Properties of Liquid Argon Scintillation Light Emission
PProperties of Liquid Argon Scintillation Light Emission
Ettore Segreto ∗ Instituto de F´ısica “Gleb Wataghin” Universidade Estadual de Campinas - UNICAMPRua S´ergio Buarque de Holanda, No 777, CEP 13083-859 Campinas, S˜ao Paulo, Brazil (Dated: December 14, 2020)Liquid argon is used as active medium in a variety of neutrino and Dark Matter experimentsthanks to its excellent properties of charge yield and transport and as a scintillator. Liquid argonscintillation photons are emitted in a narrow band of 10 nm centered around 127 nm and witha characteristic time profile made by two components originated by the decay of the lowest lyingsinglet, Σ + u , and triplet states, Σ + u , of the excimer Ar ∗ to the dissociative ground state. A modelis proposed which takes into account the quenching of the long lived triplet states through theself-interaction with other triplet states or through the interaction with molecular Ar +2 ions. Themodel predicts the time profile of the scintillation signals and its dependence on the intensity ofan external electric field and on the density of deposited energy, if the relative abundance of theunquenched fast and slow components is know. The model successfully explains the experimentallyobserved dependence of the characteristic time of the slow component on the intensity of the appliedelectric field and the increase of photon yield of liquid argon when doped with small quantities ofxenon (at the ppm level). The model also predicts the dependence of the pulse shape parameter,F prompt , for electron and nuclear recoils on the recoil energy and the behavior of the relative lightyield of nuclear recoils in liquid argon, L eff . I. INTRODUCTION
Liquid Argon (LAr) is a powerful medium to detectionizing particles and is widely used in neutrino and DarkMatter experiments since several years [1], [2], [3], [4], [5],[6]. LAr scintillation photons are emitted in the VacuumUltra Violet in a 10 nm band centered around 127 nmwith a time profile made by two components with verydifferent characteristic decay times, a fast one in thenanosecond range and a slower one in the microsecondrange [7]. The relative abundance of the two compo-nents depends strongly on the ionizing particle type andallows for a powerful particle discrimination [8] [9]. LArscintillation has been deeply studied by several authors[7], [10], [11] and a solid understanding of the main mech-anisms regulating the production, emission and propaga-tion of scintillation photons has been achieved. However,there are experimental results which can not be easilyexplained and described by the currently accepted mod-els, as the dependence of the characteristic time of theslow component on the intensity of the electric field ap-plied to LAr and the increase of the LAr photon yieldwhen doped with small quantities of xenon, at the levelof few ppm (part per million). These phenomena pointto quenching mechanisms involving the excited specieswhich are the precursors of the scintillation photons andwhich have not been investigated before.The model proposed in this work takes into account thesequenching processes and predicts the shape of the scintil-lation pulse as a function of the applied electric field andof the density of the energy transferred to the electrons ofLAr. In addition to the experimental observations men-tioned above, it also allows to explain the dependence ∗ segreto@ifi.unicamp.br of the pulse shape parameter, F prompt , on the energy ofthe incoming particle. The cases of electron and nuclearrecoils are explicitly treated and compared to data.The integral of the scintillation pulse allows to estimatethe amount of quenching due to these processes and topredict the behavior of the relative light yield of nuclearrecoils in LAr, L eff , as a function of the recoil energy,which is a quantity of central importance for Dark Matterexperiments. II. LAR SCINTILLATION MODEL
The passage of ionizing particles in LAr producesfree excitons and electron-hole pairs. The proportionbetween these two species is assumed to be independentof the ionizing particle type and energy: N ex /N i = 0,21,where N ex is the abundance of excitons and N i the abun-dance of electron-hole pairs. Free excitons and holes areself-trapped within about 1 ps from their production andresult into excited, Ar ∗ , or ionized, Ar +2 , argon dimers.Ar +2 recombines with a thermalized electron to form Ar ∗ [10] which in turn decays non-radiatively to the firstsinglet and triplet excited states Σ + u and Σ + u . Thesetwo states, whose dis-excitation leads to the emission ofthe scintillation photons, have approximately the sameenergy with respect to the dissociative ground state,while the lifetimes are very different: in the nanosecondrange for Σ + u and in the microsecond range for Σ + u [7].The scintillation photon yield of LAr depends on theionizing particle type and on the Linear Energy Transfer(LET) [12]. The highest photon yield is reached byrelativistic heavy nuclei, from Ne to La. Low LET lightparticles (e − , p) have a slightly reduced photon yielddue to the fact that a fraction of the ionization electronsescapes from recombination. Nuclear recoils and α parti- a r X i v : . [ phy s i c s . i n s - d e t ] D ec cles also have a reduced photon yield, but the quenchingmechanism is different and not fully clarified yet. Abi-excitonic interaction in the core of the track has beenproposed as a possible explanation [10] and a reasonablygood agreement is found for 5,3 MeV α particles, whilethe model is not accurate in explaining the dependenceof the photon yield for nuclear recoils [11] at very lowenergies ( ∼ tens of keV). In this case, a better agreementwith the available data is obtained when using theBirks law to account for the possible quenching pro-cesses involving excitons and excited dimers Ar ∗ [13] [14].Two recent experimental observations can not be eas-ily explained with the current understanding of the LArscintillation process. The DUNE (Deep UndergroundNeutrino Experiment) Collaboration has reported a cleardependence of the lifetime of the triplet state Σ + u of the Ar ∗ dimer on the electric field in which the LAr is im-mersed [15]. The scintillation light was produced by asample of cosmic muons crossing one of the prototypesof the dual phase DUNE detector, the 4-ton demonstra-tor [16]. The light was detected with an array of five8” photomultipliers (Hamamatsu R5912-02Mod) coatedwith a wavelength shifter, TetraPhenyl-Butadiene (TPB)[17], to convert the 127 nm photons to 430 nm and theelectric field was varied between 0 and 600 V/cm.The second experimental evidence is related to the dop-ing of LAr with small concentrations of xenon. It hasbeen reported that adding a few tens of ppm of xenonto LAr has the effect of shifting the wavelength of thetriplet component from 127 nm to 174 nm, shorteningthe signal from few µ sec to hundreds of nsec and enhanc-ing the Light Yield (LY) [9]. The enhancement of LY cannot be explained by an higher quantum efficiency of thewavelength shifter (TPB) for 174 nm than for 127 nm,since it has been measured to be almost the same [18]and should be attributed to an increase of the LAr pho-ton yield.These two effects point to quenching processes of thetriplet states and to an hidden amount of light whichhas not been described before. The proposed quenchingmechanisms for the triplet states are two: one relies onthe interaction of two excited dimers Ar ∗ and the otheron the interaction of one ionized dimer Ar +2 with oneexcited dimer Ar ∗ : Ar ∗ + Ar ∗ → Ar ∗ + 2 Ar (1) Ar ∗ + Ar +2 → Ar +2 + 2 Ar (2)In the absence of an external electric field, reaction 2 ispossible when escaping electrons prevent the completerecombination of the ionization charge and thus only forlow LET particles (electrons, muons, protons, . . . ).It is assumed that only the excited dimers in the tripletstate participate to this quenching processes, since thelifetime of the singlet state is too short. The instanta-neous variation of the number density of triplet states,N , and of the number density of ionized dimers (N + ) can be written as: dN dt = D ∇ N − λ N − σ + v + N + N − σ v N (3) dN + dt = D + ∇ N + (4)where N and N + depend on time and position in space,D and D + are the diffusion constants of Ar ∗ and Ar +2 respectively, λ is the radiative dis-excitation rate of the Σ + u state, σ + is the cross section for the process 2, v + is the relative velocity between a triplet excimer and aion, σ is the cross section for process 1 and v is therelative velocity of two Ar ∗ .In the hypotheses that the diffusion terms can be ne-glected and that Ar ∗ and Ar +2 are uniformly distributedinside a cylinder of radius r along the track, equations3 and 4 reduce to one, which depends only on time: dN dt = − λ N − σ + v + N +0 N − σ v N (5)where N +0 is the density of Ar +2 , which is a constant.The hypotheses of the model will be discussed in sectionVIII. Equation 5 can be solved analytically and gives: N ( t ) = N e − λ q t q/λ q (1 − e − λ q t ) (6)where N is the initial density of triplet states, λ q = λ + k + , k + = σ + v + N +0 , q = N σ v . The probabilitydensity of the LAr scintillation light can be written as: l ( t ) = α s τ s e − tτs + α τ e − tτq qτ q (1 − e − tτq ) (7)where α s is the initial abundance of the singlet states, τ s is the decay time of the singlet states, α is the initialabundance of triplet states, τ is the unquenched decaytime of the triplet states, and τ q = 1 / ( λ + k + )=1/ λ q .The probability density of triplet states depends on theelectric field and on the LET thorough τ q and q. The in-tegral, L, of the probability density l ( t ), which is propor-tional to the total number of scintillation photons emit-ted, is given by: L = L s + L = α s + α ln(1 + q τ q ) q τ (8)L is equal to one ( α s + α ) only when q is zero and τ q = τ . III. EXTRACTION OF THE PARAMETERS OFTHE MODEL FOR ELECTRON AND NUCLEARRECOILS
Some of the parameters of equation 7 for electron andnuclear recoils have been extracted through a fit proce-dure of experimental waveforms. The data which have
TABLE I. Parameters of the model (equation 7) extractedfrom the fit procedure. The electron recoil average waveformis constructed with events with a deposited energy between 85and 100 keV, while the nuclear recoil average waveform withevents with a deposited energy between 220 and 290 keV.gamma neutron τ (nsec) 2100 ±
20 2100 ± τ s (nsec) 5 ± ± α s ± ± α ± ± + (nsec − ) (1.3 ± × − − ) (2.3 ± × − (2.3 ± × − been analyzed were collected during the test described in[8], [19]. Within the R&D program of the WArP exper-iment [3] a 4 liters single phase LAr chamber, observedby seven 2” photomultipliers (ETL D749U), was exposedto neutron (AmBe) and γ sources. The internal sur-faces of the LAr chamber were coated with TetraPhenyl-Butadiene (TPB) [20] to down-convert the 127 nm LArscintillation photons to 430 nm making them detectableby the photomultipliers. After a selection of the eventsbased on the shape of the signals, electron and nuclearrecoil average waveforms were calculated for different in-tervals of deposited energy. The LY of the detector wasmeasured to be 1,52 phel/keV (photo-electrons/keV) forthe considered run. An average electron recoil waveform,calculated with signals containing between 130 and 150phel, and an average nuclear recoil waveform, calculatedwith signals containing between 150 and 180 phel, havebeen simultaneously fitted. Considering the LY of thedetector these correspond to an energy interval of 85 to100 keV and of 220 to 290 keV respectively. The fit func-tion contains the amplitudes of the singlet and tripletcomponents ( α s and α ), the decay time of the singletcomponent ( τ s ), the unquenched decay time of the tripletcomponent ( τ ) and the rate constants k + and q . Anadditional time component, with a decay time around50 nsec, is included to take into account the late light re-emission of TPB [17] with a constant abundance withrespect to the singlet component. The light signal isconvoluted with a Gaussian function to accomodate thestatistical fluctuations and the response of the read-outelectronics.The parameters which are assumed to be common to theelectron and nuclear recoil waveforms are the decay timeof the singlet component ( τ s ), the unquenched decay timeof the triplet component ( τ ) and the fraction of TPB latelight, while the other parameters are assumed to be par-ticle dependent. The result of the fit is shown in figure1. The main parameters of the fit are reported in tableI. -4 -3 -2 -1 γ neutron time (nesc) a m p li t ud e ( a r b . un i t s ) FIG. 1. Average waveforms for gammas and neutrons. Greenand magenta lines represent the result of the fitting procedurefor gammas and neutrons respectively.
IV. DEPENDENCE OF THE SLOW DECAYTIME FROM THE ELECTRIC FIELD AT LOWLET
The shape of the LAr scintillation waveform dependson the module of the applied electric field, E , throughthe parameters k and q (see equation 7). The chargerecombination factor, R( E ), is assumed to have the form: R ( E ) = B + A K E / E (9)where B takes into account the fraction of the chargewhich does not recombine even at null electric field, dueto escaping electrons, k E is the Birks recombination con-stant and A is a normalization constant [21].Equation 9 can be used to make explicit the depen-dence of the density of Ar +2 ions, N +0 , and of the initialdensity of triplet states, N on the electric field: N +0 = N i R ( E ) (10) N = N i α [1 − R ( E ) + N ex / N i ] (11)The parameters k + and q can be written as: k + ( E ) = k +0 (cid:104) AB (1 + k E / E ) (cid:105) (12) q ( E ) = q (cid:104) − A ( A + N ex / N i )(1 + k E / E ) (cid:105) (13)where k +0 and q are the values of k + and q at zeroelectric field and at a given value of LET. For low LETparticles, for which the phenomenon of escaping electronsis present ( k + (cid:54) = 0) and when q τ q (cid:28)
1, the scintillation
Lastoria et al.this work
Electric Field (kV/cm) τ e ff ( n sec ) FIG. 2. Variation of the decay time of the slow scintillationcomponent, τ eff as a function of the applied electric field.Magenta line represents a fit of the data with the function ofequation 14. signal of equation 7 presents only a small deviation froma purely exponential decay with a characteristic time of τ eff (cid:39) τ q and its dependence on the electric field can beexplicitly written as: τ eff (cid:39) τ q = 1 τ + k +0 + k +0 AB (1+ k E / E ) = 1 α + β k E / E (14)where α = / τ eff (0) is the inverse of the characteristictime at zero electric filed and β = k +0 A / B .The measurement of the variation of the slow decay timeof the LAr scintillation light as a function of the appliedelectric field, reported in [15], represents, substantially,a measurement of the electron-ion recombination processin LAr performed with light. Experimental points takenfrom [15] are shown in figure 2. They have been fittedwith the function of equation 14, leaving the parameters α , β and k E free and the result is shown with a ma-genta line. The agreement between data and the modelis pretty good, with the exception of one of the points atvery low electric field. The fit procedure returns a value of k E = 0,075 ± ± g / cm MeV reported in[21], when considering cosmic muons with energies be-tween 1 GeV and 4 GeV and a stopping power between1,6
MeV g / cm and 2,0 MeV g / cm in LAr. V. XENON DOPING OF LAR
It is known that adding xenon to LAr at the level ofppm (part per million) has the effect of shifting the wave-length of the slow scintillation component from 127 nmto 174 nm [9]. The complete shift of the slow componentis observed at tens of ppm of xenon concentration. It hasbeen also reported an increase of the number of detectedphotons with increasing xenon concentration, which cannot be explained with an increase of the conversion ef-ficiency of the TPB, the wavelength shifter used in theexperiment, which has been measured to be almost thesame at 127 nm and 174 nm [22].The mechanism suggested in [23] for the transfer of theexcitation energy from argon to xenon can be summa-rized with the following reaction chain: Ar ∗ + Xe → ( ArXe ) ∗ + Ar (15)( ArXe ) ∗ + Xe → Xe ∗ + Ar (16)The energy transfer process and the subsequent emis-sion of 174 nm photons compete with the radiative decayof the Ar ∗ and with the quenching processes described insection II. The net effect is the shift of the slow LArscintillation component and the partial recovery of thequenched LAr species, which both result in the emissionof 174 nm photons with peculiar time characteristics.In order to understand the gross features of the lightemission process from a xenon-argon mixture in the caseof low LET particles, it is worth making some rough ap-proximations. For a high enough Xe concentration, itshould be possible to neglect second order quenching ef-fects, such as Ar ∗ -Ar ∗ , Ar ∗ -(ArXe) ∗ , (ArXe) ∗ -(ArXe) ∗ .Assuming that the energy transfer between argon andxenon happens without losses and that the reaction ratesof the processes Ar ∗ + Xe → (ArXe) ∗ and (ArXe) ∗ + Xe → Xe ∗ are the same and equal to k Xe , the instantaneousvariation of the density of triplet argon states, N Ar (t), ofXe ∗ dimers, N Xe (t) [24] and of mixed dimers (ArXe) ∗ ,N ArXe (t) can be written as: dN Ar ( t ) dt = − λ N Ar ( t ) − k + N Ar ( t ) − k Xe [ Xe ] N Ar ( t ) (17) dN ArXe ( t ) dt = − k Xe [ Xe ] N ArXe ( t ) + k Xe [ Xe ] N Ar ( t ) (18) N Xe ( t ) dt = − λ Xe N Xe ( t ) + k Xe [ Xe ] N ArXe ( t ) (19)where λ Xe is the inverse of the characteristic time of xenon emission and [Xe] is the xenon concentra- -4 -3 -2 Pure LAr20 ppm Xe50 ppm Xe100 ppm Xe time (nsec) a m p li t ud e ( a r b . un i t s ) FIG. 3. Waveforms of xenon doped liquid argon at differentxenon concentrations as predicted by the model. The wave-forms represent the sum of LAr and xenon shifted light. tion. Assuming that λ Xe (cid:29) k Xe [Xe], equations 17, 18and 19 can be easily solved with the initial conditionthat N ArXe (0) = 0 and give the probability density forthe slow LAr scintillation component, l (t), and for Xeshifted light, l Xe ( t ): l ( t ) = α τ e − tτr (20) l Xe ( t ) = α ( k Xe [ Xe ]) τ q [ e − tτd − e − tτr ] (21)where τ r = 1 / ( k Xe [ Xe ] + / τ q ) and τ d = 1 /k Xe [ Xe ].The probability density for all the emitted photons, re-gardless of their wavelength, can be obtained by sum-ming the contributions of equations 20 and 21 to the fastLAr scintillation component, assumed to be unaffectedby xenon doping: l ( t ) = α s τ s e − tτs + l ( t ) + l Xe ( t ) (22)The total amount of emitted light is obtained by inte-grating equation 22: L = α s + α τ r τ + α ( k Xe [ Xe ])( k Xe [ Xe ]) + / τ q (23)Using the parameters of the scintillation waveform forelectron recoils found in section III, and the reaction ratek Xe = 8,8 × − ppm − nsec − (with ppm in mass) re-ported in [9] it is possible to predict approximately theshape of the scintillation signal for electron recoils andthe dependence of the LY on the xenon concentration,[Xe].Few waveforms for different xenon concentrations areshown in figure 3. The exact shape will depend on the -2 -1 Wahl et al.this work xenon concentration (ppm) r e l. li gh t y i e l d FIG. 4. Variation of the LY of LAr and xenon shifted photonsas a function of the xenon concentration in pppm (mass). Theexperimental points at zero xenon concentration have beenshifted to 10 − ppm to facilitate the visualization. The modelprediction is shown as a magenta line. precise values of the reaction rates and on the even-tual conversion of fast (and slow) LAr scintillation lightthrough photo-absorption by xenon atoms. This wouldlead to the formation of (ArXe) ∗ which would evolve inXe ∗ according to reaction 16.The variation of the total LY, which includes 127 nmand 174 nm photons, predicted by the model is shownin figure 4 together with the experimental data shownin [9]. Both model prediction and data have been nor-malized by their value at zero xenon concentration. Anincrease of the overall LY around 25% for concentrationsabove few tens of ppm in mass is observed in the dataand correctly predicted by the model. It is reasonable toconclude that the two main approximations made: ne-glecting second order quenching effects and assuming alossless transfer of energy between xenon and argon aresmall and compensate with each other, leading to a gooddescription of the experimental data. The first approxi-mation would lead to an increase of LY, since the amountof quenched Ar ∗ would be higher, while the second wouldlead to a decrease of LY, because of possible non-radiativedis-excitations of the (ArXe) ∗ states. VI. F-PROMPT FOR ELECTRON ANDNUCLEAR RECOILS
LAr allows for a powerful particle discrimination basedon the shape of its scintillation signal. In particular, therelative abundances of fast and slow components stronglydepend on the particle type [7]. This property of LAr iscrucial for discarding gamma and electron backgroundsfrom nuclear recoil events in direct Dark Matter experi-ments ([8], [3], [25], [26], [27]).The variable which is typically used to discriminate elec-tron from nuclear recoils in LAr is F prompt , which informsthe abundance of the fast component in the scintillationsignal and is defined as: F prompt = (cid:82) t ∗ l ( t ) dt (cid:82) ∞ l ( t ) dt (24)where l(t) is the scintillation waveform and t ∗ is theintegration time of the fast component which maximizesthe separation. Different values of t ∗ are used by differ-ent groups, but it is typically close to 100 nsec. F prompt depends on the recoil energy and the separation betweenelectron and nuclear recoils tends to get worse at lowerenergies. The proposed model contains explicitly thedependence of the scintillation signals of electron andnuclear recoils from the density of deposited energyand can be used to predict and explain the behavior ofF prompt observed experimentally.In order to evaluate F prompt for electrons as a func-tion of the kinetic energy, E, of the incoming electron,F eprompt ( E ), it is necessary to evaluate the parameters k + and q of equation 7. The prameter k + is proportional tothe number density N +0 of Ar +2 , whose dependence on thedensity of deposited energy is not known. It is reasonableto assume, in first approximation, that it stays constantin the region of interest, below 1 MeV. The density of de-posited energy varies very slowly down to 100 keV, whereit starts increasing more steeply. In this low energy re-gion the effect of the increased density is compensatedby the reduction of the escaping probability for the ion-ization electrons due to the increased electric field in thecore of the track.The parameter q is equal to: q e = σ v N = σ v (cid:104) dEdx r (cid:105) α e W el (cid:16) − R (0)+ N ex N i (cid:17) (25)where dE / dx is the electronic stopping power of argon,[ dE / dx r ] is the average value of the density of depositedenergy along the track, W el = 23,6 eV is the averageenergy to produce an electron ion pair in LAr [7], R(0)is the recombination factor at zero electric field and α e = 0,85 is the initial relative abundance of triplet statesfor electron recoils. For low energy electrons ( < r is proportional to γ β ,in particular: dEdx r (cid:39) dEdx ¯ ν c β γ (26)where c is the speed of light, β =v/c, γ = 1/ (cid:112) (1 − β )and ¯ ν is the average orbiting frequency of atomic elec-trons, which is related to the mean excitation potential,I, by the relation I = h ¯ ν . For argon, I = 188 eV. Usingequations 25 and 26, q is written as: q e ( E ) = k eq d e ( E ) (27) where d e ( E ) is the average value along the track of thedensity of deposited energy of equation 26 multiplied by α e : d e ( E ) = (cid:90) E dE (cid:48) dx α e ¯ ν c β γ dE (cid:48) E (28)and k eq is a characteristic constant: k eq = σ v W el (cid:16) − R (0) + N ex N i (cid:17) (29)In order to compare the model to data, the integral ofequation 28 is evaluated numerically using tabulated val-ues for the electronic stopping power [29]. d e ( E ) is well fitted, for E < d e ( E ) = 843( E − . − . E − . + 0 .
66) (30)with E in MeV and d e in MeV/ µ m . The model predic-tion is compared to experimental data from [8] and [27].F eprompt is calculated as: F e,nprompt = p L s + p L L s + L (31)where L and L are the abundances of the fast and slowcomponents respectively (see equation 8), the parametersp and p are related to the integration process for thecalculation of F e,nprompt (see equation 24) and in particularto the fraction of fast and slow component which fall in-side the integration window. Typically p is close to oneand p is close to zero. The fact of p not being exactlyequal to one is attributed to the delayed light emission ofthe wavelength shifters used to detect LAr photons [17].The parameters p , p and k eq are left free and adjustedon data. The two data sets have been fitted separately,since they show some differences in their asymptotic be-haviors at large energies, which is probably related to dif-ferent integration intervals, and in their slopes at low en-ergies. The result of the fitting procedures is shown in fig-ure 5. The values of k eq = 0 . × − nsec − MeV − µm for [8] and k eq = 1 . × − nsec − MeV − µm for [27]are obtained.In the case of a low energy nuclear recoils of an argonatom in LAr, below few hundreds of keV, a significantamount of the energy lost by the recoiling argon atomis due to elastic collisions with other argon nuclei andonly a small fraction of it is transferred to the electrons.The Lindhard theory [30] [11] gives the amount of energytransferred to the electrons of the LAr in terms of thedimensionless variable ε : ε = C ε E = a T F F A Z Z e ( A + A ) E (32)where E is the recoil energy, Z and A are the atomic andmass number of the projectile (1) and of the medium (2) Energy (MeV) f- p r o m p t Acciarri et al.Lippincott et al.this workthis work
FIG. 5. F prompt for low energy electrons in LAr measuredby two experimental groups [8] [27]. The experimental pointshave been fitted with equation 31 with fitting parameters p ,p and k q (see text). The results of the fit procedures areshown with magenta ([8]) and green ([27]) lines. and: a T F F = 0 . a B ( Z / + Z / ) / (33)a B = (cid:126) /m e e = 0 .
529 ˚ A is the Bohr radius. For Z = Z equation 32 gives C ε = 0.01354 keV − . The amount ofenergy transferred to the electrons is given by [13]: η ( ε ) = k g ( ε ) ε k g ( ε ) (34)where k = 0.133 Z / A − / = 0.144 and g( ε ) is fittedwith the function [13]: g ( ε ) = 3 ε . + 0 . ε . + ε (35)Low energy nuclear recoils deposit their energy in a veryconfined portion of space. The projected range for ar-gon ions of tens of keV kinetic energy in LAr is of theorder of few hundreds of ˚ A [31] and the transversal di-mensions of the core of the track is of the order of theBohr radius a B [11]. It is reasonable to assume that inthis range of energies, the spatial distribution of the Ar ∗ is largely dominated by the fast diffusion of excitons andholes before self-trapping and by the coulombian repul-sion of the Ar ions in the core of the track. Under thishypothesis, the recoil energy is deposited inside an ap-proximately constant volume at very low energies. A lin-ear growth of this volume with energy is expected, sincethe total stopping power is constant for energies belowfew hundreds ok keV [30] [11] and the range of the Arions is proportional to its initial kinetic energy, while the transversal dimensions continue being dominated by fastdiffusive processes. Using equations 25 and 34, the factor q for low energy nuclear recoils is written as: q n ( E ) = k nq d n ( E ) = k nq α n V η ( E )1 + k V E (36)where α n = 0.35 is the initial relative abundance oftriplet states for nuclear recoils, V is the volume insidewhich the energy deposited by nuclear recoils is containedfor very low energies, k V takes into account the increaseof the volume with the recoil energy and k nq is given by: k nq = σ v W el (cid:16) N ex N i (cid:17) (37)Equations 36 and 8 (with τ q = τ ) can be substituted inequation 31 to evaluate F nprompt for nuclear recoils andcompare it to available data.The model is fitted to the data reported in [8] and re-ferred to, as the high light yield sample, which span abroad interval of energies, from 10 keV to above 1 MeV.The conversion between the number of detected photo-electrons (N phel ) and the nuclear recoil energy in [8] isdone assuming a constant relative scintillation yield be-tween nuclear and electron recoils, L eff , equal to 0,3. Amore appropriate conversion between N phel and the recoilenergy, E n , is given by: E n = N phel LY × L eff ( E n ) (38)where LY is the Light Yield for electron recoils expressedin phel/keV. The fit procedure of the F nprompt data re-ported in [8], which takes into account equation 38 isdescribed in the following section. VII. QUENCHING FACTOR FOR NUCLEARRECOILS
The quenching of the number of emitted photons in anuclear recoil event is described as the succession of threedistinct quenching processes: (i) the quenching of theamount of energy transferred to the atomic electrons, dueto the elastic collisions of the argon ion with surround-ing argon atoms [30]; (ii) the quenching of the excitonsformed after the nuclear recoil due to bi-excitonic quench-ing [11]; (iii) the quenching of the triplet states formedafter the trapping of the excitons due to triplet-triplet in-teractions (and triplet-ion in the presence of Ar +2 ). Theoverall quenching factor for nuclear recoils can be writtenas: Q N = Q L × Q E × Q T (39)Q L is the quenching factor of process (i), which has beencalculated by Lindhard [30] and can be written as (referto equation 34): Q L = η ( ε ) ε = k g ( ε )1 + k g ( ε ) (40) Acciarri et al. (high LY)Acciarri et al. (low LY)DarkSideLippincott et al.this work
Energy (MeV) f- p r o m p t FIG. 6. F prompt for nuclear recoils in LAr measured by dif-ferent authors [8] [27] [25]. The high light yield sample (seetext) from [8] is used to fit the data. Magenta line representsthe fit result. The data sets have been aligned by applyingoverall scale factors which vary between 1.01 and 1.05. Q E is the quenching factor of process (ii). It has beenpointed out in [10] that Q E can be considered approxi-mately constant for energies below few hundreds of keV.This approximation has been proven to work well for liq-uid xenon [32]. Q T is the quenching factor for process(iii), discussed in this work. It can be written as (seeequation 8): Q T = α s + α ln (1 + q n ( E ) τ ) q n ( E ) τ (41)where α s = 0.65, α = 0.35, τ = 2100 nsec and q n (E)is given by equation 36.Experimentally, it is convenient to measure the rela-tive scintillation yield of nuclear recoils with respect toelectron recoils, L eff . Nuclear recoils are typically com-pared to the recoils of electrons emitted after the photo-absorption of the γ lines of Am (59.5 keV) and Co(122 keV). L eff can be estimated as: L eff = Q N Q el = Q L × Q T × Q E Q el (42)where Q el is the quenching factor of the referenceelectron recoil. Since Q E and Q el are both constant,the ratio Q E / Q el is an overall multiplicative constant inequation 42.In order to estimate the parameters of the model, anoverall fit of the F nprompt data from [8] and of the L eff data reported by the ARIS Collaboration in [33] isperformed, where the product k q α n / V and k V fromequation 36 and the ratio Q E / Q el of equation 42 are left as free parameters. The exact value of E n for theF nprompt set of data is obtained by inverting numeri-cally equation 38. The fit procedure returns a valueof 0,11 ± − nsec − for the product k q α n / V ,9,1 ± − for k V and 1,00 ± Q E / Q el . Taking into account the value of k eq found withthe fit of F epropmpt for electron recoils, that α n =0.35and that the recombination factor for nuclear recoils atzero electric field is zero (R(0)=0), the volume V rangesbetween 200 and 500 nm . This corresponds to a spherewith a diameter between 7 and 10 nm, which is in areasonable agreement with the diffusion of free excitonsand holes before trapping, for times of the order of1 psec and assuming a diffusion constant D = 1 cm /sec[11].A fit procedure which uses only the F nprompt data setis able to constrain pretty well the terms k q α n / V and k V ,giving results compatible with the ones reported here,while the range for the ratio Q E / Q el turns out to be quitebroad (between 0,9 and 1,1). The same ratio Q E / Q el could be estimated following the α -core approximationdiscussed in [11] giving a value close to unity, but witha quite large uncertainty (at the level of 20-30%).The result of the fit for F nprompt is shown in figure 6together with three more data sets. The data set referredas low light yield is also taken from [8], but it is obtainedwith a different experimental set-up, with a lower lightyield with respect to the high light yield one. The twoother data sets are taken from [25] and [27]The data sets have been aligned by applying overallscaling factors to the F nprompt values (one per eachdata set), which range between 1.01 and 1.05. The notperfect overlap of the different data sets before rescalingis attributed to different integration intervals to computeF nprompt and to small systematic effects.The result of the fit for L eff is shown in figure 7. Themodel prediction, together with the majority of availabledata is shown in figure 8. VIII. DISCUSSION
The triplet-triplet quenching reaction 1 has enough en-ergy to ionize one of the two interacting triplet states,since the band gap of LAr is about 14.2 eV [37]. Themodel assumes that the ionized molecule and the elec-tron recombine in a triplet state. This is a good as-sumption if the spin relaxation time of the ion-electronsystem is long compared to the recombination time. Thepossibility that the ion-electron system recombine in asinglet state has been tested by explicitly including it inthe model and a very small contribution, at the level ofpercent, has been found and it has been neglected.The triplet-ion quenching reaction 2 has enough energyto dissociate the Ar2 + ion, whose binding energy is about ARISthis work
Energy (keV) L e ff FIG. 7. Comparison of the model prediction for L eff (ma-genta line) with the data from the ARIS Collaboration [33].The factor Q E / Q el has been adjusted to data (see text). ARISSCENEMicroCLEANCreus et al.this work
Energy (keV) L e ff FIG. 8. Comparison of the model prediction for L eff (ma-genta line) with data taken from [33], [34], [35], [36]. +2 states does not change.An important hypothesis of the model is to neglect thediffusion terms in equations 3 and 4. In the case of lowLET particles this is well justified by the transversal dis-tribution (with respect to particle direction) of the den-sity of deposited energy which is grater than hundredsof nm, while the diffusion constant of Ar ∗ should be ofthe order of 10 − cm /s or less [39]. This assumptionis also consistent with the results obtained for nuclearrecoils, since the volume inside which the electronic exci- tation energy is found to be released is compatible withthe simple diffusion of free excitons and holes before self-trapping (see Sec. VII) and the contribution of Ar ∗ dif-fusion should be negligible on the time scale of LAr scin-tillation emission.The model intorduces a possible additional quenchingmechanism of the scintillation light in LAr and poses aserious question about its absolute photon yield. Thisdoes not necessarily means that the photon yield (afterthe quenching) is different from what is widely assumed,since it has never been directly measured. It is believedthat the ideal photon yield (51,000 photons/MeV [40]),estimated on the basis of the value of W el and N ex /N i , isreached by relativistic heavy ions and then scaled for theother particles and energies according to the experimen-tal results of relative measurements. This point shouldbe experimentally addressed since the technology of lightdetection is mature enough to allow precision measure-ments of the photon yield. This and other measurementsand characterizations of the LAr scintillation light prop-erties would be extremely desirable in view of the designof the next generation experiments for neutrino and DarkMatter direct detection. IX. CONCLUSIONS
This work describes a model for the production of LArscintillation light which takes into account the quench-ing of Ar ∗ through self interactions and interactions withAr +2 ions. It allows to justify two processes which couldnot be explained otherwise: the dependence of the slowscintillation decay time from the intensity of an exter-nal electric field and the increase of the photon yield ofxenon doped liquid argon, both for low LET particle in-teractions. It is possible to make an accurate predictionof the time profile of the scintillation pulse where the de-pendence on the electric field and on the density of thedeposited energy is explicit. A simultaneous fit to exper-imental average waveforms of electron and nuclear recoilevents allows to constrains some of the most relevant pa-rameters of the model such as the unquenched decay timeof the slow scintillation component, τ , which results tobe around 2100 nsec and the relative abundance of sin-glet and triplet states for electron and nuclear recoils.Knowing the shape of the scintillation pulse makes itpossible to analytically calculate the relative abundanceof slow component, F prompt , which is often used as apulse shape discrimination parameter. The expressionsof F prompt for electron and nuclear recoils need to befitted to the available data to extract some of the pa-rameters which are not precisely known, but the overallbehavior is well reproduced.The model allows to predict the shape of the relative scin-tillation efficiency for nuclear recoils in LAr, L eff , andit has been shown that it reproduces closely the experi-mental data reported by the ARIS Collaboration [33].LAr is a powerful medium for particle detection which0is being widely used in many fields of fundamental par-ticle physics. Deepening the knowledge of its propertiescan greatly benefit the design of the next generation ofdetectors. ACKNOWLEDGMENTS
The author thanks the members of the
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