Germanium response to sub-keV nuclear recoils: a multipronged experimental characterization
GGermanium response to sub-keV nuclear recoils:a multipronged experimental characterization
J.I. Collar, ∗ A.R.L. Kavner, and C.M. Lewis
Enrico Fermi Institute, Kavli Institute for Cosmological Physics, and Department of PhysicsUniversity of Chicago, Chicago, Illinois 60637, USA (Dated: February 22, 2021)Germanium is the detector material of choice in many rare-event searches looking for low-energynuclear recoils induced by dark matter particles or neutrinos. We perform a systematic explorationof its quenching factor for sub-keV nuclear recoils, using multiple techniques: photo-neutron sources,recoils from gamma-emission following thermal neutron capture, and a monochromatic filtered neu-tron beam. Our results point to a marked deviation from the predictions of the Lindhard model inthis mostly unexplored energy range. We comment on the compatibility of our data with low-energyprocesses such as the Migdal effect, and on the impact of our measurements on upcoming searches.
I. INTRODUCTION
The study of low-energy nuclear recoils (NRs) inducedby the elastic scattering of neutral particles off nucleiis an active area of research in particle physics. Untilrecently, its main motivation was the numerous experi-ments looking for Weakly Interacting Massive Particles(WIMPs), one of the most popular and well-motivateddark matter candidates. The recent experimental demon-stration of Coherent Elastic Neutrino-Nucleus Scattering(CE ν NS) [1–5], a process involving this same mechanismof interaction -albeit from particles known to exist- hasreinforced the need to understand signal generation fromthese subtle NRs in a variety of materials.For detecting media exploiting the ionization or scintil-lation generated by particle interactions, a central quan-tity is the so-called quenching factor (QF). This is theratio of observable energy expressed in one of those chan-nels by a NR, over that generated by an electron recoil(ER) of the same kinetic energy. In experimental stud-ies, the first are typically generated by fast neutrons, thesecond by gammas or x-rays. At NR energies of interest(few keV nr ) this QF is typically of order 10%, adding tothe difficulty of WIMP and CE ν NS searches. We haverecently emphasized the importance of dedicated QF cal-ibrations able to discern its energy dependence [6]. Thesestudies are fundamental in order to access the many devi-ations from the Standard Model, involving new physics,that are testable via CE ν NS. Experimental efforts shirk-ing QF characterization are subject to uncertainties insignal significance and interpretation [7, 8].Low-noise p-type point contact (PPC) detectors [9] canregister the ionization from sub-keV energy depositionsin large ( > ν NS [15–18], and exotic modes of particle decay [19].This work concentrates on the characterization of the QF ∗ [email protected] for ionization-sensitive germanium detectors in the sub-keV nr NR energy region. This realm remains essentiallyunexplored for most materials. The information obtainedstrongly impacts future germanium searches for CE ν NSfrom reactor antineutrinos and for low-mass WIMPs.In order to access the tiny energies involved in thisstudy, down to ∼
50 eV in deposited ionization, we em-ploy a small (1 cm , 78 eV FWHM noise) GL0110 LEGe(Low Energy Germanium) detector [20]. For crystals thissize the use of n-type germanium is possible while pre-serving the good charge collection and energy resolutionseen to rapidly degrade for larger n-type point-contactconfigurations [9, 21]. This choice removes the nuisanceparameters introduced in the analysis by the O(1) mmthickness of inert surface electrode layers in p-type ma-terial [22, 23]. At sub-micron thickness for this device,this surface structure can be safely neglected. In addi-tion to this, continuous advancements in noise reductionfor point-contact detectors [24] allowed us to reach a 200eV ionization energy analysis threshold in this device andeven smaller ( ∼
100 eV) for externally-triggered signals.This is in contrast to the 1 keV threshold achieved in ourlatest germanium QF study [22]. A reduced noise also re-sults in excellent energy resolution (Fig. 1). Lastly, multi-ple scattering in this small LEGe involves just a 4%-17%of interactions, depending on calibration technique. Mul-tiple scatters dominated previous studies using ∼
100 cm PPCs [22, 25], limiting the modes of analysis available.This paper is organized chronologically. It tells a storyof experimentation spanning three years, where the un-expected results from a first QF study led to a total offour calibration techniques being used in an attempt tovalidate or refute those. The process involved a varietyof neutron sources selected to populate the sub-keV nr region sought. The net outcome is a strong case for asharply increasing ionization yield in germanium with de-creasing NR energy below ∼ nr , in clear departurefrom the Lindhard model [26] typically assumed for thismaterial. We conclude by briefly commenting on severalphysical processes able to lead to our observations and ontheir impact on upcoming searches for rare-events, whileencouraging further QF characterization work by others. a r X i v : . [ nu c l - e x ] F e b II. PHOTO-NEUTRON SOURCES: Y/BE
Profiting from the factor of five improvement in en-ergy threshold in the new LEGe compared to the largerPPC employed in [22], we revisited the technique imple-mented there, proposed in [27] and previously also usedfor silicon QF characterization [28]. Briefly reviewed, aphoto-neutron Y/Be source generates monochromatic152 keV neutrons from beryllium photo-disintegrationaccompanied by a much more intense high-energy gammaemission, which can nevertheless be blocked by 15-20cm of lead while causing only a minimal degradation ofneutron energies. Additional data taken with a Y/Alsource configuration isolate any remaining events from anotherwise unchanged gamma component: for Y emis-sions, the gamma stopping of Al and BeO are equiv-alent, while no photo-disintegration is possible for Al.The residual spectrum from the difference of both runscontains NR contributions only [27].For germanium, the maximum recoil energy producedby the elastic scattering of these neutrons is 8.5 keV nr .As in [22] a 5 mCi Y source [29] was evaporated intoa triple-sealed container, placed within a BeO ceramicgamma-to-neutron converter. Over the one month ofLEGe exposure to the source ( T / =106.6 d), its averageneutron yield was 848 n/s. This was measured indirectlyvia gamma spectroscopy followed by a calculation involv-ing a revised Be( γ, n ) Be cross section [30], and directlywith a He counter surrounded by moderator. In thiscase, both methods agreed within ∼ µ s shap-ing time. This results in an optimal noise performancefor detectors with low leakage current. The ionizationenergy scale and energy resolution were measured usingalpha-induced x-ray emission from a number of samples,benefiting from a 25 µ m beryllium entrance window tothe LEGe cryostat (Fig. 1, inset).An immediately evident, reproducible feature in all Y/Be runs is a “kink” in the spectrum at ∼ FIG. 1. Y/Be and Y/Al LEGe spectra for individualdaily runs (top) and cumulative (bottom). The grayed regionindicates the noise pedestal. The color scale allows to visualizethe decay of the source during data-taking. A solid line showsthe mean triggering efficiency in the top inset and a fit to theenergy resolution, expressed as in [32, 33], in the bottom inset. nent by the lead shield, as per comparison with back-ground in the absence of a source (Fig. 1).Simulations of neutron response were performed usingMCNP-PoliMi [34]. Attention was paid to including allinner detector and source components in full detail, aswell as the effect of impurities in the lead down to ppmlevel, measured via ICP-MS. The effect of energy resolu-tion was included in the simulations assuming the sameFano factor [35] applies to both ERs and NRs. These sim-ulations confirmed the impossibility to explain the low-energy rapid spectral rise in the Y/Be- Y/Al residualwhile embracing the best-fit Lindhard model presentedin [22], with or without the presence of the adiabatic fac-tor discussed in that publication. The top panel in Fig.2 shows a direct comparison of the experimental residualwith simulated predictions using this Lindhard QF andthe nominal average neutron yield of the source. Themagnitude of ENDF neutron cross-sections for lead andfor materials around the germanium crystal within itscryostat (aluminum and steel) was varied by up to ± FIG. 2. Y/Be- Y/Al residual containing NR contribu-tions only. Statistical errors include the effect of triggeringefficiency for the lowest-energy datapoint. Red lines are best-fit simulated models discussed in the text. The inset showsthe fractional QF from the model-independent fit vs. NR en-ergy. Neutron yields from the source are shown as a fraction Y of its nominal value. A gray band is delimited by twoLindhard fits to data in previous work (Fig. 6 in [22]). to the Y/Be- Y/Al residual in the 0.5-3 keV interval,with the maximum NR energy of 8.5 keV nr . This ini-tial match is possible only if multiple neutron scatteringwithin the detector is infrequent, as is the case (12% ofsimulated neutron histories for this source and geome-try). The running integrals of measured interaction ratevs. ionization energy and of simulated rate vs. NR energyare compared for all energies below their respective end-points: the dependence of QF on NR energy is inferredfrom their matching projection onto each other [28]. Theneutron yield Y of the source is left as a single free pa-rameter able to provide an optimal fit to the residual.This best-fit, capable of reproducing the low-energy ex-cess, is shown as a red line in the middle panel of Fig. 2.The corresponding fractional neutron yield of the sourceis Y = 0 . ± Y = 1) is rep-resentative of our past ability to measure it, as mentionedabove. Interestingly, Y ∼ .
85 also provides the bestmatch to the Lindhard model above ∼ nr up to the8.5 keV nr endpoint (Fig. 2 inset). The lowest NR energythat can be explored with this method is Y -dependent:however, for all values tested, a trend for a rapid QFincrease below ∼ nr is noticeable. Recent phenomenological work has focused on theMigdal effect [36] and its potential impact on rare eventsearches [37–42]. This phenomenon would account for aprompt emission of excess ionization (“electron shake-off”) following the sudden perturbation to the centralatomic potential caused by a NR. While this process hasnot yet been confirmed for NRs, it has been observedfor other atomic perturbations (e.g, following nuclear β ± decay [43–45]). For some detector materials, this excessionization would significantly increase their sensitivity tolow-mass dark matter and CE ν NS [37–42]. The bottompanel in Fig. 2 shows that our observations can in princi-ple be understood by invoking a toy model for this pro-cess, with free parameters fine-tuned for a good fit. Wereturn to this interesting possibility in Sec. VII.
III. PHOTO-NEUTRON SOURCES:
SB/BE
A similar procedure was followed using a
Sb/Besource. The gamma emitter ( T / =60.2 d) was obtainedby activation of a sample of high-purity antimony metalat the North Carolina State University reactor. The av-erage neutron yield during these runs was ∼ × n/s.Fig. 3 shows the response to the dominant 23 keV neutronemission from this source [28, 46], expected to produceNRs carrying a maximum of 1.3 keV nr . A few weeksprevious to these tests, an intentional neutron activationof the germanium detector was performed to obtain a1.3 keV (ionization) energy calibration peak from GeL-shell electron capture (EC, T / =11.4 d), part of thebottom inset in Fig. 1. A small contamination with theremaining activity under this peak is visible. FIG. 3. LEGe exposure to
Sb/Be and
Sb/Al. Errorbars in their residual (inset, see text) are one-sided for clarity.
A full analysis of these data was not attempted dueto the small usable energy range spanned by NR signalsabove the presently-achieved detector threshold and thedifficulties in determining a precise ionization endpointfor the NR distribution. Those derive from the presenceof a considerable (3%) branch of higher-energy 378 keVneutrons for this source [47] and the larger fraction (17%)of multiple scatters expected from simulations. For com-parison, the Y/Be source generates just 0.5% of neu-trons at a higher 963 keV energy [27, 46]. The insetin Fig. 3 shows the
Sb/Be-
Sb/Al residual, fittedby two exponentials representing both neutron branches.Vertical arrows indicate the position of the expected end-point (1.3 keV nr ) for single-scatter NRs from the domi-nant 23 keV branch, for the values of the QF indicated. IV. RECOILS FROM GAMMA EMISSIONFOLLOWING THERMAL NEUTRON CAPTURE
While the use of monochromatic low-energy neutronsfrom photo-neutron sources is a convenient way to pro-duce few-keV nr NRs in the laboratory, this method trustsneutron transport simulations to accurately predict theNR energies being generated. Seeking to remove this pos-sible source of uncertainty while testing the unexpectedresults in Sec. II, we revisited a technique first put for-ward in 1975 [48] able to produce fixed-energy germa-nium recoils carrying a mere 0.254 keV nr , with negligi-ble spread ( ∼ nr ). These are, to our knowledge, thelowest-energy NRs thus far characterized in any material. FIG. 4. Spectrum from a 2.9 hour LEGe exposure to theOSURR thermal beam at 1 kW reactor power, correspondingto ∼ × n/cm s. The inset expands the “ γ +NR” peakfrom Ge(n, γ ) Ge (see text). As expected from the highpurity of this beam, an excellent Gaussianity is observed.
This alternative approach exploits the thermal neutroncapture reaction Ge(n, γ ) Ge whenever it populates a6784.2 keV excited nuclear state. If the reaction takesplace within a small germanium detector, short-lived in-termediate decays to the lowest ∼ Ge excitedlevel will generate a cascade of high-energy gammas thatescape the crystal with high probability, while inducingthe net NR energy listed above. This step is dominatedby the emission of a single high-energy gamma [48]. Onthe other hand, the ∼ τ = 0 . µ sof this state both energy depositions are effectively si-multaneous. By separately measuring the energy of thisgamma alone (e.g., with an auxiliary silicon detector nextto an inert germanium target exposed to thermal neu-trons) it is possible to isolate the ionization energy of theNR and its corresponding QF [48].An ideal beam for this mode of calibration is avail-able at The Ohio State University Research Reactor (OS-URR). Single-crystal sapphire and polycrystalline bis-muth are used for fast neutron and gamma filtering ofcore radiations, respectively, resulting in a high-puritythermal neutron beam ∼ -10 n/cm s. Its cadmium ratiois 266, i.e., there are just 3.76 neutrons with energieshigher than 0.4 eV for every 1,000 thermals [49].The LEGe detector was placed in the path of thisbeam. A 7.5% lithium-polyethylene 1 cm collimator [50]was used to confine the beam to the region immediatelyaround the germanium crystal, reducing capture gammabackgrounds. As in [48], energy calibration peaks weregenerated continuously during beam irradiation using thedominant gamma emission from a Am source, and leadfluorescence x-rays from a 1 mm Pb disk placed blockingthe LEGe Be window, while holding a Co source (Fig.4). To avoid the introduction of any bias in the analy-sis, the same nominal energies as in [48] were assignedto these calibration peaks. All peaks in the vicinity ofthe sought signal ( Ge “ γ +NR”) are shown in Fig. 4.A skewed peak at ∼ µ s amplifier shaping time on a Ge de-excitationcascade involving the emission of a 53.4 keV gamma fol-lowed by 13.3 keV from a level with a half-life of 2.9 µ s.A peak at 60.916 keV [51] originates from activation ofan indium electric contact on the surface of the crystal.Fig. 5 compares the position of the sought γ +NR peakwith that previously obtained in [48]. The presentlyachieved uncertainty is considerably smaller than in theoriginal 1975 measurement. Our result is also incom-patible with it, pointing in the direction of a larger QF.Prompted by this observation, an attempt to indepen-dently measure the energy of the isolated gamma wasmade. During a second OSURR visit, a 0.5 g sample of >
96% isotopically-enriched Ge oxide [52] was exposedto the collimated beam, using an Amptek XR-100SDDsilicon x-ray detector in close proximity to collect thegamma emissions from this target. To avoid an exces-sive activation of the detector itself by neutrons scatteredfrom the irradiated sample, a 3 mm-thick blanket of LiFpowder (95% isotopically-enriched in Li [53]) compactedto a density of 1.1 g/cm was inserted between both. Thepowder was held in place by aluminum foils glued to asupporting acrylic ring. This blanket is ∼
95% transpar-ent to 68.75 keV gammas, while it reduces thermal neu-tron transmission by close to four orders of magnitude.A gamma-less neutron capture via Li(n,t) He avoids abackground penalty. Unfortunately, even with this pre-caution, several peaks traceable to activation of tungstenin a multilayer collimator internal to this silicon detectorencumbered the sought signal.
FIG. 5. Comparison of Ge(n, γ ) Ge measurements con-taining simultaneous gamma and NR energy depositions, withthose involving the gamma energy alone, as a function of am-plifier shaping time. Their difference corresponds to the ion-ization energy deposited by 0.254 keV nr NRs in germanium[48]. The same data acquisition system (amplifier, multi-channel analyzer) was used for all present measurements.
An alternative route to this ancillary measurement, fi-nally successful, is depicted in Fig. 6. Thermal neutronsfrom moderated
Cf activate a germanium metal sam-ple consisting of two wafers adding up to 1 mm thickness,placed in proximity to the LEGe detector Be window.The thermal flux reaching the LEGe crystal is abated bythe same LiF blanket as above, in addition to a cadmiummetal sheath surrounding the cryostat. The residual be-tween spectra obtained with and without the presence ofthe germanium wafers, shown in Fig. 7, reveals a peak atthe position of the “ γ alone” emissions from this sample,with expected width (341 ±
49 eV FWHM, compare withFig. 4). The energy scale was continuously defined asabove, assisted by
Am and Pb fluorescence peaks. Our“ γ alone” peak position is seen to be compatible withinerrors with the original 1975 measurement (Fig. 5). Thepresent uncertainty was limited by the weak intensity (4 × n/s) of the available Cf source. A discussioncontained in [48] points at an even better agreement withour peak position when their Ge( α , α ’ γ ) Ge “ γ alone”datapoints [54], identified in [48] as outliers, are dropped. FIG. 6. Geometry employed for the “ γ alone” measure-ment described in the text: 1) 20.3 cm-diameter polyethylenesphere housing a Cf source at its center, 2) Pb disc (fis-sion gamma shield, fluorescence source), 3) cylindrical acrylicholders, 4) Ge wafers, 5) LiF blanket, 6) polycarbonate ring,7) Al crystal holder, 8) PTFE field-effect transistor holder,9) layered Cd sheath, 10) acrylic ring, 11) Be window, 12)Ge crystal, 13) Al cold finger, 14) stainless steel cryostat. A
Am source was positioned near the crystal, outside the Cd.FIG. 7. Normalized spectra obtained using the arrangementof Fig. 6, in presence and absence of Ge sample (wafers).Their residual and Gaussian fit are shown in the inset. The“shark tooth” structure above ∼ LiF blanket [55, 56]. Neighbouring
Am and Pb x-rayenergy calibration peaks fall outside the range of this figure.
We considered the possibility that the ∼ µ s shapingtime used in [48] (based on the 4 µ s peaking time quoted)might have been insufficient to account for the τ = 0 . µ lifetime of the 68.75 keV Ge excited level, shifting theposition of their γ +NR peak to a lower value. This hy-pothesis was tested by removing Ge wafers, LiF blanket,and cadmium sheath from the configuration in Fig. 6, al-lowing the LEGe to capture thermalized
Cf neutronsunimpeded, while varying the shaping time of our am-plifier. As can be seen in Fig. 5 (“moderated
Cf”),no significant dependence on shaping time can be con-cluded. However, the mean of these γ +NR measurements(68.808 ± ± ± γ +NR and energy cali-bration peaks in OSURR data, compared to those in [48],should be emphasized at this point.All in all, our present revival of the thermal capturetechnique first described in [48] points at 77 ±
20 eV ofionization being produced by a 0.254 keV nr germaniumNR, corresponding to a 30.3 ± ± Y/Be model-independent best-fit ( Y =0.86) to lower recoil energies. V. MONOCHROMATIC 24 KEVIRON-FILTERED NEUTRONS
In view of the apparent consistency of our thermal neu-tron capture and photo-neutron results, we embarkedon a final characterization effort able to provide ad-ditional QF values at discrete recoil energies below 1keV nr . To that effect, the LEGe was exposed to a highly-monochromatic 24 keV ( ± ∼ nr [9], however using a large de-tector ( ×
90 the LEGe volume) prone to multiple scatter-ing, with ∼ ∼ FIG. 8. Normalized residual spectra for all KSU iron-filterruns, labelled by neutron scattering angle φ , nominal NR en-ergy, and Ti-filter on/off status. Signals produced by NRsfrom 24 keV neutron scattering are visible only in Ti-filterabsence. Red histograms show a simulated response, for thebest-fit QF listed. A small increase in signal rate with de-creasing φ is expected from scattering kinematics. Error barsin the insets are encumbered by data point size. taminations, by inserting an additional thin (1.25 cm)titanium post-filter [57]. This exploits a resonance in theTi cross-section precisely at ∼
24 keV neutron energy. Acomparison of Ti filter on/off runs can provide convinc-ing evidence that low-energy signals assigned to NRs doindeed originate from the scattering of 24 keV neutrons.Data were acquired using the system in [22], able todigitize LEGe preamplifier and backing detector signalsat 120 MS/s. A trigger was provided by a single-channelanalyzer centered around the neutron capture peak of alarge (1.5 cm × LiI[Eu] scintillator, used to detect neutrons scattered offthe LEGe. The virtues of this choice of backing detectorfor this application are discussed in [57]. The scintillatorwas mounted on a support arm able to pivot around theLEGe cryostat, 15 cm away from the germanium crys-tal. A goniometric table installed on the LEGe Dewarallowed the selection of scattering angles and their cor-responding NR energy. A best effort was made to cen-ter the small germanium crystal in the Gaussian-profile( ∼ ∼ Fe source, for a total ofsix reference peaks in the 4-7 keV energy range (Fig. 8inset). No measurable energy drift was observed over thetwo days of data-taking: all twelve calibration points areused for the linear fit in the figure.During data analysis, events in coincidence and in anti-coincidence with the neutron capture signal from thebacking detector were inspected using an edge-finding al-gorithm, illustrated using this same LEGe detector in[19]. It is able to identify the rising edge character-istic of radiation-induced pulses in preamplifier traces,while rejecting low-energy noise nuisances. The signalacceptance (SA) of this algorithm was measured (Fig.8 inset) using programmable electronic pulser signals ofsame rise-time as calibration x-rays. The energy of eventspassing this data-quality cut was determined using a dig-ital implementation of a 36 µ s zero-area cusp filter [58–60]. The residual difference of energy spectra from co-incident and anti-coincident events, normalized to sameexposure, is expected to be dominated by the NR signalssought and free of significant low-energy noise by virtueof the subtraction. These residuals, corrected for SA, areshown in Fig. 8. Error bars are statistical: by makingthe anti-coincidence time window a factor of fifty longer than that for coincidence, the uncertainty on steady-statelow-energy backgrounds is greatly reduced. As expected,clear low-energy excesses are observed only in the absenceof the Ti-filter, confirming their origin in the elastic scat-tering of 24 keV neutrons.The distribution of MCNP-PoliMi simulated energydepositions by NRs in the germanium crystal also pro-ducing a capture in the backing detector was convertedinto an ionization equivalent using QF values in the range15%-40%, including the effect of energy resolution andmultiple scattering (4.1% to 7.6% of all events), for eachof the four scattering angles tested. The resulting spectrawere compared to the Ti-off residuals of Fig. 8, obtainingbest-fit QF values via log-likelihood analysis. The QFerrors shown in Figs. 8 and 9 combine the uncertaintyextracted from this procedure with that from the energycalibration. The energy spread (HWHM) of simulatedevents is utilized as horizontal error bars in Fig. 9. FIG. 9. Present QF results, labelled by calibration tech-nique. A red band shows the 95% C.L. region for the model-independent fit of Fig. 2. A dotted line is the Lindhard modelwith a default germanium value of k =0.157 [22]. Previousmeasurements are shown in gray: circles [54], squares [9, 25],diamonds [61], triangles [62], and inverted triangle [48]. VI. SYSTEMATICS AND COMPATIBILITY
In this section we elaborate on possible systematic ef-fects able to have a moderate impact on our measure-ments, as well as on the compatibility of these measure-ments with each other and with previous work in [22].The normalization of the two datasets shown in Fig. 7is based on a matching of their backgrounds in the regionsabove and below the peak of interest. The same normal-ization factor of 1.51 was found to apply to both regions.An alternative method of normalization based strictlyon the difference in exposure between the datasets woulduse a factor of 1.57. This modest difference is due to thesmall yet finite shielding of backgrounds that the thingermanium sample produces. If this alternative normal-ization is employed, the “ γ alone” peak position is mini-mally shifted by 11 eV to a lower energy. The net resultis an increase in the QF derived from the thermal cap-ture method to 34.6 ± ±
11 eV due to the excellentlinearity observed (Pearson’s R=0.99998). Use of lower-energy calibration data points (e.g., via alpha irradiationof PVC, producing Cl x-rays as in Fig. 1) was not pos-sible due to the longer exposures required for those andthe limited beam time available at KSU. The quality ofthis fit is not in doubt, as non-zero independent termsof this magnitude are commonplace and traceable to thealgorithms used for energy determination. It is howeverworth mentioning that making this independent termequal to zero would bring the iron-filter data points inFig. 9 to near-perfect agreement with the photo-neutronbest fit shown there and its extrapolation to low energy.Nevertheless, as mentioned in Sec. V, the effect of theknown uncertainty in the energy scale is already includedin the vertical error bars for iron-filter data.The photo-neutron QF should be considered an ap-proximation, as its model-independent method is pred-icated on a total absence of multiple scatters. Still, amore complex analysis leaving both source yield Y andionization endpoint in the Y/Be- Y/Al residual as freeparameters might be able to produce an improved fit tothe low-energy excess in Fig. 2, bringing the derived QFcloser to iron-filter results. In lieu of this analysis, weassess the agreement between both techniques by assum-ing a QF model consisting of a no-frills linear fit to theiron-filter QF datapoints in Fig. 9 for energies below 1.35keV nr . At this energy this fit intersects the standard k =0.157 Lindhard line in the same figure. For higher en-ergies the QF model switches to Lindhard. When thissimple model is applied to the interpretation of photo-neutron data, a fair quantitative and qualitative agree-ment is obtained (Fig. 10). This test illustrates the in-ternal consistency of the ensemble of our measurements.Finally, we have examined the compatibility of the newmeasurements presented here with our previous photo-neutron dataset in [22]. As mentioned in Sec. I, the detec-tor used for that study had a threshold five times largerthan presently achieved. This derived from a combina-tion of higher intrinsic electronic noise and an issue withthe internal gain of the digitizer employed, later resolved.As a result, signals from NRs below ∼ nr could onlybe detected as part of events involving multiple neutronscattering, dominant for that large crystal. In addition FIG. 10. Comparison of the QF model described in thetext, based on iron-filter QF measurements and Lindhard the-ory, with the Y/Be- Y/Al residual of Sec. II. The adoptedsource yield is Y = 0 .
95. No other free parameters are used. to this, two free parameters had to be included in theanalysis to account for the thickness of dead and tran-sition surface layers [23] in that p-type diode. Addedto the yield of the source and the two free parametersallowed for the Lindhard model (the mentioned k andone to account for a possible adiabatic factor [63]), thisresulted in a total of five free parameters being used tomatch simulation to data.Two QF models based on present experimentation,both devoid of free parameters, have been tested againstthe Y/Be- Y/Al residual from [22]. As in that pub-lication, we used a popular Markov Chain Monte Carlo(MCMC) ensemble sampler [64, 65] to explore the param-eter space available to the fits. The first model is thatdescribed in the discussion above regarding Fig. 10. Thesecond corresponds to the photo-neutron best-fit line inFig. 9, linearly extrapolated to lower energy. Three freeparameters were adopted, the yield of the source Y , andthe independent thicknesses of dead and transition layers.As a first cross-check, we reproduced the results in [22]for the Lindhard model, finding similar best-fit values forall five free parameters. However, when using the new QFmodels, the quality of their fits to the residual is compa-rable to that using Lindhard. What is more, both presentQF models result in a combined depth for surface layers ∼
40% shallower than in [22]. We consider this to be morereasonable, as the values obtained in [22] would have re-sulted in a sizable degradation of the energy resolution,not observed. The new models also favor Y ∼ Y =1.37 [22]. We conclude thatwhile the dataset in [22] was sufficient to constrain freeparameters in a model (Lindhard) embraced as an articleof faith, it is inadequate to exclude QF deviations takingplace at energies well-below detector threshold. This is incontrast with the model discrimination possible with thecurrent detector and dataset, illustrated in Fig. 2, whileusing an economy of free parameters (one). In all, weconsider that the present effort supersedes that in [22]. VII. COMMENTARY AND CONCLUSIONS
Our measurements strongly suggest that a new phys-ical process (or processes), absent from Lindhard’s clas-sical treatment of ion slowdown, dominates the produc-tion of ionization by NRs below ∼ nr in a ger-manium semiconductor drastically enhancing the low-energy quenching factor of this material. Unexpected asthis may seem, a similar behavior has been observed be-fore for the production of light by sub-keV proton recoilsin organic scintillators [63, 66]. This property was re-cently confirmed [67]. In this section we briefly commenton possible origins for our observations, and on their im-plications for upcoming rare-event searches.A plausible explanation for the observed behavior isthe Migdal effect already invoked in Sec. II. The toymodel adopted for this process, shown in Fig. 2, assumesthat Migdal-style electron shakeoff takes place for a sig-nificant fraction of NR episodes. A shakeoff probability P = 50% is a first free parameter fine-tuned to obtain theagreement with data shown in the figure. A second freeparameter (cid:15) = 0.35 keV describes the kinetic energy E ofthe single electron ejected, distributed as ∝ e − E/(cid:15) up toa maximum constrained by the magnitude of the atomicperturbation (simulated initial NR energy) and conser-vation of energy. This functional form used to sample E is a very crude approximation to formal Migdal differen-tial ionization probabilities in [37]. Following this samepublication, the ejected electron is assumed to be pref-erentially originating from the germanium M-shell. Theenergy invested in breaching an electron binding energyof 35 eV is taken to be returned as ionization, followingatomic orbital relaxation via radiative or Auger emis-sions. The remaining energy up to that of the simulatedinitial NR is dispersed assuming that a Lindhard QF asin [22] applies to the slowing-down of the recoiling ion.NRs not producing shakeoff are Lindhard-governed.The chosen values of P and (cid:15) can be further adjustedto obtain similar fits to photo-neutron data. The nom-inal Y = 1 neutron yield from the source was adoptedfor the fit shown in Fig. 2. A value of P = 50% mayseem arbitrarily large, being a factor of approximatelyseven above the integrated ionization probabilities calcu-lated for Migdal shakeoff from atomic germanium in [37].However, recent work [40] indicates that this probabilityis significantly enhanced for the present case of a germa-nium semiconductor. It is nevertheless hard to quantifythe magnitude of this probability increase using the in-formation in [40], specific to low-mass WIMPs. Presentdata offer a benchmark against which phenomenologicalMigdal predictions can be contrasted, possibly confirm-ing the presence of this process.Alternative paths leading to an enhanced ionizationyield from low-energy NRs in semiconductors have beenrecently put forward in [68–70]. All these interpretationsrapidly complicate when the secondary NRs abundantlyproduced in the wake of a low-energy primary NR are folded in. As a reference, a 1 keV nr germanium recoilproduces on average 43 displaced secondaries, each car-rying just a few tens of eV nr , adding up to 92% of the pri-mary energy, with essentially every atom within a (25˚A) lattice voxel containing the full trajectory of the primarybeing perturbed [71]. From this perspective, focus on theprimary as in our Migdal toy model should be replacedby an understanding of how a densely-packed cloud ofsecondaries might collectively or individually contributeto a higher ionization yield. Examined from this point ofview, differences between crystals in their spatial concen-tration of lattice excitation at the NR site may becomevery relevant: we observed a reduction in QF with respectto Lindhard for a lighter, lower-density semiconductor,silicon, at least down to 0.7 keV nr [28]. A possible con-trast between germanium and silicon in their response tosub-keV NRs is an incipient area of study [72].The importance of developing material-specific modelsof response to sub-keV NRs, solidly anchored on experi-mental characterization data, cannot be overemphasized.If our observations are confirmed, ionization-sensitivegermanium detectors with thresholds below ∼
200 eV[14, 15, 73] should enjoy a sizeable improvement in theirsensitivity to low-mass WIMPs and to CE ν NS signalsfrom reactor antineutrinos (for a 300 eV threshold [8] ourobservations have a comparatively minor impact). Whilethis may be advantageous in the second context whensuch a detector is simply used as a neutrino counter forreactor monitoring [74, 75], an imperfect understandingof the sub-keV QF would severely hamper the opportu-nities for probing new physics that CE ν NS otherwise af-fords [6]. Aware of both promise and perils, we concludeby encouraging further phenomenological predictions inthis low-energy frontier, and welcoming innovative exper-imental techniques capable of their verification [76].
ACKNOWLEDGMENTS
We are indebted to Dan Baxter, Yoni Kahn and Gor-dan Krnjaic for calling our attention to the Migdal effectas a possible origin for our observations and for manyuseful conversations on the subject. Similarly, to AlvaroChavarria for proposing the use of the model-independentmethod for Y/Be analysis, and to Simon Knapen andTongyan Lin for helpful input. Our gratitude also goes toJim Colaresi and Mike Yocum at Canberra for detector-related consultations. Three experimental reactors wereinvolved in this work: we thank Lei Cao, Andrew Kauff-man, and Susan White at Ohio State University, ScottLassell at North Carolina State University, and Alan Ce-bula at Kansas State University for their generous sup-port of our operations at their facilities. This work wasfunded by NSF awards PHY-1806722 and PHY-1812702,and by the Kavli Institute for Cosmological Physics atthe University of Chicago through an endowment fromthe Kavli Foundation and its founder Fred Kavli.0 [1] D. Z. Freedman, Phys. Rev. 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