectric field can be used to separate electrons frompositive ions produced by ionization. Charge collectionprovides another valuable channel for identifying thenature of the radiation stopped in the liquid.Liquid helium possesses some of these attributes of theheavier noble gas liquids but suffers as a potential darkmatter detector for a number of reasons. 1) Scintillationfrom helium occurs at a higher energy, consisting ofa broad distribution peaking at 16 eV. No materialexcept for helium itself is transparent at this energy,requiring for detection of scintillation either the use of awavelength shifter or the placement of the EUV detectordirectly within the containment volume of the helium.Additionally, because of the larger W-value of heliumthan the heavier noble gases, fewer photons are emittedper unit of energy deposition. 2) The use of liquidhelium requires operating at much lower temperatureswith the attendant technical cryogenic complexities.3) Because of its low density liquid helium providespoor self shielding of background radiation. 4) Sinceelectrons form bubbles and positive ions form snowballsin liquid helium[9], the mobility of charges is significantlydifferent in helium compared as to that of heavier noblegas liquids. Not withstanding the challenges presentedby these features, the potential benefit that helium,because of its low mass, brings to a search for light-massdark matter particles makes the study of its propertiesin this regard of some interest. Hence this discussion a r X i v : . [ a s t r o - ph . I M ] A ug of the scintillation efficiency of helium to low-energynuclear recoils. We are unaware of any measurementsor estimates of the emission efficiency of helium for lowenergy incident particles and use what is known abouthelium-helium scattering and the scintillation of liquidhelium upon stopping energetic alpha particles to fillthis gap.The knowledge of the effect of a helium atom withlow recoil energy, from 1 to 100 keV, in liquid helium issparse. There are two types of information that is usefulin developing an understanding what happens should aWIMP scatter from a helium atom. 1) Measurementsexist on the ionization produced by He-He scatteringin gases at the relevant energies. From this data it ispossible to make some estimates of the likely conse-quence of WIMP scattering in the liquid. 2) From whatis known about processes that occur along the track ofan energetic alpha particle stopped in liquid helium it ispossible to evaluate the likely consequences of processesthat lead to quenching at lower energies.This paper is organized as follows. Section II isfocussed on a discussion of the production of ionizationand excitation as a consequence of recoil from a WIMP.In Section III we review what is known about theinteractions that occur along the track of an ionizingparticle in liquid helium that are important for anunderstanding of the scintillation. Section IV containsa calculation of the expected scintillation efficiency of ahelium recoil as compared to the scintillation from anelectron of the same energy, while Section V discussesthe scintillation from electrons and compares chargecollection from electron events and nuclear recoils.Section VI summarizes the results and limitations of thecalculations. II. Helium-helium scattering
If a WIMP were to scatter off a helium nucleus andthe recoil energy is low, the resulting recoil projectilewould be expected to be the uncharged helium atom,He . A calculation of the probability that the re-coil atom is in such an un-ionized state is shown inFig. 1. However, if the recoil energy is high the recoilwould likely be in a charged state, either the singlycharged ion, He , or the doubly charged, bare nucleusHe . The crossover from uncharged to charged stateoccurs on a scale determined by the atomic velocity, v = e / (cid:126) = 2 . × cm s − . On average whathappens subsequently to the projectile and the mediumwith which it interacts is independent of initial chargedstate, depending only on energy. A neutral atom canionize target atoms or be stripped of an electron therebybeing converted into a He ion. Numerous other pro-cesses such as charge exchange, electron capture, doubleionization, ionization plus excitation, etc. can also occur.The consequence of the energy dependence of various p r obab ili t y FIG. 1: color online) Energy dependence of the probability that ahelium recoil atom is in the neutral ground state, He , ascalculated by Talman and Frolov[10]. processes such as charge exchange and stripping is thatthe equilibrium probability of a projectile being in a par-ticular charge state differs strongly with energy as illus-trated in Fig. 2, where the experimentally measured equi-librium charge fraction for the three states is plotted as afunction of energy. Since we are primarily concerned withrecoil energies less than 100 keV, at which energy thecharge fraction of He , F , is less than 1%, this chargestate makes essentially no contribution to the expectedscintillation signal from low mass WIMPs. Even thecharge fraction F for He is less than 30% at 100 keV. c ha r ge f r a c t i on FIG. 2: (color online) Equilibrium charge fraction as a function ofprojectile energy for three states of helium. He is predominantat low energy and He at high energy. Circles Ref.[11]; crossesRef.[12]; triangles Ref.[13]; squares Ref.[14]. Lines are empiricalfits to the data. A. Ionization and excitation
There is a considerable body of information in the lit-erature for the various processes that can occur when anenergetic charged or neutral helium atom collides withanother helium atom. The direct ionization processHe i + + He → He i + + He + + e − , (2)where a projectile in charge state i , which remains un-changed, ionizes a neutral target atom has been studiedextensively. (The He with the superscript i + denotesthe projectile.) The experimentally measured cross sec-tions for the three charge states are plotted in Fig. 3,and the effective ionization cross section for the variouscharge states of a helium projectile in helium is plotted inFig. 4. The effective cross section is the ionization crosssection multiplied by the respective charge fraction; thatis, the plotted quantities are the products F i σ i,ion for thethree charge states. i on i z a t i on c r o ss s e c t i on ( - c m ) He FIG. 3: (color online) The energy dependence of the ionizationcross section for the three charge states of a helium projectileincident on a helium target. Lines: empirical fits to the data.Data for He : triangles Ref.[15], circles Ref.[16], squares Ref.[17],diamonds Ref.[18], crosses Ref.[19]. Date for He : trianglesRef.[20], squares Ref.[17], circles Ref.[21], diamonds Ref.[19]. Datafor He : triangles Ref.[21], circles Ref.[22], squares Ref.[19]. While processes in which the target atom is singlyionized dominate, double ionization can occur as well.The cross sections for double ionization are an orderof magnitude smaller than for single ionization. Theeffective cross sections for the three charge states of theprojectile are shown in Fig. 5.In addition to direct ionization, ions are generated byprocesses in which the projectile changes its charge state– called variously, exchange, capture, or stripping – pos- e ff e c t i v e c r o ss s e c t i on s ( - c m ) FIG. 4: (color online) Plots of the effective direct ionization crosssections, F i σ i,ion for the three charge states of helium as functionof energy. Lines: empirical fits; dashed line: He ; dot-dash line:He ; dotted line: He . sibly accompanied by single or double ionization of a tar-get atom. At low energies the most important of theseprocesses, having a cross section normally labeled σ , isone involving charge exchange in which a He projectileis neutralized,He + He → He + He + . (3)In charge equilibrium the rate at which this process oc-curs, F σ , must be the same as the rate of the processwhere a neutral projectile is ionizedHe + He → He + He + e − , (4)namely, F σ ,ion , since viewed from the center of massof the two interacting atoms there can be no distinctionin Eq. (4) between which of the two atoms is ionized.The two processes together results in a target atombeing ionized and hence doubles the overall cross sectionfor ionization in the energy range where He is thedominant charge equilibrium species.Based on measurements in the literature we have plot-ted in Fig. 5 the consequences of two other charge ex-change processes. These make smaller contributions athigher energies to the overall ionization. The effectiveionization cross section F σ is associated with the pro-cess He + He → He + He , (5)which in equilibrium must have the same rate asHe + He → He + He + e − . (6)Similar expressions describe the process related with F σ plotted in Fig. 5.Scintillation also results from atoms that are pro-moted directly to excited states without having first e ff e c t i v e c r o ss s e c t i on ( - c m ) FIG. 5: (color online) The effective ionization cross sections (crosssection times charge fraction) for various double ionization andcharge exchange processes. Long dashed line: simultaneousionization of both target atom and He projectile, data fromRef. [18]. Solid line: double ionization of target atom by He ,data from Ref. [21, 25, 26]. Short dashed line: double ionizationof target by He , data from Ref. [22, 26]. Dot-dashed line:exchange σ , data from Ref. [22, 23]. Dotted line: exchange σ ,data from Ref. [23]. been ionized. The cross sections for excitation of heliumby helium are not as well studied as for ionization.The only measurement of the excitation cross section ofHe on He of which we are aware is that of Kempter etal. [27]. Their results, plotted for the specific transition1 S to 2 P in Fig. 6, are scaled as recommended byKempter [28] to agree with theoretical predictions [29].The necessity for the scaling is the consequence of howthe measurements were made, involving the detectionof the UV radiation at 58.4 nm for the transition backto the ground state using a scintillator in conjunctionwith a photomultiplier whose calibration was not knownto within a factor of 3. In the low energy range below10 keV, where the equilibrium charge state is predom-inantly 0+, excitations make a larger contribution toscintillation than ionization, as can be observed bycomparing the excitation cross section for He in Fig. 6with the cross section for ionization in Fig. 3.Although the transition from 1 S to 2 P state is byfar the most likely excitation to occur, other transitionsare non-negligible. Based on the measurements ofKempter[27] and calculations by others[29–31] for exci-tations created by helium in the 1+ state, we estimatethat for every transition having a 2 P final state thereare 0.4 transitions to other states of which half are spinsinglets and half triplets. This rough estimate results ina singlet to triplet ratio of .86/.14.Experimental data for the excitation transition 1 Sto 2 P by He in the 1+ charge state is also plottedin Fig. 6. The two maxima for excitation by resultfrom different processes[32], the maximum at higherenergy being related to the excitation of the target e xc i d t a t i on c r o ss s e c t i on ( - c m ) FIG. 6: (color online) The cross sections for excitation of the 1 Sto 2 P transition by He and by He in helium. Data for He :solid circles, Ref. [27] scaled to fit theory of Refs. [29, 31]; squares,theory Ref. [30]. Data for He : triangles, Ref. [32]; diamondsRef. [33]; open circles, Ref. [34]. Data for charge exchange plussimultaneous excitation by He : crosses, Ref. [35]. Lines areempirical fits to data and theory. atom and the lower related to charge exchange andexcitation of the projectile. We are unaware of anymeasurements of excitation of target He atoms byHe . The scaling dependence for excitation by chargedprojectiles at high energies given by σ/Z = f ( v /Z )[36], suggests that it does not make a significant contri-bution to energy dissipation. Yet another process hasbeen measured, which we do include in our analysis.Folkerts et al. [35] have measured the combined crosssection for a number of processes where the projectileis He . These include the process of charge exchange,with either the resulting He projectile or target ionsimultaneously promoted to an excited state, and theprocess of ionization and simultaneous excitation ofthe target. While these cross sections for excitationby He and He are comparable to that of He ,they make little contribution to the scintillation yieldbecause the respective charge fractions are small in theenergy range where the cross sections are large. Again,in estimating the effects of these excitation processes wemultiply the cross section of the 1 S to 2 P transitionby 1.4 to approximate the total production of excitations.There exist other processes, for example, a change incharge or excitation state of both projectile and targetatom, but these have even smaller cross sections and wedo not consider them further.
B. Secondary Electrons
One other mechanism is important for producingions and excited-state atoms in helium, namely, thesecondary electrons created by an initial ionization thathave recoil energies greater than the ionization potentialor the first excitation level. This process is unimportantat primary projectile energies below 100 keV, but it isa significant contributor at higher energies. An under-standing of what happens at high energies is importantin calculating quenching at low energies, so we discussionization by secondary electron here.At high projectile energies, the energy distribution ofsecondary electrons is such that they can produce addi-tional ionization. In a review Rudd et al. [37] have givena semi-empirical expression for the single differentialionization cross section (SDCS) of secondary electrons,which depends on the projectile energy and the energy ofthe secondary electron. Their expression was generatedfrom measurements of the differential cross section forthe production of secondaries by protons stopped inhelium. This expression can be modified to calculatethe secondary distribution for a helium projectile byscaling the energy of the projectile by 4 to account forthe difference in mass between a proton and heliumand scaling the magnitude of the distribution by afactor of 4 (square of charge ratio) when computing thecontribution of He compared to that of the proton.Garibotti and Cravero[38] have measured the SDCS for4 and 7.36 MeV He ions in helium and find the scaledRudd expression fits their data well but is slightly lowat the higher electron recoil energies.There exist accurate electron impact cross sectionsfor helium for both ionization[39, 40] and excitation[41]so that one can perform calculations, such as a MonteCarlo simulation, of the generation of ions and excitedatoms starting from the energy distribution of recoilelectrons. However, given the uncertainty in the energydistribution and the limited use to which the the resultsare to be put, we calculate the number of ionizations bydetermining the total energy in the secondary electronspectrum with energies above the ionization potentialand dividing by the W-value of electrons in helium of 43eV. The number of excitations is estimated by assigning33 eV to each ionization, this number being the sum of24.6 eV for the ionization and 8.4 eV for the averagerecoil energy of electrons below the excitation threshold.The remaining 10 eV is presumed associated withexcitations, which have an estimated weighted averageenergy of 22 eV, yielding the number of excitations toionizations of 10 /
22 = 0 .
45. This estimate is made forboth the 1+ and 2+ charge states, multiplied by therespective charge fractions and converted into effectivecross sections.In the absence of any measurements of the secondaryelectron distribution from other ionization processes, theRudd expression is assumed to be applicable to thesecases as well, scaled for the appropriate charge state. Analternative procedure is to presume, as discussed by manyauthors, see for example, Manson and Toburen[43], thatfor kinematic reasons the direction of the velocity of re-coil electrons is sharply peaked in the forward directionand the magnitude of the velocity is peaked around thatof the projectile. Hence, the ratio of the average recoil en- ergy to the projectile energy is approximately m e /M He .The two approaches yield comparable results, given theapproximations involved in the estimates. e ff e c t i v e c r o ss s e c t i on ( - c m ) FIG. 7: (color online) Sum of the effective ionization andexcitation cross sections from Eq. (7). Solid line: ionization withsecondary electron contribution. Dashed line: ionization withoutsecondary electron contribution. Dot-dashed line: excitation withsecondary electron contribution. Dotted line: excitation withoutsecondary electron contribution.
The sum of the effective cross sections for ionizationand excitation processes with and without the inclusionof the contribution of secondary electrons at high energyis plotted in Fig. 7. The sums of the effective cross sec-tions for ionization and excitation are S ion,eff = (cid:88) i (cid:88) j F i σ ion,i,j and S exc,eff = (cid:88) i (cid:88) j F i σ exc,i,j (7)where the subscript i refers to the three charge statesand the subscript j to the specific processes. Onenoteworthy feature of this plot is the very low effectiveexcitation cross section in the energy range between10 and 100 keV. While the excitation cross sectionsfor He and He are of a size comparable to thecorresponding ionization cross sections, the maxima ofthe cross sections for excitations occur at a lower energywhere the charge fractions are considerably less. III. Stopping power
The stopping power is the average energy loss of aprojectile per unit path length due to all scatteringprocesses occurring in the target material and is usuallyexpressed in units of MeV cm /g [44] or, more conve-niently for this discussion, in eV cm [42]. The stoppingpower is normally divided into an electronic component,due to Coulomb interactions creating ionizations andexcitations, and a nuclear component, the result ofelastic collisions. The division is not without ambiguity,however, since it can be dependent on the time at whichit is made [46].To calculate the electronic stopping power from thecompilation of effective ionization and excitations crosssections discussed in the previous section requires knowl-edge of the energy loss associated with each process. Theelectronic stopping power is SP = (cid:88) j S j,eff Q j (8)where the sum is over all the processes involvingionization and excitation and Q j is the energy loss ofthe particular process labeled by the subscript j . Theproblem, then, to compute the stopping power is oneof estimating the energy loss for each of the processesinvolved.The energy loss for a simple excitation event istaken to be 22 eV on average. The energy loss for anionization event initiated by either a He or He isthe sum of the ionization energy plus the average kineticenergy of the recoil electron, which can be computedusing the empirical expression of Rudd[37], discussedearlier. The electron recoil energy resulting from otherionization processes is similarly treated. The energyloss of an interaction involving charge exchange requiresthe addition of the energy of the process by which theprojectile returns to its initial state.The result of computing the stopping power fromEq. (8), including all the microscopic processes discussedabove, is shown in Fig. 8. Also plotted in Fig. 8 is theelectronic stopping power of an alpha particle in heliumtaken from the ASTAR tables[44] converted from unitsof MeV cm /g to 10 − eV cm using the density of liq-uid helium. As illustrated in the graph, the two stoppingpowers are in reasonable agreement in some energy re-gions but differ considerably in others. Between 5 keVand 150 keV the two values are certainly within the ac-curacy of the calculations, but the difference of 50% at500 keV seems large even given the uncertainties associ-ated with the various ionization and excitation processesthat contribute at this energy, as shown in Fig. 5. Above1 MeV the calculated stopping power is close to 20% lessthan that measured, a difference which is hard to explainas being the result of uncertainties or approximations inthe calculations. The only mechanism, discussed in theliterature, responsible for energy dissipation in this en-ergy region is the direct ionization of the target heliumby He . Double ionization is more than a factor of 10 less at 5 MeV. Direct excitation processes by He havenot been reported in this region. As a means of bringingthe calculated stopping power into agreement with themeasured value, one might consider a very slight modi-fication of the Rudd empirical expression for the singledifferential cross section for the recoil electrons. Such amodification would be consistent with the electron recoildata of Garibotti and Cravero[38] as well, but it would have other consequences. As discussed below, the W-value calculated for alphas in helium using the effectivecross sections is 38 eV at 5.5 MeV, well below the knownvalue of 43 eV. Any modification of the secondary elec-tron energy distribution to fit the stopping power makesthe disagreement in the W-value larger. We are left toconclude that the absence of the direct excitation of tar-get atoms by He , not included in the measurements ofFolkerts, et al. [35] is missing. s t opp i ng po w e r ( - e V c m ) FIG. 8: color online) The stopping power as function of energy.Solid line: calculated using Eq. (8). Dotted line: from ASTAR[44].Long dashed line: calculated from Lindhard theory[46].
Below 5 keV the difference between the calculatedstopping power and that given by the ASTAR tables hasother origins. The electronic stopping power for alphasin helium has not been measured below 100 keV[45],and it is therefore not surprising that the tables at lowerenergies based on an empirical relationship betweenelectronic stopping power and projectile energy of theform SP ∝ E . does not agree particularly well below5 keV. The difference at 1 keV between the computedstopping power and the ASTAR values, larger than afactor of two, is also related to the fact that not allof the contributions to the stopping power calculatedfrom Eq. (8) are included in the electronic stoppingpower from ASTAR. The total ASTAR stopping powerconsists of two components, electronic and nuclear,where the nuclear stopping power is the average rateof energy loss per unit path length due to the transferof energy to recoiling atoms in elastic collisions. Butneutral recoiling helium atoms can excite the electronicsystem as evidenced by the measured ionization andexcitation cross sections for He . This is the compo-nent missing from the ASTAR electronic stopping power.The stopping power calculated using Eq. (8) alsomay miss the fraction of the energy that ends up asexcitations at low energies. For example, should a He projectile elastically scatter giving the target atomsufficient energy to create an excitation, this would notbe accounted for in Eq. (8). However, the magnitude ofthis effect is not expected to be large.The Lindhard theory [46] of stopping power makes adifferent division of energy E = η + ν, (9)between nuclear, ν , and electronic, η , components, a di-vision that includes in the electronic term all the energythat ends up in ionization and excitation no matter ifthe origin scattering involved elastic collisions. At highprojectile energies where elastic scattering is unimpor-tant, dη/dx is the same as the ASTAR electronic stop-ping power, but at low energies it is not since it con-tains ionizations and excitations produced by secondaryrecoiling He atoms. At low energies Lindhard, on thebasis of models for the various scattering cross sections,developed an analysis of the dependence of η and ν onenergy, from which it is possible to estimate how muchionization and excitation – and, hence, the scintillation –are enhanced over what would be calculated on the basisof the electronic scattering power alone. The Lindhardanalysis[46] develops a semi-empirical expression for νν = (cid:15) k g where (cid:15) = 11 . E (in keV) /Z / is a reduced energy, theexpression being valid when the charge of the projectilenucleus is the same as that of the target material.The constant is k = . Z / A − / , where A is thenucleon number. The parameter g is a function of thereduced energy, which is given graphically in Ref. [46]and in Ref. [47] by the empirical analytical expression g = 3 (cid:15) . + . (cid:15) . + (cid:15) .From these considerations one can calculate η , theso-called nuclear quenching factor f n = η/E and dη/dx .These terms can be considered the Lindhard nuclear andelectronic stopping power, respectively. The quantity dη/dx is plotted in Fig. 8. The agreement of the Lind-hard theory with the results of Eq. (8) is quite good.The difference between the two curves at low energies,25% at 1 keV, could be due to approximations in thetheory or an overestimate of the excitation stoppingcross section for He , which involves the use of a scalingfactor[27, 28].The agreement between the stopping calculated froma consideration of microscopic processes and obtainedby other means, lends credence to our approach toestimating the numbers of ionizations and excitationsproduced by a low energy nuclear recoil in liquid helium. IV. Interactions and scintillation in liquid helium
The scintillation from helium not only depends on themechanisms for production of ions and excitations; italso is influenced by nonradiative quenching processesthat can occur in the liquid. For this we turn to a discussion of what is known about the difference inscintillation produced by energetic electrons and alphaparticles in liquid helium.In dense helium gas or in the liquid a helium ion He + or atom in an excited state He ∗ will quickly combine witha ground state atom to form an excimer.He + + He → He +2 . He ∗ + He → He ∗ . (10)The excimer He ∗ or He +2 has an inter nuclear distanceof 0.12 nm and a binding energy of approximately 1.9eV. In the liquid, positive ions form excimers quickly ( ∼ ( a Σ u )for triplets and He ( A Σ u ) for singlets. The radiativelifetime of the lowest singlet excimer to the dissociativeground state ( X Σ g ) is the order of 10 − s and accountsfor the prompt scintillation signal. While the energyof the ( A Σ u ) state is roughly 20 eV above the wellseparated, ground-state helium pair, the fact that thetransition satisfies the Franck-Condon principle and theenergy of the ground state rises rapidly with decreasinginter nuclear separation accounts for the emissionspectrum being a broad peak centered at 16 eV. Thetriplet state ( a Σ u ) has a measured lifetime of 13 sin liquid helium[48], and its radiative decay does notcontribute to the particle identification unless use ismade of afterpulsing[49, 50].Because of the long lifetime of the lowest lying tripletexcimer, the prompt scintillation signal depends on theratio of the number of singlets to triplets produced byan ionizing particle. It also depends on the number andtype of excited state atoms that are generated. Theonly estimates of the number of excitations produced forparticles with energies above 100 keV, of which we areaware, are those of Sato et al. [51, 52] who calculate that0.45 atoms are promoted to excited states for every ionproduced. Of the excited atoms 85% are predicted to bein spin singlet states and 15% in triplet states.However, on the basis of our present estimates ofthe effective cross section we are now able to estimateindependently the ratio of excitations to ionizations,which can be found by a numerical integration of theplots in Fig. 7. This ratio, obtained from a summationof the contributions from the various microscopic pro-cesses, has a magnitude and energy dependence that issomewhat different from that obtained theoretically bySato et al. [51], who did not consider the variation of thecharge state of the projectile.Using our value of 0.34 for the ratio of excitations toionizations at 5.5 MeV and assuming the ratio of sin-glet to triplet excitations is .85/.15, we can estimate thefraction of energy that should appear as prompt scintilla-tion. If the ions recombine in proportion to the number ofavailable states, then 3/4 recombine in triplets and 1/4 insinglets, and the fraction of deposited energy appearingas prompt scintillation should be1643 × (1 / . × .
85) = . , (11)where the first term arises from singlet excimers createdon recombination and the second from singlet excita-tions. Calorimetric measurements of the scintillationfrom 5.5 MeV alphas indicate that, instead of 20%,only 10% of the energy appears as photons[53, 54], incontrast to measurements of 364 keV electrons for whichthe prompt scintillation yield is 35%[53, 54].The origin of the difference in scintillation yield foralpha particles and electrons lies in the very differentstopping power of helium for electrons and alphas.For a 5.5 MeV alpha an ionization occurs on averageevery few nanometers along the track, but for anenergetic electron the separation between ionizations isthe order of 1000 nm. The mean distance a secondaryelectron with energy below the excitation threshold of19.8 eV travels by diffusion before becoming localizedby forming a bubble is estimated to be approximately60 nm[50, 55]. Consequently, for ionization by electronsthe recombination is primarily geminate, and the spinsof the recombining ion and electron can be correlated. Itis estimated from the 35% scintillation yield that morethan 50% of the excimers formed on recombination inthis case are singlets rather than the 25% expected onthe basis of number of available states.Along an alpha particle track, the recombination is de-cidedly not geminate, and the ratio of singlets to tripletsshould be 1 to 3. The decrease in scintillation yield by afactor of 2 from that calculated on the basis of this ratiois attributed to the nonradiative destruction of excimersby the exothermic Penning process,He ∗ + He ∗ → S ) + He + + e − , or → S ) + He +2 + e − . (12)In either case, two excimers are destroyed and anew one is formed upon the recombination of theelectron and ion. Keto et al. [56, 57] were the first tomeasure the rate coefficient for this bimolecular process, dn/dt = − α n ( t ), for triplet excimers in liquid helium.These measurements have been repeated and extendedby others[58, 59], but no direct observation of this Pen-ning process has been observed for singlets. Nonetheless,it is presumed to be the cause of the quenching of thescintillation from a highly ionizing particle in liquid helium.The same type of exothermic process illustrated inEq. (12) can occur if one or both of the interactingspecies are not excimers but atoms in excited states,which may not have formed excimers prior to encoun-tering another excited species.This quenching of the scintillation signal, observedfor energetic alpha particles, will also occur for lowenergy scattering by WIMPS if the density of excimersand excited atoms along the recoil track is comparableto that for an alpha. Ito et al. [50] have made a roughestimate of quenching and its dependence on density ofsinglet atoms and excimers along an alpha track, and weuse that approach to predict what is likely to occur fora low energy recoil.The absence of knowledge about possible differences inthe rates at which bimolecular Penning processes occuramong the different excimers and excited state atoms,makes any rigorous calculation of electronic quenchingimpossible. Instead, we lump all the species togetherinto a single differential equation for the rate of changein the total density, n , of all the species. dndt = − γn − r nτ . (13)The bimolecular rate γ is taken to be the same for all in-teracting pairs while the radiative decay governed by thetime constant τ is restricted to the singlet species by set-ting the constant r to the value r = (1 / . × . / (1 + .
34) = .
40 . Since we are only interested in quenching ofthe prompt signal with time constant of 10 − s, diffusionof excimers and excited atoms out of the dense cloudabout the primary track can be neglected. The highlysimplified Eq. (13) is useful for demonstrating the de-pendence of quenching on concentration. The quenchingfactor f is defined as f = 1 n (cid:90) ∞ r nτ dt = ln (1 + ξ ) ξ . (14)It is the fraction of singlet species that radiatively decayrather than are destroyed by bimolecular interactionsand is related to n through ξ = n γτ /r . The value forthe quenching factor as determined calorimetrically fora 5.5 MeV alpha particle is f = . /.
20 = .
50 (ratio ofmeasured scintillation to predicted value[54]), resultingin ξ = 2 .
3. Because of the different ratio of excitations toions used here compared to that we used previously[50],the value of f also differs.The use of Eq. (13) to estimate the effect of quenchingthrough Eq. (14) involves another simplification necessi-tated by a lack of information at the microscopic level.It assumes that the density of the interacting speciesis uniform both along the track and perpendicularto it, which is certainly not the case. It is relativelystraightforward to account for the density variationalong the track by assuming the number of excimersand excited atoms produced is proportional to thestopping power (or stopping cross section). The effectof allowing for this variation along the track is to adda multiplicative term in the relationship between ξ and n or, alternatively, to change the value of the ratecoefficient, γ , by a corresponding amount. For the caseof a 5.5 MeV alpha the change is only about 15%, andwe do not consider it further. The variation in densityperpendicular to the track is not so easily treated. Itis possible that the spatial distribution of excimersformed on the subsequent recombination of ions andelectrons may differ from that for excited state atoms orexcimers formed from them. All we can assume is thatthe distribution is the same, independent of the energyor the primary particle and that relationship betweenquenching factor and density discussed above remainsvalid in comparing different energy depositions. V. Calculation of scintillation yield
The average number of ions and excited atoms perunit length deposited along the track as a function ofthe initial energy of a helium projectile is plotted inFig. 9. The number per unit length of all species thatcan partake in the Penning process changes significantlywith energy, varying from 6 . × per cm at 5.5 MeVwhere the electron quenching is measured to be 0.50to 1 . × per cm at 10 keV. At this latter averagenumber per cm the quenching factor, using Eq. (14)and the assumption of the same spatial distribution ofparticles perpendicular to the track, is calculated to be0.80, so that there is still a 20% reduction in scintillationfor nuclear recoils resulting from nonradiative decays.The quenching factor for prompt scintillation is plottedin Fig. 12.One consequence of the variation in number of ioniza- nu m be r pe r c m FIG. 9: (color online) Average number of ions and excited atomsper cm along track of projectile. Dotted line: excitations. Dashedline: ions. Solid line: sum of the two species. W v a l ue ( e V ) FIG. 10: The calculated W-value for a helium projectile in liquidhelium. tion and excitations with energy of the primary particleis the W-value does not remain a constant with energyas shown in Fig. 10. At 5.5 MeV the calculated W-valueis 38 eV, below the known value, increases slightlyat 1 MeV and then drops to a minimum of 30 eV at100 keV. But at 5 keV it is has a value of 160 eV. Thisrise in the W-value at low projectile energy is the resultof the decrease in probability of ionization comparedto excitation of helium atoms in this low energy range.The trend of increasing W-value for helium recoils withdecreasing energy is what is expected theoretically[51],and has been observed to occur in a number of puregases and mixtures[60, 61]. However, we can find noreport of a measurement of this phenomenon in helium.The W-value for electrons in helium remains es-sentially constant above 1 keV, since the W-value isinsensitive to the energy of the projectile as long asits velocity is much higher than that of the valenceelectrons[62, 67].While the dependence of the scintillation yield onrecoil energy is affected by electronic quenching, it is alsostrongly influenced by the change in ratio of excitationsto ionizations at low energies. Ionization are expectedto produce singlets to triplets in the ratio of 1 to 3,whereas excitations, as discussed above in consideringtheir creation by stopping of He are presumed tocreate singlets far more copiously, the ratio of singlets totriplets being 0.86 to 0.14.The number of prompt UV photons from singletexcimers and excited state atoms is plotted as a functionof the recoil energy helium atom in Fig. 11. The data forthis graph was obtained by summing the effective crosssections discussed in Section II, adding the contributionof secondary electrons to account for behavior above100 keV, and correcting for electronic quenching. Theproduction of excitations by secondary electrons wasincorporated following Sato[52].0 nu m be r o f pho t on s FIG. 11: (color online) Number of prompt UV photons producedby primary particle stopped in liquid helium. Dashed line: heliumatom recoil/alpha particle. Solid line: electron. L e ff and quen c h i ng f a c t o r FIG. 12: (color online) Solid line: relative scintillation efficiency, L eff as a function of He projectile energy. Dotted line: quenchingfactor due to bimolecular processes. The expected UV scintillation when electrons are theprimary particles stopped in helium is also plotted inFig. 11. Since the ionization density along the trackof an electron is so low the recombination is geminate.No quenching occurs. The G-values for ionization andfor excitation of helium by electrons is independent ofenergy above 1 keV[52, 62], so the scintillation yield isexpected to be linear in electron energy. As discussedabove, Adams[54] has measured that 35% of the initialkinetic energy of an electron stopped in liquid heliumappears as photons. The corresponding number foralphas at 5.5 MeV is 10%. Hence the number of UVphotons is 3.5 times larger for electrons than for alphasat high energies and remains larger, although by alesser amount, in the range below 100 keV. The relativescintillation yield, L eff , is simply the ratio of the twocurves in Fig. 11[63]. The energy dependence of L eff isplotted in Fig. 12. VI. Discrimination
The utility of any medium as a dark matter detectoris dependent on the ability to distinguish the signalproduced by nuclear recoil from that of produced bybackground, principally electrons from beta decay andCompton scattering. Hence we discuss the difference intime dependence of the scintillation response to electronsand nuclear recoils stopped in helium and the use ofcharge collection for discrimination.
A. Charge collection
Ito et al. [50] have measured the field dependence of thescintillation from alpha particles in helium to fields up to45 kV/cm. At this field the scintillation is decreasedby 15% from the zero field value. They used Kramerstheory[68] of columnar recombination to fit the field de-pendence of the ionization current generated by alphasstopped in liquid helium as measured by Gerritsen[69].A cylindrical Gaussian distribution of the initial chargesabout the track of the form, n ( r ) = N πb re − r /b , (15)where N is the total number of ionizations, produceda reasonable fit with b = 60 nm. However, this distri-bution does not provide a good approximation to theionization current for an alpha particle at low appliedfields as measured by Williams and Stacey[70]. Theirdata is reproduced in Fig. 13, normalized to the totalnumber of ionizations produced. We take this datato be a better measure of the charge separation withfield expected for low energy He nuclear recoil. Thelower initial charge density along the track of a lowenergy recoil as compared to that of an alpha is likely toincrease the field dependence somewhat for low energynuclear recoils, but without a more realistic model ofthe distribution of charge along the track any estimateof the change is unwarranted.For electrons in helium, Guo et al. [55] accounted forthe variation of geminate recombination and the decreasein scintillation with electric field by fitting a sphericaldistribution in separation of an electron from the positiveion from which it originated by the expression n ( r ) = N π ξ r e − r /πξ . (16)In the absence of diffusion a pair will recombine in a field E depending on the initial separation r and orientation θ of the separation with respect to that of the field. Whenthe condition e π(cid:15) r [1 + tan ( θ/ > E. (17)is valid, then the pair will recombine, otherwise theywill not. The effect of diffusion can be accounted for1
10 100 1000 10000 100000field (V/cm)10 -5 -4 -3 -2 -1 f r a c t i on o f c ha r ge c o ll e c t ed FIG. 13: (color online) Fraction of ion/electron pairs that areseparated and do not recombine as a function of applied electricfield in liquid helium. Solid line: α particles, from Ref. [70].Crosses: electrons, from Ref. [73]. Open circles: electrons, fromRef. [74]. Dashed line: electron distribution given by Eq. (16)with ξ = 56 nm, separation constrained as in Eq. (17) and nodiffusion. Dot/dashed line: Monte Carlo simulation includingdiffusion. numerically by performing a Monte Carlo simulation or,alternatively, by using the analytic expressions developedby Que and Rowlands[71] for the Onsager theory[72]of geminate recombination. Guo[55] found that thechange in scintillation at a field of 2700 V/cm could befit with ξ = 56 nm in Eq. (16). A plot of the fractionof charge that would be extracted as a function of fieldfor the distribution chosen by Guo is shown in Fig. 13both with no diffusion and with diffusion appropriatefor electron bubbles and positive ion snowballs in liquidhelium at 2.5 K. Such a distribution fails to fit themeasured charge collection as measured by Ghosh[73]and Sethumadhavan[74].Ghosh[73] in experiments on electron bubbles in he-lium used a Ni (beta emitter with an end point of66 keV) as a source. In the course of those experimentshe measured the current created in a pair of electrodesas a function of field in the liquid. He also measuredthe current in helium gas so as to obtain the saturationcurrent, that is, the complete charge separation of ioniza-tion events. His results obtained at 2.5 K for the liquidare plotted in Fig. 13. He found a small dependence ofcurrent on temperature but not sufficient to warrant dis-cussion here. Also, plotted is an extension of Ghosh’sresults to higher fields by Sethumadhavan[74]. What isclear is that the model assumed by Guo does not predictcharge separation properly at low fields. We do not havea theoretical understanding of the initial ion distribu-tion produced by electrons that is adequate for explain-ing their subsequent separation by an applied field. It iscoincidental that the charge separation for alpha parti-cles and for electrons as illustrated in Fig. 13 is the samefor fields less than 200 V/cm. If instead of using Ghosh’s data taken at 2.5 K, the data at 4.2 K were plotted, thecurves would differ by more than 50%. What is presum-ably also coincidental is that the field dependence of thecharge separation in the low field region below 200 V/cmis consistent with the expression
Q/Q = aE ln (1 + 1 /aE ) , (18)which is the form of the field dependence derived byThomas and Imel[75] with the constant a = 10 − cm/V.Above 200 V/cm the expression given by Eq. (18) bearsno relation to the measurements. B. Afterpulsing
Scintillation resulting from metastable tripletsexcimers ( a Σ u ) with a lifetime of 13 s has beendiscussed[49, 77, 78] as a means of discriminatingbetween electron and nuclear recoils. The number ofsingle photon events in the first few microseconds afterthe prompt signal with a 10 − decay time dependson the density of ions and excitations along the trackof the projectile. The delayed, discrete single-photonscintillation, called ”afterpulsing”, is not the result ofthe radiative decay of triplet excimers, a much tooinfrequent process to explain the observed rate, but israther the consequence of the Penning annihilation of apair of triplet excimers that results in the creation of asinglet that immediately radiatively decays[76].A calculation of the magnitude of the afterpulsing andits time dependence for nuclear recoils and for electronsstopped in helium is complicated by the dependence ofthe Penning bimolecular process on distribution of inter-acting species about the track of the primary projectileand the diffusive motion that leads to their encounter.McKinsey et al. [49] showed at 1.8 K that the magnitudeof afterpulsing normalized to the size of the promptscintillation signal was five time greater for a 5.3 MeValpha particle than for a 1 MeV electron. However,the highly ionizing products of the capture reaction He(n,p) H, with a combined recoil energy of 764 keV,produce only three times more afterpulsing than elec-trons. This variation of afterpulsing with energy cannotbe explained without a more detailed knowledge than iscurrently available of the parameters and mechanismsaffecting the process of afterpulsing. What happens atnuclear recoil energies below 100 keV is an open question.Afterpulsing is also dependent on temperature of theliquid. For alphas the magnitude rapidly decays below1 K as the quasiparticle density decreases and diffusionof the excimers away from the track is enhanced[49].Also, an electric field decreases afterpulsing[50].
VII. Discussion
The large increase in the relative scintillation effi-ciency, L eff , by more than a factor of two between100 keV and 5 keV as illustrated in Fig. 12, qualitatively2corresponds to the behavior of this quantity in theother liquefied noble gases. Both in neon[66] and inargon[79] the relative scintillation efficiency increaseswith decreasing energy below 50 keV down to 10 keV.This increase is primarily the result of the growth inexcitation relative to ionization at low energy. Bezrukov et al. [80] recently predicted the relative scintillationefficiency for liquid xenon using the electronic andnuclear stopping powers together with an analysis ofrecombination. They, too, note an increase in L eff withdecreasing recoil energy. There are no observations inthe other noble gas liquids of the decrease in relativeefficiency below 5 keV predicted for helium, as seen inFig. 12. This decrease is associated with the increasingfraction of energy going into the nuclear recoil channel.An analysis similar that performed here on helium doesnot appear possible for the heavy noble gases given theabsence of data on cross sections for both ionization andexcitations by nuclear recoils.This discussion of the scintillation yield of liquidhelium for low energy nuclear recoils is based to a largedegree on measurements of ionization, charge exchangeand excitation processes by helium ions in various chargestates. This approach is not without its problems. Themeasured cross sections have, in many cases, consider-able uncertainty. The energy deposition associated withthem is even less well known. Theory of these atomiccollisions with many electrons is not of help except incertain cases at low or high energies. Nonetheless, thereasonable agreement between the energy dependence ofthe stopping cross section obtained from a summationof microscopic processes and that generated from thenuclear and electronic stopping powers suggests theapproach has validity. As discussed earlier, the mostprominent difference between the stopping power curvesin Fig. 8, occurring in the energy range from 200 keVto 1 MeV, is to be due to the improper estimate of theenergy deposition of charge exchange processes. Thelow value of the calculated W-value and the stoppingpower above 1 MeV is more likely the result of excitationprocesses that have not been accounted for. Fortunately,these deficiencies are not of serious concern in predictingthe scintillation behavior for nuclear recoils below100 keV from WIMPS.The overall agreement between the stopping powercalculated as a sum of all the contributing processesand that from ASTAR [44] is 20% or better. Fromthis and the consideration of the uncertainty in thesinglet to triplet ratio, we estimate that the overalluncertainties of our calculated scintillation yield for lowenergy nuclear recoils is 30%. At energies below 10 keV,where excitations are the dominant component of theinteracting species, the uncertainty may be larger dueto the potential inaccuracy of the assumption that theradial spatial distribution of all interacting species is thesame. The radial spatial distribution of excimers formed Beam neutron Recoil helium atom Recoil neutron Liquid helium volume viewed by PMTs Neutron detector θ
FIG. 14: (color online) Layout of a possible experiment tomeasure the liquid helium scintillation yield for low-energynuclear recoils. on recombination of ions and electrons is dictated by thediffusion of electrons [50, 55], whereas this is not the casefor excitations. A different radial spatial distributionresults in a different number density of interactingspecies, affecting the quenching factor from bimolecularprocesses and thus the scintillation yield.The use of cross sections measured in the gas phase ofhelium, can be reasonably be assumed to be applicableto what happens in the liquid. Cooperative effectsin the liquid occur in what happens along the trackwith bimolecular Penning processes and with chargeseparation in an applied electric field. These phenomenacan depend upon density and diffusion and hence exhibita temperature dependence.Prior to any serious consideration of the use of liq-uid helium to detect low-energy nuclear recoils, it wouldbe highly desirable to perform an experimental measure-ment of scintillation yield as a function of recoil energy.This can be achieved by introducing a neutron beam ofknown energy into liquid helium and detecting the scat-tered neutrons at a known angle (see Fig. 14). In fact,the scintillation yield of heavier noble gas liquids (neon,argon, and xenon) have been studied in this manner usingneutrons from a D-D generator (see e.g. Refs. [5, 64–66]).Neutrons from a D-D generator, however, have a energyof ∼ VIII. Summary and conclusions
This paper contains a discussion of the scintillationproperties of liquid helium for low energy nuclear recoilsin the context of the possible use of this medium asa dark matter detector. We first review the availablecross section data on ionization and excitation of heliumatoms due to collisions with helium atoms and ions. Asa confirmation of the validity of our understanding ofthe ionization and excitation processes in liquid helium,the stopping power is calculated for a helium atom orion incident on helium as a sum of all the contributingmicroscopic processes. The resulting calculated stoppingpower is in reasonable agreement with the widely usedempirically determined stopping power. We then turnto what is known about scintillation processes theliquid helium generated by 5 MeV alpha particles.Nonradiative processes that quench the scintillation are also considered. Combining this information, wecalculate the liquid helium scintillation efficiency for lowenergy nuclear recoils. The prompt scintillation yieldthus obtained in the range of recoil energies from a fewkeV to 100 keV is somewhat higher than that obtainedby a linear extrapolation from the measured yield for an5 MeV alpha particle. Furthermore, we compare boththe scintillation yield and the charge separation by anelectric field for nuclear recoils and for electrons stoppedin helium. We also discuss a possible experiment to testthe results of our calculations.
Acknowledgments
We appreciate receiving a copy of a preprint of a paperby W. Guo an D.N. McKinsey[82] that covers some ofthe material discussed here. This work was supported bythe US Department of Energy and the National ScienceFoundation. [1] P. Benetti et al.
Astropart. Phys. , 495 (2008).[2] E.Aprile et al. (XENON100 collaboration)arXiv:1207.5988.[3] D.S. Akerib et al. (LUX collaboration) Nuc. Instum.Meth. A , 1 (2012).[4] D.N. McKinsey and J.M. Doyle, J. Low Temp. Phys. ,153 (2000).[5] W.H. Lippincott, K.J. Coakley, D. Gastler, E. Kearns,D.N. McKinsey, and J.A. Nikkel, Phys. Rev. C ,015807 (2012).[6] R.E. Lanou, H.J. Maris and G.M. Seidel, Phys. Rev. Lett. , 2498 (1987).[7] R.E. Lanou, H.J. Maris and G.M. Seidel, in Dark Matter ,Proc. XXIII Ren. de Moriond, ed. J. Adouze and J. TranThanh Van, p. 79, Editions Frontieres (1988).[8] Y.H. Huang, R.E. Lanou, H.J. Maris, G.M. Seidel, B.Sethumadhavan, W. Yao, Astropart. Phys. , 1 (2008).[9] J. Wilks, The Ptoperties of Liquid and Solid Helium ,Clarendon Press, Oxford (1967).[10] J.D. Talman and A.M. Frolov, Phys. Rev. A , 032722(2006).[11] S.K. Allison, Rev. Mod. Phys. , 1137 (1958).[12] W. Meckbach and I.B. Nemirovsky, Phys. Rev. , 13(1967).[13] L.I. Pivovar, V.M. Tubaev and M.T. Novikov, Sov. Phys.JETP , 20 (1961).[14] A. Itoh and F. Fukuzawa, J. Phys. Soc. Jap. , 632(1981).[15] N. Noda, J. Phys. Soc. Jap. , 625 (1976).[16] C.F. Barnett and P.M. Stier, Phys. Rev. , 385 (1958).[17] E.S. Soloviev, R.N. Il’in, V.A. Oparin and N.V. Fe-dorenko, Sov. Phys. JETP , 342 (1964).[18] J.M. Sanders, et al. , Phys. Rev. A , 062710 (2007).[19] L.J. Puckett, G.O. Taylor and D.W. Martin, Phys. Rev. , 271 (1969).[20] M.E. Rudd, T.V. Goffe, A. Itoh and R.D. DuBois, Phys.Rev. A , 829 (1985).[21] R.D. DuBois, Phys. Rev. A , 4440 (1989).[22] M.B. Shah and H.B. Gilbody, J. Phys. B. , 899 (1985).[23] R.D. DuBois, Phys. Rev. A , 2585 (1987). [24] D.W. Martin, R.A. Langley, D.S. Harmer, J.W. Hooper,and E.W. McDaniel, Phys. Rev. , A385 (1964).[25] R.M. Wood, A.K. Edwards and R.L. Ezell, Phys. Rev. A , 4415 (1986).[26] J.L. Forest, J.A. Tanis, S.M. Ferguson, R.R. Haar, K.Lifrieri, and V. L. Plano, Phys. Rev. A , 350 (1995).[27] V. Kempter, F. Veith and L. Zehnle, J. Phys. B. , 1041(1975).[28] V. Kempter, The Physics of Electronic and Atomic Col-lisions , J. Risley & R. Geballe ed.; p. 327 U. Washington,Seattle (1975).[29] R.E. Olson, E.J. Shipsey and J.C. Browne, J. Phys. B. , 905 (1975).[30] K.E. Banyard, B.J. Szuster and G.J. Seddon, J. Phys. B. , 2109 (1975).[31] J.P. Gauyacq, J. Phys. B. , 3067 (1976).[32] R. Okasaka, Y. Konishi, Y. Sato and K. Fukuda, J. Phys.B , 3771 (1987).[33] V. Pol, W. Kauppila, and J.T. Park, Phys. Rev. A ,2990 (1973).[34] R. Hippler, K-H. Schartner and H.F. Beyer, J. Phys. B , L337 (1978).[35] H.0. Folkerts, F.W. Bliek, L. Meng, R.E. Olson, R. Mor-genstern, J.M. von Hellermann, H.P. Summers and R.Hoekstra, J. Phys. B: , 3475 (1994).[36] W. Fritsch, J . Phys. B: , 3461 (1994).[37] M.E. Rudd, Y.K. Kim, D.H. Madison and T.J. Gay, Rev.Mod. Phys. , 441 (1992).[38] C.R. Garibotti and W.R. Cravero, Phys. Rev. A , 2012(1993).[39] M.B. Shah, D.S. Elliott, P. McCallion and H.B. Gilbody,J. Phys. B , 2751 (1988).[40] R. Rejoub, B.G. Lindsay, and R.F. Stebbings, Phys. Rev.A Helium: Stop-ping Powers and Ranges in All Elemental Matter , Vol.1;Pergamon Press, New York (1985). [43] S.T. Manson and L.H. Toburen, Phys, Rev. Lett, , 529(1981).[44] M. J. Berger, J. S. Coursey, M. A. Zucker, and J. Chang,ESTAR, PSTAR, and ASTAR: Computer Program forCalculating Stopping-Power and Range Tables for Elec-trons, Protons and Helium Ions, National Institute OfStandards and Technology, Gaithersburg, MD (2005),http://physics.nist.gov/Star [Originally published as: M.J. Berger, NISTIR 4999, National Institute of Standardsand Technology, Gaithersburg MD (1993)].[45] J.F. Ziegler, Helium: Stopping Powers and Ranges inAll Elemental Matter , Vol.4; Pergamon Press, New York(1977).[46] J. Lindhard, V. Nielsen, M. Scharff and P.V. Thomsen,Mat. Fys. Medd. Dan. Vid. Selsk. , no.10 (1963).[47] D.M. Mei, Z.B. Yin, L.C. Stonehill, A. Hime, Astropart.Phys. , 12 (2008).[48] 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).[49] D.N. McKinsey et al. , Phys. Rev. A , 062716 (2003).[50] 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).[51] S. Sato, K. Kowari and K. Okazaki, Bul. Chem. Soc.Japan , 933 (1976).[52] S. Sato, K. Kowari and K. Okazaki, Bul. Chem. Soc.Japan , 2174 (1974).[53] S.R. Bandler, S.M. Bro¨uer, C. Enss, R.E. Lanou, H.J.Maris, T. More, F.S. Porter, and G.M. Seidel, Phys. Rev.Lett., , 3169 (1995).[54] J.S. Adams, Ph.D. thesis, Brown University (2001).[55] W. Guo, M. Dufault, S.B. Cahn, J.A. Nikkel, Y. Shin,and D.N. McKinsey, JINST, , P01002 (3012).[56] J.W. Keto, F.J. Soley, M. Stockton, and W.A. Fitzsim-mons, Phys. Rev. A , 872 (1974); ibid , 887 (1974).[57] J.W. Keto, M. Stockton, and W.A. Fitzsimmons, Phys.Rev. Lett. , 792 (1972).[58] V.B. Eltsov, A.Ya. Parshin, and I.A. Todoshchenko, Zh.Exper. Teor. Fiz. , 1657 (1995); [Sov. Phys. JETP , 909 (1995)].[59] V.B. Eltsov, S.N. Dzhosyuk, A.Ya. Parshin, and I.A. To-doshchenko, J. Low Temp. Phys. , 219 (1998).[60] W.P. Jesse, Phys. Rev. ,1195 (1961).[61] H. Tawara, N. Ishida, J. Kikuchi and T. Doke, Nucl.Instr. Meth. B , 447 (1987). [62] D.A. Douthat, Radia. Res. , 1 (1975).[63] There appear to be two different definitions of the relativescintillation yield in the literature. One is L eff ( E ) = Y nr ( E ) /Y er (122 keV), where Y nr and Y er represent thescintillation yields for nuclear recoils and electron recoils,respectively. This definition is seen to be used for liquidxenon [64, 65]. The other is L eff ( E ) = Y nr ( E ) /Y er ( E ),which is used for argon and neon [5, 66]. We employ thelatter definition in this paper.[64] G. Plante, E. Aprile, R. Budnik, B. Choi, K.-L. Giboni,L.W. Goetzke, R.F. Lang, K.E. Lim, and A.J. MelgarejoFernandez, Phys. Rev. C , 045805 (2011).[65] A. Manzur, A. Curioni, L. Kastens, D.N. McKinsey,K. Ni, and T. Wongjirad, Phys. Rev. C , 025808(2010).[66] D. Gastler, E. Kearns, A. Hime, L. C. Stonehill, S. Seib-ert, J. Klein, W.H. Lippincott, D.N. McKinsey, andJ.A. Nikkel, Phys. Rev. C , 065811 (2012).[67] M. Inokuti, Radia. Res. , 6 (1975)[68] H.A. Kramers, Physica , 665 (1952).[69] A.N. Gerritsen, Physica , 407 (1948).[70] R.L. Williams and F.D. Stacey, Canad. J. Phys. , 928(1957).[71] W. Que and J.A. Rowlands, Phys. Rev. B , 10500(1995).[72] L. Onsager, Phys. Rev. , 554 (1938).[73] A Ghosh, Ph.D. thesis, Brown University (2005).[74] B. Sethumadhavan, Ph.D. thesis, Brown University(2007) and unpublished.[75] J. Thomas and D.A. Imel, Phys. Rev , 614 (1987).[76] T.A. King and R. Voltz, Proc. Roy. Soc. London A, ,424 (1966).[77] K. Habicht, Ph.D. thesis, Technische Universitat Berlin(1998).[78] G. Archibald et al. , AIP Conf. Proc. Low TemperaturePhysics: 24th International Conference on Low Temper-ature Physics ,
143 (2006).[79] C. Regenfus, Y. Allkofer, C. Amsler, W. Creus, A.Ferella, J. Rochet, and M. Walter, J. Phys. Cof. Series , 119 (2011).[81] LANSCE User Guide, http://lansce.lanl.gov/media/WNRUserGuide.pdf[82] W. Guo and D.N. McKinsey, Phys. Rev. D87