aa r X i v : . [ nu c l - e x ] J u l A New Neutron Lifetime Experiment with ColdNeutron Beam Decay in Superﬂuid Helium-4
E-mail: [email protected]
Kellogg Radiation Laboratory, California Institute of Technology, Pasadena,California 91125, USAMay 2020
The puzzle remains in the large discrepancy between neutron lifetimemeasured by the two distinct experimental approaches – counts of beta decays in aneutron beam and storage of ultracold neutrons in a potential trap, namely, the beammethod versus the bottle method. In this paper, we propose a new experiment tomeasure the neutron lifetime in a cold neutron beam with a sensitivity goal of 0.1%or sub-1 second. The neutron beta decays will be counted in a superﬂuid helium-4scintillation detector at 0.5 K, and the neutron ﬂux will be simultaneously monitoredby the helium-3 captures in the same volume. The cold neutron beam must be ofwavelength λ >
Keywords : neutron lifetime, neutron lifetime discrepancy, beam neutron lifetime,neutron beta decay spectrum, superﬂuid helium-4 scintillator
J. Phys. G: Nucl. Phys.
A precise measurement on the neutron lifetime is important to many fundamentalquestions in particle physics, astrophysics and cosmology, such as CKM unitarity andprimordial helium abundance in Big Bang Nucleosynthesis (BBN).[1, 2, 3] So far, itsvalues obtained from the two distinct methods signiﬁcantly diﬀer from each other[4, 5, 6],possibly due to unaccounted systematic eﬀects in either or both of the methods; yetotherwise it implies new physics[7, 8, 9, 10, 11, 12, 13, 14], many theories of whichremain controversial[15, 16, 17]. On one side, the measurement is done in a neutronbeam by counting the number of neutrons undergoing beta decay when the neutron ﬂuxpasses through a deﬁned volume. It is thus called the beam method. The weightedaverage of the recent two beam lifetime measurements with a proton quasi-Penning
New Beam Neutron Lifetime Experiment in Superﬂuid Helium-4 τ n = 888 . ± . τ n = 879 . ± . . ± . . σ ). Many further experimental eﬀorts are on the way to addressthe discrepancy. While existing experiments are upgrading to improve their statisticsand searching for hidden systematic eﬀects, new experimental strategies with a distinctset of systematic eﬀects are being proposed and carried out. For instance, researchersin J-PARC started a new measurement in the pulsed CN beam by characterizationof the electron recoils in the beta decay events and the helium-3 capture events ina Time Projection Chamber (TPC) ﬁlled with gaseous mixture of helium and carbondioxide.[27, 28] It is a revival of the beam experiment originally proposed by Kossakowski et al. in 1989 . Researchers at Los Alamos National Laboratory are prototyping abeam/bottle hybrid experiment, named UCNProBe, to measure the number of decaysand helium-3 captures via detection of scintillation in a UCN storage box.In this paper, we propose a new experimental method with a diﬀerent combinationof existing technologies, in order to resolve the neutron lifetime enigma. Theproposed experiment is essentially a beam lifetime measurement. It counts the decayproduct – electrons, rather than protons, via detection of electron recoil scintillation insuperﬂuid helium-4 at 0.5 K. In order to eliminate neutron scattering with superﬂuidhelium, the CN beam must be of wavelength λ > .
2. Experimental Method
Suppose the decay volume is cylindrical with a length L of 75 cm and a diameter D of7.5 cm, i.e. the length to diameter ratio is L/D = 10. The CN beam of 3 cm diameterpasses the decay volume along the axis of the cylinder. The detectable neutron decayrate in the volume is given as˙ N β = τ − β ǫ β L Z A b da Z v dv I ( v, ⇀ r ) 1 v (1)where τ β is the neutron lifetime, ǫ β is the detection eﬃciency of the beta decay in New Beam Neutron Lifetime Experiment in Superﬂuid Helium-4 A b is the cross sectional area of the beam, and I ( v, ⇀ r ) is cold neutronﬂuence rate with respect to the neutron velocity v and cross sectional distribution ofpositions ⇀ r .When the CN beam passes through the decay volume, the He nuclei in superﬂuidhelium-4 capture neutrons via nuclear reaction n + He → p + t +764 keV. The detectablecapture rate is given as˙ N p + t = ǫ He σ th He v thn n He L Z A b da Z v dv I ( v, ⇀ r ) 1 v , (2)where ǫ He is the detection eﬃciency of the capture events, σ th He is the absorptioncross section of He nuclei for thermal neutrons at a velocity v thn = 2200 m s − , and n He is the He density. The neutron lifetime τ β can be obtained from the ratio of theobserved neutron He capture rate to the beta decay rate. τ β = ˙ N p + t ˙ N β · ǫ β ǫ He · σ th He v thn n He (3)Eqn. (3) is the key expression in this experiment. It explicitly shows themeasurement of τ β is independent of the neutron ﬂux as well as the geometry ofthe decay volume. The overall accuracy relies on that of the observed ratio ofevent rates κ = ˙ N p + t / ˙ N β , the helium-3 density n He in superﬂuid helium-4, and thedetection eﬃciency of scintillation events ǫ He and ǫ β . The former two quantities willbe experimentally acquired, and the detection eﬃciencies will be determined throughsimulations considering the calibration and background discrimination.Here, we provide an estimate of count rates based on the published performance ofthe Fundamental Neutron Physics Beam Line (FnPB) in the Spallation Neutron Source(SNS) at Oak Ridge National Laboratory (ORNL), as shown in Fig. 1 . The neutronﬂux at 17 ˚A is about 2 . × Hz ˚A − cm − MW − . With a time-averaged proton powerof 1.8MW at 60Hz of double-chopper, the incident rate of 17 ˚A neutrons with 0.5 ˚A pulsewidth is about 1 . × Hz. In this estimate, neutron lifetime is taken as the PDGsuggested value τ β = 880 s . It takes about 3.2 ms for the 17 ˚A neutrons to pass75 cm long decay volume, and the neutron decay probability is 3 . × − for a CNbeam with a cross section of 3 cm diameter. There are an average of 55.9 Hz of neutrondecay events. The natural abundance of He in liquid helium is X He = 5 × − infractional concentration. Near isotopically pure He with X He < . × − has beenproduced as reported by Hendry and McClintock. Assuming superﬂuid helium-4with X He = 2 × − can be prepared, an average of 252.1 Hz neutron capture eventswill occur simultaneously when the CN beam passes the decay volume.
3. Scintillation Signals in Liquid Helium
The number of beta decays and neutron captures will be counted via scintillation signalsin liquid helium. Liquid helium is an ideal scintillator that has been proposed and
New Beam Neutron Lifetime Experiment in Superﬂuid Helium-4 > 16.5 Å Wavelength ( Å ) N eu t r on F l u x ( H z Å - c m - M W - ) x10 Figure 1.
Spectrum of cold neutron beam at the SNS FnPB beamline with choppers(a reprint of Figure 5 in Fomin 2015 ). The portion of wavelength λ > . CN Beam Superﬂuid Helium at 0.5KeTPB ﬁlmLead/Tungsten ShieldLead/Tungsten ShieldLithium-6 Enriched Neutron AbsorberLithium-6 Enriched Neutron AbsorberPTFE Reﬂector and HolderDECAY VOLUME WLS ﬁbersDiamondWindow DiamondWindow
Schematic of the conceptual detector (non-scaled)
New Beam Neutron Lifetime Experiment in Superﬂuid Helium-4 ∗ molecules of excited singlet state He ( A Σ + u ) in liquid helium.The singlets radiatively decay in less than 10 ns and emit about 22 extreme ultraviolet(EUV) photons per keV of electron recoil energy KE e , with a spectrum spanning from13 eV to 20 eV and centering at 16 eV. It forms the prompt pulse of scintillationlight. There are about 1 . × photons per decay event at the end point energy of782 keV in the neutron beta decay spectrum. On the other hand, the neutron capture ispurely a nuclear recoil event, the scintillation process of which is similar but of diﬀerentfeatures. About 13% of the recoil energy of 764 keV converts into a prompt light pulseand results in about 6 . × photons per capture event. The stopping power dE/dx for a recoiling nucleus in liquid helium of a density ρ = 0.145 g cm − is 2 × eV µ m − .The typical stopping range for a 800 keV recoiling nucleus is 40 µ m. By contrast, thestopping power for a 800 keV recoiling electron is only 40 eV µ m −  on average, andits stopping range can reach up to 2 cm. Therefore, a diameter of 7.5 cm is suﬃcientto prevent almost all of the recoiling electron born in the 3 cm diameter CN beam fromtouching the inner surface of the decay volume. It nearly guarantees no quenching of theprompt scintillation on the wall. Because of the dramatic diﬀerence in track length, thescintillation light of decay events is much more dispersed spatially than that of captureevents. The former appears as a line of chained point sources, whereas the latter as asingle point source.In addition, both electron and nuclear recoils also generate a large amount of tripletHe ∗ excimers ( a Σ + u ), which has a 13 s lifetime in liquid helium. The radiative decayof the triplet excimers is forbidden as it requires a spin ﬂip; yet it can occur via thebimolecular Penning ionization that converts a portion of the triplet into singlet, mostlikely in a high density of triplet excimers along the recoil track. This type of scintillationlight appears as a large number of after-pulses of EUV photons, following the promptpulse, and temporally scattered over tens of micro-seconds. Each of them is muchweaker than the prompt pulse, and therefore mostly registered as pulses of single or afew photo-electrons in the same detector. The occurrence rate of after-pulses decreasesas to a combination of two components dependent exponentially and inversely on time,respectively, g ( t ) = Ae − t/τ s + B/t + C . It has been experimentally demonstrated thatthe 1 /t component of the electron recoils is much weaker than that of the nuclearrecoils.[42, 43] This feature oﬀers an important tool to distinguish the decay eventsfrom the capture events. New Beam Neutron Lifetime Experiment in Superﬂuid Helium-4
4. Detection of Scintillation
A standard method has been well developed to detect the EUV scintillation in liquidhelium by many experiments.[39, 40, 42, 43, 44, 45] Based on the known technologies,we describe a conceptual design as a baseline for a quantitative analysis. A schematicof the detector is shown in Fig. 2. The EUV scintillation light is ﬁrst converted into ablue spectrum near 400 nm by an organic ﬂuor – tetraphenyl butadiene (TBP). A thinlayer of evaporated TPB (eTPB) can be coated on an acrylic ﬁlm and wrapped into acylinder as the boundary of the decay volume. The eTPB coating faces the inside ofthe decay volume. Optical ﬁbers can be molded with a structural support as if woundon the outside of the ﬁlm cylinder to collect light. The ﬁbers cover the full length ofthe decay volume so as to maximize the light collection. Wavelength shifting (WLS)ﬁbers are a common option to convert the emitted blue light into a green spectrum near500 nm along with a redistribution of photon phase space. A portion of the shiftedlight can be trapped inside the ﬁber by total internal reﬂection and transmitted to thephoton sensors.The overall light conversion eﬃciency is estimated as follows. Owing to the large
L/D ratio of the decay volume, more than 96% in solid angle of the scintillation lightcan be converted by TPB for the events occurring in the central region, as shown inFig. 3. The conversion eﬃciency of eTPB has been demonstrated to be greater thanunity. Since a thick eTPB coating often appear opaque for visible light due to itssurface roughness, the blue photons heading inwards the decay volume might reﬂect anddiﬀuse on the eTPB coating. It is thus diﬃcult to characterize the distribution of theseinwards-going photons that are collected by WLS ﬁbers upon multiple scattering in theeTPB coating. As a moderate estimate, we only take into account the 50% of eTPB re-emitted blue photons that travel outwards to the adjacent WLS ﬁbers. Approximately90% of them can impinge on the ﬁber cores with the help of a Polytetraﬂuoroethylene(PTFE) reﬂector, which is also a structural holder clamped on the outside, and thenabout 80% is absorbed and shifted into green light. The double cladding WLS ﬁbersmade by Kuraray have a trapping eﬃciency of 5.4% in one direction. When read onboth ends, 10.8% of the shifted green light can be conveyed towards the 2 photon sensors.Since the ﬁber has a bending loss of about 4% per turn on a 7.5 cm diameter curve andan attenuation length longer than 7.5 m, the ﬁber length must be constrained. In eachdetector unit, a round WLS ﬁber of 1 m long and 1 mm diameter is helically woundaround the decay volume by 3 turns, and the extra length on each free end is routedto a separate photon sensor. It needs 250 units in a tight packing to cover the wholelength of the decay volume and set up an axial resolution. The average transmissioneﬃciency of such a conﬁguration is about 90% along the ﬁber. With regard to thediﬃculty of making large amount of superﬂuid-leak-tight ﬁber feedthroughs, there mustbe two optical breaks at the windows of the liquid helium vessel, each of which has a 90%transmission. As for the 500 photon sensors, we may employ silicon photomultipliers,which are compact in size and have a typical quantum eﬃciency of 34% for the versions
New Beam Neutron Lifetime Experiment in Superﬂuid Helium-4 η tot is 0.9%, i.e. an averageof 9 photo-electrons ( PE ) can be detected per 1 × EUV scintillation photons. Theaverage prompt PE numbers for decay events with the spectrum peak energy at 245 keVand the endpoint energy at 782 keV are N ( peak ) PE = 49 . N ( endpt ) PE = 158 .
9, respectively.Every PE corresponds to about 5 keV of electron recoil energy. On the other hand, theaverage prompt PE number for the neutron capture events of recoil energy at 764 keVis N ( p + t ) PE = 57 .
7. It coincides with beta events of 283.8 keV, close to the peak of thebeta spectrum.
5. Detector Response and Event Reconstruction
We perform a preliminary study on the response of detectors by Monte Carlosimulations. As listed below, several assumptions have been adopted to simplify themodel but present the essential physics as a proof of principle. Further modelling withmore details is needed.(i). Only the prompt scintillation signals are recorded for all the events. It meansthe decay and capture events cannot be distinguished among the simulated data.Yet in real experiment, they are distinguishable by the diﬀerence in the 1 /t time-dependent occurrence rates of the after-pulses. This additional information willimprove data analysis and understanding of systematic eﬀects(ii). The scintillation light for the capture events is emitted from a point source as theirtrack length is tens of microns, whereas that for the beta events is from an energy-dependent straight tracks of length up to 2 cm. For electron recoils, more energydeposits in the vicinity of the track end as it slows down. The spatial energydeposition approximately follows dE/dr ∝ r , where r is the geometric distancefrom the starting point of electron recoil. Further studies can be performed onsimulated scattering tracks with productions of secondary δ -electrons.(iii). Only the outward-going portion of the eTPB converted light can be collected bythe adjacent ﬁbers, but none of the inward-going, as the latter reﬂects and diﬀuseson the coating into a broader distribution over all the detectors, yet much weakerin intensity than the former.(iv). The EUV light converted by eTPB will be collected by the ﬁbers tightly woundagainst the thin ﬁlm at the same axial position. It means the solid angle of lightfrom an event projecting on the section of the eTPB ﬁlm is equivalent to that onthe detector lying against the ﬁlm.(v). All the detectors have the same eﬃciency. In reality, the eﬃciency of detectors arediﬀerent and may vary with time. Calibrations are necessary and will be discussedin Subsection 5.3.(vi). There is no timing information in this simulation. We assume all the eventsare in the coincident time window as the cold neutron beam passes the decay New Beam Neutron Lifetime Experiment in Superﬂuid Helium-4 ⇀ x of events as to a uniform distribution function, P r ( ⇀ x ) = const. , is carried out in the beam-occupied volume. ‡ × random events aregenerated and assigned as either capture or decay according to a preset ratio, κ = 4 . ⇀ x , P r ( n det | ⇀ x ), where n det ∈ [1 , − .
65 cm, and sum ofthat over all the detectors for each event at various axial positions z , respectively. Theevents ending in the beam path are highlighted in blue. It shows the total acceptedsolid angle has a weak dependence on the radial ending position, which correlates withthe recoil energy of electrons. With all the detectors functioning, it can cover more than96% of solid angle for events in a central region spanning 42 cm in the axial direction,as shown by the upper plot in Fig. 3. The events lying within the axial edges of aunit detector have about 4% of chance to be registered by this detector, as shown inthe lower plot in Fig. 3. The overall detector hit probability P r ( n det ) is derived by theintegral over the entire volume V , P r ( n det ) = R V d ⇀ x P r ( n det | ⇀ x ) P r ( ⇀ x ), and plotted inthe Fig. 4. For detectors in the central region, the overall hit probability is about 0.40%for scintillation light of an event at any position to be registered; while for those closeto the ends, the chance is naturally much less. With the registered PE numbers N PE ( n det ) from an event on a series of detectors n det , n det V ( z k ) centered at z k is derived in Eqn. (4) by the Bayes’ theorem. It is on theassumption that every detector is independent, i.e. no cross talk. The probability ofan event within a certain region of interest, e.g. the 42 cm long central region, can becalculated as the accumulated probability, P z k P r (∆ V ( z k ) | N PE ( n det )). Fig. 5a showsan example of the simulated electron recoil event with 77 observed PE s distributedon several detectors. The entire volume V is divided into sub-volumes ∆ V ( z k ) asdisks of 1 cm thick, and the distribution probability of event position for each sub-volume is calculated and plotted in Fig. 5b. The accumulated probability for thisevent to occur within the 42 cm long central region is 93.7%. In general, the more PEs observed, the more accurate the reconstructed position of the event. However, sincethe events near the ends of the decay volume lose a signiﬁcant portion of scintillationlight on the end windows, i.e. information is truncated, the reconstructed positions arebiased towards the center. Therefore, the accumulated probability of positions inside ‡ The cross sectional distribution of the beam ﬂuent rate I ( v, ⇀ r ) is independent of ⇀ r in this simulation. New Beam Neutron Lifetime Experiment in Superﬂuid Helium-4 -40 -30 -20 -10 0 10 20 30 40 Event Axial Position z (cm) R a t i o i n S o li d A ng l e beam +1cm< r < R in R beam < r < R beam +1cmin r < R beam Events ending one detectorall detectors
The coverage of solid angle for events ending at diﬀerent axial positions inthe decay volume. The upper plot is the total coverage of solid angle by the sum of allthe detectors; and the lower plot is that by one unit detector centered at − .
65 cm.Events ending in the beam pass (r Rbeam) are marked with blue, within 1cm awayfrom the beam pass (Rbeam < r Rbeam+1 cm) marked with light grey, and outsidethe regions above (Rbeam+1 cm < r R0) marked with dark grey, where R0 is themaximum radius that a recoiling electron can reach. the central region of 42 cm generally performs better in identifying events thereof thanthat of reconstructed positions, especially for low PE events. Yet neither is satisfactoryin selecting events in a region with relatively identical position distribution and uniformratio of capture-to-decay event rates. Later, we ﬁnd a combination of both actuallyforms a good ﬁducial cut. It is demonstrated in Fig. 6. Figs. 6a and 6b show the eventsare selected by a combination of the following criteria, (i). reconstructed positions within ±
15 cm, and (ii). accumulated probability of more than 80% inside the central regionof 42 cm. The number of selected events is about 42% of the total. Fig.6c shows thedistributions of the original axial positions between the decay and capture events arealmost identical; and Fig.6d shows the capture-to-decay ratio κ has a ﬂat plateau andrelatively sharp edges in the selected region with respect to the original axial positions. New Beam Neutron Lifetime Experiment in Superﬂuid Helium-4 -40 -30 -20 -10 0 10 20 30 40 Axial Position z (cm) H i t P r obab ili t y
The overall hit probability on each of the 250 detectors
P r (∆ V ( z k ) | N PE ( n det ) , n det Z ∆ V ( z k ) d ⇀ x P n det =1 N PE ( n det ) P r ( n det | ⇀ x ) P r ( ⇀ x ) P n det =1 N PE ( n det ) P r ( n det ) (4) The spectrum of the selected events is plotted against the PE number in Fig. 7. Thecapture events overlap with the decay events in the mid range and the spectrum cutsoﬀ at a lower bound of 4 PE s, equivalent to a recoil energy KE e = 20 keV, on purposeto exclude random backgrounds of few photons. In order to resolve the ratio κ ofcapture-to-decay event rates from the acquired spectrum, a theoretical model for ﬁttingis constructed. It consists of three components: the neutron decay spectrum, the singlecapture peak and the background. In this study, we only simulate signals of the formertwo, but omit the eﬀect of the background, because it will be poorly deﬁned withoutthe knowledge of the actual system. A discussion on the possible backgrounds will bepresented in Section 6.The neutron beta decay spectrum is formulated as New Beam Neutron Lifetime Experiment in Superﬂuid Helium-4 -40 -30 -20 -10 0 10 20 30 40 Axial Position z (cm) PE N u m be r N PE = 77 -40 -30 -20 -10 0 10 20 30 40 Axial Position z (cm) P r obab ili t y Pr c = 0.937, Pr tot = 0.99fit position: 9.82 0.88 cmoriginal position: 10.78 cmx0.01 (b) (a) Electron Recoil
An example on the deduced possibility of event axial position based on theobserved distribution of 77 PE s. (a) the spatial distribution of observed PE numberson diﬀerent detectors; (b) the spatial distribution of probabilities on diﬀerent eventaxial positions. The total probability Pr tot is 0.99 and the probability in the centralregion Pr c is 0.937. The ﬁt position based on the observed PE s is 9 . ± .
88 cm, andin comparison, the original position fed in the simulation is 10.78 cm. d Γ n dE e ∝ F n ( E e ) pE e ( E ( endpt ) − E e ) (5)where E ( endpt ) is the endpoint energy, E e is the total electron energy, and p =( E e − m e ) / is the momentum of electron. m e is the electron mass. F n ( E e ) is the Fermifunction for neutrons deﬁned as F n ( E e ) = 4 exp ( παE e /p ) | Γ(1 − iαE e /p ) | Γ(3) (6)where α = 1 /
137 is the ﬁne structure constant, and Γ( z ) is the gamma function. New Beam Neutron Lifetime Experiment in Superﬂuid Helium-4 R a t i o -40 -30 -20 -10 0 10 20 30 40 Original Axial Position z (cm) (d)024 P r obab ili t y -40 -30 -20 -10 0 10 20 30 40 Original Axial Position z (cm)0.01 decay eventscapture events(c)
Original Axial Position z (cm) F r equen cy -40 -30 -20 -10 0 10 20 30 40decay eventscapture events(b)-40 -30 -20 -10 0 10 20 30 40-20-1001020 Original Axial Position z (cm) R e c on s t r u c t ed A x i a l P o s i t i on z ( c m ) decay eventscapture events(a) Figure 6.
Analysis of the combined ﬁducial cut on the simulated data. (a) thereconstructed axial position z of both the simulated decay events (blue dots) andcapture events (red dots) plotted against their original position; (b) and (c) frequencyand probability distribution of the selected decay events (blue solid line) and captureevents (red dashed line) by the ﬁducial cut as to their original positions, respectively;(d) the ratio κ calculated within each bin of original position z . New Beam Neutron Lifetime Experiment in Superﬂuid Helium-4 Simulated PE counts at the central regionFitted PE counts at the central region
PE Numbers -505 R e s i dua l ( ) F r equen cy Figure 7.
Simulated neutron beta decay spectrum in addition to the neutron capturepeak (black solid line) and the result of the ML ﬁt (red dotted line). The residual ofﬁtting with respect to the PE numbers is shown in the lower plot. χ /ndf = 1 . In reality, the spectrum contains a Poisson smearing due to the random process inthe light transportation and conversion into PE s. Though the deposited energy fromneutron capture events is single-valued at 764 keV, it appears as a much broadened peak.The Poisson probability function, P ois [ N PE , λ ], with the PE number N PE and the mean λ , is thus embedded in the probability function f ( N PE ) of the observed spectrum as f ( N PE ) = 11 + ˜ κ ǫ cut ( N PE ) C e X M PE P ois [ N PE , M PE ] · F e ( M PE )+ ˜ κ κ C p + t P ois [ N PE , λ p + t ] + B (7)where ˜ κ is the observed ratio of capture-to-decay event rates, C e and C p + t are thenormalization factors for the spectrum bins above the lower bound of 4 PE s. The thirdterm B represents the background. The second term is the broaden peak of the neutroncapture events, where λ p + t is the mean PE number for the capture events. The ﬁrst termis the modiﬁed neutron beta decay spectrum. In a Poisson process, events of diﬀerentrecoil energies and their corresponding deterministically-converted PE number M PE allcontribute to a given bin of observed PE number N PE . The share of contribution from New Beam Neutron Lifetime Experiment in Superﬂuid Helium-4 PE Number I ne ff i c i en cy o f D e t e c t i on MC Simulated DataPiecewise Fit
Ineﬃciency of detection, 1 − ǫ cut ( N PE ), related to the combined ﬁducialcut of reconstructed positions within ±
15 cm and 80% accumulated probability within ±
21 cm. The simulation data is plotted in black circles, and the piecewise ﬁt is plottedin red dashed line. Piecewise ﬁt can better capture the features of ineﬃciency as asharp rise in low PE region and a slow tilt in the high PE region. each M PE is actually the probability of events with a deterministic M PE in the neutronbeta decay spectrum that follows Eqn. (5), namely F e ( M PE ), and given by F e ( M PE ) = Z ( M PE +0 . /η e Ω( M PE )( M PE − . /η e Ω( M PE ) d Γ n dE e dE e (8)where η e · Ω( M PE ) is the KE e -to- PE s conversion coeﬃcient for electron recoils. Asindicated in Fig. 3, the coverage ratio of solid angle Ω( M PE ) has a weak dependenceon the electron recoil energy, i.e. , on the PE number M PE in the central region of thedecay volume. The particular coeﬃcient ǫ cut ( N PE ), in the ﬁrst term of Eqn. (7), is theeﬃciency of detection related to the ﬁducial cut. It comes from a signiﬁcant positionuncertainty due to few PE s and long tracks for the low and high PE events, respectively.It inevitably introduces a detection ineﬃciency, 1 − ǫ cut ( N PE ), and hence, a distortionof spectrum. Such an eﬀect is extracted from the simulation and ﬁtted as shown in Fig.8. There remain 3 ﬁtting parameters in Eqn. (7): η e , λ p + t , and most importantly, ˜ κ .Maximum Likelihood (ML) method is employed to acquire the best ﬁt and associatederrors of the above 3 parameters. κ in Eqn. (7) is marked as ˜ κ , as the ﬁtting only New Beam Neutron Lifetime Experiment in Superﬂuid Helium-4 Table 1.
Result of ML ﬁt to the simulated spectrum. κ = ∆ N p + t / ∆ N β η e λ p + t Preset Value 4.4975 0.2032 57.254Monte Carlo 4.4997 0.2032 57.292ML Fit 4 . ± . . ± . . ± . acquires the observed capture-to-decay ratio within the data bins. ˜ κ must be correctedfor the actual κ with the detection eﬃciencies ǫ He = 1 and ǫ β = 0 . ǫ β < PE s. The result is listed in Table1, and the plots in Fig. 7 show the result of the ML ﬁt (in red dotted line) on top ofthe acquired spectrum (in black solid line). It demonstrates the ﬁt can extract κ valueat an accuracy well within 0.1%. Reconstruction of event energy will be performed in calibration of all the detectorsas a cross reference. In the previous subsection, the conversion eﬃciency η e forelectron recoils is ﬁtted through the analysis of the neutron decay spectrum, and thereconstructed energy is obtained as E e = N PE /η e Ω( N PE ). In reality, each photonsensor has a diﬀerent quantum eﬃciency η ( i ) SiPM , and each ﬁber has a variation intransmission eﬃciency η ( i ) fiber . The overall detection coeﬃcient η tot ( ⇀ x ) thus varies fordiﬀerent sub-volumes, due to the variation of solid angles upon detectors of diﬀerentquantum eﬃciency. It is a common approach to use conversion electron sources, suchas Cd (63, 84 keV),
Ce (127, 160 keV),
Sn (364, 388 keV),
Bi (481, 975,1047 keV) for calibrations in between production runs. These sources can be placedin many designated positions to map out the response of diﬀerent detectors. During theproduction runs, the calibration can also be done with the neutron capture peak, andadditional deposits of α source or lithium neutron capture ﬁlm on the end windows ofthe decay volume.
6. Background Suppression
In order to achieve a highly accurate measurement on the ratio of capture-to-decay rates,background signals must be properly suppressed, discriminated or subtracted. Cosmicray muons can be easily identiﬁed by coincidence in the veto detectors surrounding theapparatus. The static radioactive backgrounds from materials of the apparatus can beshielded by a thick layer of lead or tungsten as shown in Fig. 2 and characterized inthe background runs. The gamma rays from the cold neutron source can be greatlysuppressed by bending the beam direction out of sight with proper neutron optics.The most harmful type of backgrounds are the gamma rays produced by the neutron-induced activation near the decay volume and undergoing Compton scattering on liquid
New Beam Neutron Lifetime Experiment in Superﬂuid Helium-4 PE spectrum and modelled as B in Eqn. (7). The spectrum of theCompton electrons produced by gamma rays above 4 MeV is mostly ﬂat in the region ofneutron decay spectrum. Some of the delayed gamma rays can be characterized duringthe intervals between the CN beam pulses, such as the 1.6 MeV gamma rays emitted ata half-life of 11.16 s from the neutron activated ﬂuorine. In this paper, we focus on twotypes of prompt gamma rays due to the neutron captures by the window material, andby hydrogen, p + n → d + γ (2 . etc. They are believed to be the major contributors tothe backgrounds.The ﬁrst measure to suppress neutron-induced gamma rays is to reduce the captureand scattering of neutrons on the window materials. Polycrystalline CVD diamond isa good option, because carbon has relative small scattering and capture cross sections,and a thin window of 5 cm diameter and 1 mm thickness  is commercially availablewith a good mechanical strength. The capture cross section of carbon for the 17 ˚A coldneutrons is 0.033 barns and the capture fraction is 5 . × − . For a cold neutron ﬂuxof 1 . × CN s − , about 8 . × Hz of neutrons are captured with an emission ofprompt gamma rays mostly at energies of 1.3, 3.7 and 4.9 MeV. A simulation showsthe intensive prompt gamma rays result in more than 600 Hz Compton events in liquidhelium inside the decay volume, and more than 100 Hz in the polystyrene WLS ﬁbers.Most events distribute spatially near the windows, and temporally at the moments whenthe neutron ﬂux passes the windows. Although the number of the window-originatedCompton events greatly overwhelms that of the decay events, they can be separated intime if the neutron beam can be chopped into sharp pulses both in time and energy.It requires the decay volume to be set up close to the source. The neutron decays willthen appear as scattered single events in time sequence between two intensive burstsof Compton events when neutrons pass the entrance and exit windows, respectively.Considering it takes about 3.2 ms for the 17 ˚A neutrons to pass the 75 cm long decayvolume, the middle 1.6 ms is the time interval when the beam pulse passes the centralregion. The typical recovery time for the SiPM sensors are hundreds of nano-seconds,whereas the rate of after pulses following each prompt signal decays in tens of micro-seconds. Therefore, such a time cut can eﬀectively distinguish the events occurringin the central region, which are crucial in construction of the energy spectrum, andeliminate impacts of the window-originated Compton events. In return, the brightbursts of Compton events can be used as a calibration reference of the beam ﬂux andspectrum.The scattering of cold neutrons on the windows at 0.5 K is dominated by incoherentscattering, which is an s-wave scattering independent of the incident velocity. Theincoherent cross section of carbon for the 17 ˚A cold neutrons is 0.001 barns, and thescattered fraction is 1 . × − . For the same cold neutron ﬂux as above, about 270 Hzof neutrons are scattered isotropically from both the windows into the delay volume New Beam Neutron Lifetime Experiment in Superﬂuid Helium-4 Li has a large neutron absorption crosssection of 8887 barns for 17 ˚A cold neutrons, and there is no associated emission ofgamma rays in the reaction, Li + n → α + t + 4 .
78 MeV.A simulation on neutron scattering and capture is carried out on the geometry ofa 1 mm thick polystyrene ﬁbers around the decay volume, a 5 mm thick PTFE holderclamped on the ﬁbers, and a suﬃciently thick lithium absorber at the outermost shellthat absorbs all the stray neutrons, as illustrated in Fig. 2. It is found about 4.3% ofthe scattered neutrons are captured by the ﬁbers and 0.4% by the PTFE holder. Theneutron captures on hydrogen in the ﬁbers do not induce any signiﬁcant scintillationin the polystyrene, as the kinetic energy of deuterium is merely about 1.3 keV. Theresultant prompt gamma rays of 2.2 MeV contribute a background of Compton eventsat 0.28 Hz in liquid helium inside the decay volume, and 0.31 Hz in the polystyreneWLS ﬁbers. As shown in Fig. 9, the Compton events have a higher chance to occurnear the windows. Since the majority of Compton electrons are at energies near theCompton peak of 2.0 MeV, it adds about 0.05% to the total counts of neutron decayevents with a ﬁducial cut of ±
15 cm on the central region. It can be characterized andcorrected in data analysis.
7. Accurate Measurement of Helium-3 density with UCNs
The last quantity crucial to determine the neutron lifetime in Eqn. (3) is the He density n He = 2 . × X He , where the fractional concentration X He of about 2 × − needsto be prepared and characterized in high precision. One possible way of measurementis to employ the Atom Trap Trace Analysis, which has been demonstrated to measurethe abundance of rare isotope Ar at the level of 10 − . Similar technology may bedeveloped for detection of the He concentration at a precision well below 0.1%. Then,the uncertainty of the combined term σ th He v thn n He in Eqn. (3) will be dominated by thatof the He capture cross section of thermal neutrons, σ th He = 5333 ± He density, which takes advantage of the scintillation rates correlated to the neutroncaptures on He nuclei in the sample liquid helium. These deployed neutrons are notcold neutrons in a beam but rather ultracold neutrons (UCN) stored in a materialbottle ﬁlled with the sample liquid helium. Since UCNs uniformly distribute in thestorage volume and scintillations can partially quench on the walls, it is impractical toconstruct a well-deﬁned spectrum as in Subsection 5.2, and therefore, the diﬀerence inthe 1 /t responses of after-pulses will be the key tool to distinguish the decay and capture New Beam Neutron Lifetime Experiment in Superﬂuid Helium-4 -40 -30 -20 -10 Axial Position z (cm) P r obab ili t y o f C o m p t on E v en t s Figure 9.
Probability of Compton electron events in the axial axis of the decay volumeinduced by the prompt gamma rays due to hydrogen-captures of neutrons scatteredfrom the beam inlet and outlet diamond windows. events. We will characterize the purity of He and accurately measure He concentrationin the later prepared helium mixture via the time-dependent rates of both decay andcapture scintillation events. Furthermore, it will be shown that value of the combinedterm σ th He v thn n He will be directly obtained in experiment. Therefore, our goal sensitivityin neutron lifetime of below 0.1% can be achieved with enough statistics, and is nolonger limited by the uncertainty of 0.13% in σ th He . Details will be articulated in thefollowing subsections. Suppose the sample liquid helium ﬁlls a storage volume made of UV transmitting acrylictube of 75 cm long, 7 cm ID and 7.5 cm OD. It is sealed at both ends and coated withdeuterated ﬁlms on the inside so that it is hermetic and friendly to UCNs. It can beinstalled inside the detector setup as described in Section 4, except the PTFE reﬂectorcan be as thin as 50 µ m because of the following two reasons: (i). the structuralsupport can be loaded to the storage tube; and (ii). PTFE generates a high level ofbackground due to the delayed gamma rays from neutron-activated ﬂuorine as discussedin Subsection 7.6. New Beam Neutron Lifetime Experiment in Superﬂuid Helium-4 N UCN ( t ) = − N UCN ( t ) τ tot , (9) N UCN ( t ) = N exp (cid:18) − tτ tot (cid:19) , (10)where N is the initial number of UCNs and τ tot is the storage time constant. Severalfactors contribute to τ tot of this volume, τ − tot = τ − He + τ − β + τ − up + τ − loss . (11) τ He is the neutron He capture time constant of interest as given in Eqn. (12).For X He = 2 × − , τ He is about 195.2 s, which dominates the total storage time,compared to the neutron lifetime τ β ≈
880 s.1 τ He = n He σ th He v th He = 2 . × X He [s − ] (12)UCNs suﬀer a loss from captures or up-scattering of the wall nuclei. Such aneﬀect can be described by Schr¨odinger equations with one-dimensional potential andcharacterized by a loss probability per bounce, f ( E UCN ). The rate of wall collisions isgiven by ( vA/ V ), where v is the UCN velocity, A is the area of the storage chamber,and V is its volume. The contribution of wall losses to the storage time is then given as1 τ wall = f ( E UCN ) (cid:18) vA V (cid:19) . (13)Generally, the hydrogen in the organic materials has a large up-scattering crosssection for the UCNs. Therefore the hydrogen in contact with UCN must be replacedwith deuterium. The inner wall of the storage volume needs to be coated with a layerof deuterated polystyrene (dPS), whose Fermi potential is about 160 neV. The TPBconverter coating also needs to be deuterated. These technologies are under developmentand tests by the SNS nEDM collaboration. With a loss probability per bounce of f ( E UCN ) = 10 − , the same requirement as the SNS nEDM UCN storage cells , thetime constant due to wall loss τ wall is about 1672.0 s. τ up is the loss rate due to upscattering of neutrons by quasi-particles, phononsand rotons, in superﬂuid helium. It is greatly suppressed by a Boltzmann factor. At T < . τ up = T
100 [s − ] . (14)At T = 0 . τ up = 12800 s. With all the contributions above included in Eqn. (11),the storage time constant τ tot is about 144.2 s. This is merely an estimate. τ tot will beaccurately measured in experiments in order to acquire a high-precision determinationon the He concentration X He in Subsection 7.5. New Beam Neutron Lifetime Experiment in Superﬂuid Helium-4 The UCNs for this measurement are produced in situ in the neutron decay volume bysuper-thermal process: a 8.9 ˚A beam of CNs are down-scattered inelastically into UCNsvia exciting a single phonon in superﬂuid helium. The UCN density can build up inthe decay volume with the time constant τ tot . The accumulated UCN density in liquidhelium exposed to the CN beam is given by ρ UCN ( t fill ) = Rτ tot (cid:20) − exp (cid:18) − t fill τ tot (cid:19)(cid:21) . (15) τ tot is the storage time constant as given in Eqn. (11), and the production rate perunit volume R is given by R = 2 . × − (cid:18) d Φ dE (cid:19) [cm − s − ] (16)where an incident ﬂux spectrum of ( d Φ /dE ) is in units of (cm − s − ˚A − ), and theproduction of UCNs is up to the maximum storage Fermi energy of 160 neV.[57, 58] Asshown in Fig. 1, the 8.9 ˚A CN ﬂux in the SNS FnPB is about 5 . × Hz cm − ˚A − .The production rate per unit volume is deduced to be R ≈ . − s − . So thesteady state UCN density ρ UCN can reach an average of about 141.4 UCNs cm − with200 s of beam ﬁlling. i.e. a total number N ≈ . × of UCNs can be ﬁlled in thestorage cell with a beam-occupied volume of 530.1 cm . However, the initial UCN ﬁllingnumber N varies in diﬀerent runs depending on the CN beam intensity and stability,and is of little use in the data analysis due to the large uncertainty. The estimate aboveis for the sake of presenting the order of magnitude. The observed capture and decay rates of UCNs in the storage volume as well as theirratio are given by ˙ N ( p + t ) UCN ( t ) = − ǫ ′ He N τ He exp (cid:18) − tτ tot (cid:19) , (17)˙ N ( β ) UCN ( t ) = − ǫ ′ β N τ β exp (cid:18) − tτ tot (cid:19) , (18)˜ κ UCN = ˙ N ( p + t ) UCN ( t )˙ N ( β ) UCN ( t ) = ǫ ′ He ǫ ′ β τ β τ He exp (cid:18) − t − t τ tot (cid:19) , (19)where ǫ ′ He and ǫ ′ β are the detection eﬃciencies of UCN capture and decay eventsthat can be obtained via simulation. Since the He capture events can be distinguishedfrom the decay events via the diﬀerence in the 1 /t decay rate of after-pulses, ˙ N ( p + t ) UCN ( t )and ˙ N ( β ) UCN ( t ) can be directly acquired in the measurement. The ratio ˜ κ UCN can then be
New Beam Neutron Lifetime Experiment in Superﬂuid Helium-4 i.e. t i = t = t , ˜ κ UCN ( t i ) = ǫ ′ He ǫ ′ β τ β τ He = ǫ ′ He ǫ ′ β τ β n He σ th He v th He . (20)˜ κ UCN ( t i ) should statistically ﬂuctuate around its true value. In this scenario, theneutron lifetime enigma can be examined by simply comparing τ β n He obtained aboveby Eqn. (20) in the UCN storage volume with that by Eqn. (3) in the beam decayvolume. The real value of He density n He is no longer necessary. But there is a caveat:the success of this ”shortcut” trick greatly depends on how well one can characterizethe background events, especially the Compton events induced by the gamma rays, andseparate them from decay events through analysis and modelling. The production of isotopically pure He can be carried out in a puriﬁer similarto that designed by Hendry and McClintock. For a residual He concentration, X (0) He < . X He ≈ × − , the He capture time constant τ (0) He is expected tobe more than 2 × s. With the property of the UCN storage volume described inSubsection 7.1, the total storage time τ (0) tot is expected to be about 551.7 s, and the UCNdensity ρ (0) UCN can reach about 219.2 UCNs cm − . With 200 s of beam ﬁlling, it mayachieve a ﬁll of N (0) ≈ . × UCNs in the storage volume.Because of the scarcity of the He atoms in the isotopically pure He liquid, a totalnumber of neutron capture events ∆ N ( p + t, UCN may be counted over a period of storage time∆ t via identiﬁcation on the after-pulses of scintillation events. So is the total numberof the neutron decay events ∆ N ( β, UCN . Integration on Eqns. (17) and (18), respectively,gives ∆ N ( p + t, UCN = − ǫ ′ He N (0) τ (0) tot τ (0) He (cid:20) − exp (cid:18) − ∆ tτ (0) tot (cid:19)(cid:21) , (21)∆ N ( β, UCN = − ǫ ′ β N (0) τ (0) tot τ β (cid:20) − exp (cid:18) − ∆ tτ (0) tot (cid:19)(cid:21) . (22)With the ratio of two equations above, the residual He concentration, X (0) He , isobtained via the following equation,2 . × X (0) He [s − ] = τ (0) − He = 1 τ β ǫ ′ β ǫ ′ He ∆ N ( p + t, UCN ∆ N ( β, UCN . (23)It is expected the counts of background Compton events might be comparableto or even larger than that of capture events with the residual He concentration. Itundoubtedly introduces a large uncertainty on X (0) He , but may not appear as large afterpropagated to that of the measured X He in the later prepared helium mixture. The New Beam Neutron Lifetime Experiment in Superﬂuid Helium-4 τ β used in Eqn. (23). Despite simply fed with the PDGvalue, the error should be negligible after propagated to that of X He .Meanwhile, since τ (0) − He is negligible, the storage time τ (0) tot in isotopically pure Heis expressed as τ (0) − tot = τ − β + τ − up + τ − loss . (24) τ (0) tot can be accurately measured by ﬁtting the time-dependent decline of ˙ N ( β ) UCN ( t )as in Eqn. (18) with normalization to an arbitrary time zero after the beam stops. Theuncertainty of τ (0) tot greatly depends on a good understanding of the Compton event rate. Once the isotopically pure He is characterized, the desired He concentration of about X He = 2 × − can be prepared by mixing with natural helium of a known Heabundance as to a preset volume ratio. The He atoms may be expelled from thevolume via heat ﬂush to ﬁne tune its concentration.[35, 59] It is noteworthy that Eqns.(19) and (23) are equivalent but neither can be used in the accurate determination of X He , because they proportionally link τ β to τ He . Therefore, X He should be extractedfrom the diﬀerence between τ (0) tot in the isotopically pure He and τ tot in the preparedhelium mixture, according to the following expression,2 . × X He [s − ]= n He σ th He v th He = τ − He = τ − tot − τ (0) − tot . (25)In a storage measurement, both the decay and capture event rates, ˙ N ( p + t ) UCN ( t ) and˙ N ( β ) UCN ( t ), decline with an identical time dependence on the total storage time constant τ tot of the volume, as in Eqns. (17) and (18). τ tot can be measured via both channelsindependently, though the decay rate channel might have a higher uncertainty due tothe contamination of background Compton events. The detection eﬃciencies, ǫ ′ He and ǫ ′ β , are irrelevant as long as they remain stable in one measurement cycle. However,it has been experimentally found the UCN loss rate in a material bottle storage isnot completely described by an exponential decline as in Eqns. (10), (17) or (18).A dual exponential ﬁt often works much better than the single exponential ﬁt in astorage volume. It is possibly due to the change of UCN spectrum during the storage,especially as UCNs of higher energy have more chances to be up-scattered and absorbedon the wall, or to leak out at some spots of wall materials with lower Fermi potentials.As a result, it may impose systematic uncertainties on the measurement of both τ tot and τ (0) tot , which is tightly related to the energy spectrum of the UCNs generated in thestorage volume and quality of the wall material.Nevertheless, X He given by Eqn. (25) is not a good option to calculate thebeam neutron lifetime via Eqn. (3), because the prefactor 2 . × has inheritedan uncertainty of 0.13% from σ th He . In fact, Eqn. (3) asks for the combined term n He σ th He v th He , which can be directly acquired once and for all by Eqn. (25). Therefore, σ th He by itself is no longer in the path of ﬁnal calculation of the beam neutron lifetime, New Beam Neutron Lifetime Experiment in Superﬂuid Helium-4 τ tot and τ (0) tot , aswell as the ratio of event rates κ = ˙ N p + t / ˙ N β discussed earlier in Subsection 5.2. Similarly, gamma rays generated from the neutron captures on the surrounding materialsare the major contributor to the background, but the UCN storage measurement ismostly sensitive to the delayed components rather than the prompt. In addition to thecapture and scattering by the windows, the 8.9 ˚A CN beam has a scattering cross sectionof about 0.025 barns on liquid helium at 0.5 K.[60, 61] 4% of CNs will be inelasticallyscattered by the phonons in liquid helium. A very small portion is down-converted intoUCNs and trapped in the storage volume, whereas majority of the scattered neutronsproject at angles of around 84 degrees oﬀ the incident direction. About 12% of thescattered neutrons are captured by hydrogen, on top of those captured by the diamondwindow. The resultant prompt gamma rays are intensive but vanish right after thebeam halts. Therefore, it doesn’t aﬀect the measurement of UCN storage time τ tot .More troublesome are the delayed gamma rays from the 0.002% of scattered neutronscaptured on the ﬂuorine in the 50 µ m thick PTFE reﬂector. After 200 s of UCN ﬁllingwith the 8.9 ˚A CN beam, the neutron-activated ﬂuorine saturates. The resultant delayedgamma rays of 1.6 MeV induce Compton scintillation at 2.5 Hz in liquid helium and1.7 Hz in the ﬁbers at time-zero when the beam turns oﬀ. Since the half-life of theactivated ﬂuorine is 11.16 s, the rate of Compton scintillation in liquid helium dropsbelow 1 Hz after 15 s. At the meantime, the rate of decay events is 76.8 Hz, assuming thestorage time is 144.2 s and the initial UCN number is N ≈ . × . The delayed gammaray background discussed above can be eliminated by replacing the PTFE reﬂector witha polymer reﬂector, such as Vikuiti VM2000 , but others may still remain due toimpurities in the surrounding materials.
This proposed experiment has great potential to reach a sensitivity of 0.1% or sub-1second in neutron lifetime measurement. It oﬀers an entirely diﬀerent set of systematicuncertainties from the existing beam experiments. As explicitly expressed in Eqn. (3),the calculation of τ β is independent of the neutron ﬂux and the geometry of the decayvolume. Most uniquely, it does not require any magnetic ﬁeld, and may be set up totest the hypothesis of neutron-mirror neutron n − n ′ oscillations, where the intensityof magnetic ﬁeld plays an important role.[7, 8, 63, 64] Its apparent disadvantage is theﬂux of a CN beam at the wavelength λ > . New Beam Neutron Lifetime Experiment in Superﬂuid Helium-4 κ in the CN decay volume, as well as the storage time constants τ tot with liquid helium mixed with a proper concentration of helium-3, and τ (0) tot withisotopically pure liquid helium-4 in the UCN storage volume. Both the volumes canbe connected and share the same batch of prepared liquid helium, where statistics ontwo measurements can be gained simultaneously. The 8.9 ˚A CN beam can be extractedfrom the main beam by a monochromator [33, 65] and sent to the UCN storage volume,whereas the 17 ˚A CNs can be selected by a set of double or triple choppers in the mainbeam and sent to the CN decay volume. Hopefully, a good design of the double or triplechoppers can eﬀectively ﬁlter the contamination of neutrons in the unwanted spectrum.The Atom Trap Trace Analysis can be set up to characterize the helium-3 concentrationin samples as a veriﬁcation of the result oﬄine. Furthermore, while the intrinsic UCNstorage time τ (0) tot is being measured in the storage volume ﬁlled with isotopically pureliquid He, the neutron beta decay spectrum can be simultaneously obtained in thedecay volume at a good resolution and accuracy with this detector. It may provide ameasurement on the Fierz interference term b in the energy dependent neutron betadecay rate at a potentially high precision compared to the most recent results [66, 67].This paper only covers the proof of principle for the proposed experiment. Manyunknown technical issues will unsurprisingly emerge as the engineering designs andprototyping tests proceed. We hope it can eventually help resolve the neutron lifetimeenigma. Acknowledgement
This paper greatly beneﬁts from the publications, notes and data shared by the SNSnEDM collaboration. The author acknowledges Vince Cianciolo, Bradley W. Filippone,Roy J. Holt, Humphrey J. Maris, Jeﬀrey S. Nico, George M. Seidel, Christopher M.Swank, Fred E. Wietfeldt and Liyuan Zhang for help and discussions on many topicspresented in this paper. This work is supported in part by the National ScienceFoundation under Grant No. 1812340.
References  Wietfeldt F E and Greene G L 2011
Rev. Mod. Phys. Eur. Phys. J. C Rev. Mod. Phys. Nucl. Instrum. Meth. A Phys. Proced. Atoms Eur. Phys. J. C Phys. Lett. B Phys. Rev. Lett.
New Beam Neutron Lifetime Experiment in Superﬂuid Helium-4  Ivanov A N, Hollwieser R, Troitskaya N I, Wellenzohn M and Berdnikov Y A 2019 Nucl. Phys. B J. High Energy Phys.
Phys. Rev. D Phys. Rev. Lett.
Phys. Lett. B Lett. High Energy Phys.
Europhys. Lett. Phys. Rev. Lett. et al
Phys. Lett. B Phys. Lett. B Phys. Rev. C Phys. Lett. B et al Phys. Rev. C et al Science et al
J. Exp. Theor. Phys. Lett. et al
Proc. Sci.: 26th Int. Nucl. Phys. Conf. (INPC2016) 191 Hirota K et al
Nucl. Phys. A et al Bull. Am. Phys. Soc.  Greene G L and Geltenbort P 2016 Sci. Am.
Nucl. Instrum. Meth. A et al (Particle Data Group) 2018 Phys. Rev. D Cryogen. Nucl. Instrum. Meth. A Phys. Rev. D Phys. Rev. D J. Instrum. C09008 Ito T M, Clayton S M, Ramsey J, Karcz M, Liu C-Y, Long J C, Reddy T G and Seidel G M 2012
Phys. Rev. A NIST Stand. Ref. Database (http:// webbook.nist.gov) McKinsey D N et al
Phys. Rev. A Ph.D. thesis (Harvard University) McKinsey D N, Brome C R, Butterworth J S, Dzhosyuk S N, Golub R, Habicht K, Huﬀman P R,Mattoni C E H, Yang L and Doyle J M 2004
Nucl. Instrum. Meth. A J. Instrum. P01002 McKinsey D N, Brome C R, Butterworth J S, Golub R, Habicht K, Huﬀman P R, Lamoreaux SK, Mattoni C E H and Doyle J M 1997
Nucl. Instrum. Meth. B Ph.D. thesis (Brown University) Plaster B et al
Phys. Rev. C New Beam Neutron Lifetime Experiment in Superﬂuid Helium-4 et al Phys. Rev. Lett.
Neutron News et al Nucl. Instrum. Meth. A et al to be published; Tang Z et al to be published Ahmed M W et al J. Instrum. P11017 Golub R and Pendlebury J M 1977
Phys. Lett. A et al Phys. Lett. A Phys. Rev. B Phys. Rev. Nucl. Instrum. Meth. A Phys. Rev. Lett. Phys. Lett. B et al Phys. B: Conden. Matt. et al
Phys. Rev. C101