measurement in the big gravitational spectrometer

The method of UCN “small heating” measurement in the big gravitational spectrometer (BGS) and studies of this effect on Fomblin oil Y-HVAC 18/8 V. V. Nesvizhevsky, A. Yu. Voronin, A. Lambrecht, S. Reynaud, E. V. Lychagin, A. Yu. Muzychka, G. V. Nekhaev, and A. V. Strelkov

Citation: Review of Scientific Instruments 89, 023501 (2018); doi: 10.1063/1.5010974 View online: https://doi.org/10.1063/1.5010974 View Table of Contents: http://aip.scitation.org/toc/rsi/89/2 Published by the American Institute of Physics

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REVIEW OF SCIENTIFIC INSTRUMENTS 89, 023501 (2018)

The method of UCN “small heating” measurement in the big gravitational spectrometer (BGS) and studies of this effect on Fomblin oil Y-HVAC 18/8 V. V. Nesvizhevsky,1 A. Yu. Voronin,2 A. Lambrecht,3 S. Reynaud,3 E. V. Lychagin,4 A. Yu. Muzychka,4 G. V. Nekhaev,4 and A. V. Strelkov4 1 Institut

Laue-Langevin, 38042 Grenoble, France Institute, 119991 Moscow, Russia 3 Laboratoire Kastler Brossel, UPMC-Sorbonne Universit´ e, CNRS, ENS-PSL Research University, Coll`ege de France, Campus Jussieu, 75252 Paris, France 4 Joint Institute for Nuclear Research, 141980 Dubna, Russia 2 Lebedev

(Received 28 October 2017; accepted 17 January 2018; published online 5 February 2018) The Big Gravitational Spectrometer (BGS) takes advantage of the strong influence of the Earth’s gravity on the motion of ultracold neutrons (UCNs) that makes it possible to shape and measure UCN spectra. We optimized the BGS to investigate the “small heating” of UCNs, that is, the inelastic reflection of UCNs from a surface accompanied by an energy change comparable with the initial UCN energy. UCNs whose energy increases are referred to as “Vaporized UCNs” (VUCNs). The BGS provides the narrowest UCN spectra of a few cm and the broadest “visible” VUCN energy range of up to ∼150 cm (UCN energy is given in units of its maximum height in the Earth’s gravitational field, where 1.00 cm ≈ 1.02 neV). The dead-zone between the UCN and VUCN spectra is the narrowest ever achieved (a few cm). We performed measurements with and without samples without breaking vacuum. BGS provides the broadest range of temperatures (77-600 K) and the highest sensitivity to the small heating effect, up to ∼10 8 per bounce, i.e., two orders of magnitude higher than the sensitivity of alternative methods. We describe the method to measure the probability of UCN “small heating” using the BGS and illustrate it with a study of samples of the hydrogen-free oil Fomblin Y-HVAC 18/8. The data obtained are well reproducible, do not depend on sample thickness, and do not evolve over time. The measured model-independent probability P+ of UCN small heating from an energy “mono-line” 30.2 ± 2.5 cm to the energy range 35–140 cm is in the range (1.05 ± 0.02stat )×10−5 −(1.31 ± 0.24stat )×10−5 at a temperature of 24 ◦ C. The associated systematic uncertainty would disappear if a VUCN spectrum shape were known, for instance, from a particular model of small heating. This experiment provides the most precise and reliable value of small heating probability on Fomblin measured so far. These results are of importance for studies of UCN small heating as well as for analyzing and designing neutron lifetime experiments. Published by AIP Publishing. https://doi.org/10.1063/1.5010974

I. INTRODUCTION

The phenomenon known as “small heating” of ultracold neutrons (UCNs) was discovered over 15 years ago.1,2 It is the inelastic reflection of UCNs from a surface accompanied by an energy change (increase or decrease) comparable with the initial UCN energy. UCNs whose energy increases are referred to as “Vaporized UCNs” (VUCNs). An analogy with the evaporation of molecules from a liquid surface justifies the choice of terminology. (a) This is a thermodynamically non-equilibrium process. (b) A UCN and a molecule may both increase and decrease their energy depending on the volumes of available phase space. (c) The energy change is approximately equal to the initial energy. (d) Before evaporation, the particle is below a certain level in the gravitational field. (e) After evaporation, it is above it, and the particle leaves the system. In the literature, this process is also referred to as (a) “cooling,” when measuring a UCN energy decrease, or (b) quasi-elastic UCN scattering for both increasing and decreasing UCN energy as well as (c) a Doppler shift in UCN energy. In any event, the experimental values of the probability of this 0034-6748/2018/89(2)/023501/13/$30.00

process are much greater than those estimated within theoretical models which consider the process of neutron reflection from bulk materials.3–5 The observation of small heating (“cooling,” quasi-elastic scattering, and Doppler shift) of UCNs has been reported by various experimental groups: in the reflection of UCNs from solids1,2,6–10 and in the reflection of UCNs from liquids.1,2,9–12 The values of UCN small heating probability obtained for the same materials by different teams vary noticeably from each other. This discrepancy could be the result of actual differences in the probability process due to poorly controlled parameters as well as of insufficient knowledge of spectral characteristics and hence insufficiently accurate estimations of VUCN detection efficiency. For solid surfaces, no contradictions have been observed thanks to the work 13 which showed that probability essentially changes as a function of sample treatment history, in particular the degree of prior sample heating. The results of a series of experimental studies7,8,13 and theoretical studies14–16 indicate that the small heating of UCNs is due to their scattering on nanoparticles adsorbed on the surface.17 Most nanoparticles are tightly bound to the surface, while others are weakly bound

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and thus move along the surface or oscillate. UCN scattering at weakly bound nanoparticles is the reason for their small heating. All experimental data available for solid surfaces are in good agreement with the predictions of this model. For liquid surfaces, there are still contradictions. In general, the small heating probability is measured from a certain initial UCN energy range to a range of larger energies available for spectroscopy; both ranges are specific for particular experimental setups. To be able to compare the different results correctly, both the experimental details and the efficiency of VUCN detection as a function of energy must be known. In all previous experiments, the calculated efficiency depended on assumptions. The first observation of the small heating phenomenon1,2,6 showed a probability value equal to 10 5 per UCN bounce and a VUCN energy approximately equal to the initial UCN energy. Reference 12 revealed instances of energy increase and energy decrease (“cooling”) in the UCN reflection from an oil surface. The energy change was also approximately equal to the initial UCN energy, but the probability was ∼10 6 . The authors of Ref. 9 confirmed the existence of small heating and cooling of UCNs and estimated the probability at 3 × 10 6 , while in Ref. 10 it was found to equal 5 × 10 6 . All these measurements involved the same temperature and the authors measured or simulated VUCN spectra. This dispersion of the results therefore greatly exceeds the estimated experimental accuracy. Despite the number of related publications, there has been no detailed study of UCN small heating on Fomblin. In particular, no one has previously shaped a UCN spectrum precisely and/or measured a VUCN spectrum. The physical origin of small heating on liquid surfaces may differ from that on solid surfaces. The authors of Refs. 18 and 19 consider UCN scattering on surface capillary waves as a reason for small heating and provide reasonable estimations for the observed parameters. Investigation of this phenomenon on liquid surfaces is also interesting in view of its contribution to systematic errors in the measurements of the neutron lifetime.20 Recent work 21 revises the results of some experiments involving UCN storage in Fomblin traps from this point of view. Hydrogen-free oils of various types are used in UCN experiments. They can be subdivided into two groups: lowtemperature and high-temperature oils. The pour temperatures of low-temperature oils are lower and saturation vapor pressures are higher; such oils can operate at boiling nitrogen temperature and provide the lowest UCN loss coefficients.22 High-temperature oils are widely used due to ease of operation and the absence of oil mass transfer at ambient temperatures. In the present work, we study the high-temperature hydrogenfree oil Fomblin Y-HVAC 18/8 used in neutron lifetime experiments.23–28 In the following paper, we describe the experimental setup, the Big Gravitational Spectrometer (BGS), which was specifically designed and built for the study of the small heating of UCNs. In addition, we describe the samples of the hydrogen-free oil Fomblin Y-HVAC 18/8 and the powder of diamond and sapphire nanoparticles used. Then, we present in detail the measurement procedure for this setup. We present the efficiency of VUCN detection as a function of energy and

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describe the procedure for its evaluation. We point out residual systematic effects limiting the accuracy of the evaluation of VUCN spectra. Finally, after describing the experimental and data-treatment procedures, we present the experimental results. Additional information is covered in Appendices. II. EXPERIMENTAL SETUP

We optimized the Big Gravitational Spectrometer (BGS)13 for the study of the small heating of UCNs. The BGS provides the narrowest UCN spectra of a few cm and the broadest “visible” VUCN energy range of up to ∼150 cm (UCN energy is given in units of height of its maximum rising in the Earth’s gravitational field, where 1.00 cm ≈ 1.02 neV). The dead-zone between them is the narrowest achieved (a few cm). The BGS provides their precise shape and direct measurement and hence the most precise and reliable measurements of small heating probability. We performed measurements with and without samples without breaking vacuum. The BGS provides the broadest range of temperatures (77-600 K) and the highest sensitivity to the small heating effect, up to ∼10 8 per bounce, i.e., two orders of magnitude higher than the sensitivity of alternative methods. The BGS is a vertical cylindrical UCN trap with a moving absorber in its upper part. A gravitational barrier divides the inner space of the trap into two sections: an internal and an external part. Figure 1 shows a BGS scheme of principle and a photo of its internal structure. A sample1 is placed in the bottom of the spectrometer inside the internal part defined by a cylinder 2 with the diameter 40 cm. In the experiment presented in this paper, the cylinder height is 35 cm; in other experiments (for instance, in those described in Ref. 13), it may be different. The cylinder plays the role of gravitational barrier for UCNs with an energy insufficient to jump over its upper edge and thus to the outside. UCNs fill the internal volume through an input neutron guide and then a valve3 traps them inside. A detector 4 monitors the UCN flux density at the bottom of the spectrometer. A valve (a cover that can move up and down) closes the exit from the storage volume to the monitor detector. The exit guide diameter is 90 mm. A calibrated hole8 in the valve with the diameter 5 mm is permanently open. Samples can be placed in the central part of the storage volume and extracted from it using a lift9 (shown in the photo, but not in the scheme). The lift consists of two horizontal disks connected to each other by three vertical pillars. The lift makes it possible to measure with or without a sample without breaking vacuum. An absorber 5 at a certain height H abs shapes the upper edge of the UCN spectrum in the spectrometer. Neutrons with energy sufficient to jump over the height H abs are lost in the absorber with a certain characteristic time. The absorber consists of two parts. The central part, a flat polyethylene disk, can move inside the internal part of the storage volume. In this case, the external part of the absorber, a flat polyethylene ring, lies on the gravitational barrier edge. Neither part of the absorber touches the walls of the gravitational barrier, thus leaving a gap of up to several millimeters. The detector,6 installed outside the gravitational bar2 rier, counts only those neutrons with an energy higher than the barrier (E bound ). If the absorber is in its upper position

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FIG. 1. A scheme of the setup. 1—sample; 2—gravitational barrier; 3—input valve; 4—monitor; 5—absorber; 6—detector; 7—exit valve; 8—calibrated hole; 9—“lift” for samples; 10—input neutron guide.

(H abs > E bound ), neutrons with an energy higher than E bound can overcome the gravitational barrier and reach the detector 6 by passing through an open input valve.7 In this work, the gravitational barrier height is always equal to 35 cm. The upper edge of the input neutron guide10 is 25 cm above the bottom of the storage volume. Thus, the initial UCN energy spectrum is quite narrow. Taking into account the suppression of the UCN spectrum just above the input guide edge as well as its suppression by the absorber above 32.5 cm, we estimate it to be 30.2 ± 2.5 cm. The small spectrum width makes it possible to minimize the spurious effects of the UCN spectra on the data. Appendix B describes the UCN spectra in detail; Fig. 14 illustrates the spectrum shape. The detector6 and the monitor 4 are equivalent UCN detectors with the same UCN detection efficiency, close to 100%. The spectrometer design provides setting and measuring temperatures in the range from 77 K (boiling liquid nitrogen) to 600 K. In measurements above ambient temperature, a titanium absorber replaces the polyethylene absorber to avoid its thermal degradation. Sample temperature is defined by the temperature of the spectrometer walls; adding helium gas triggers the heat transfer from the walls to the sample for the period needed to achieve thermal equilibrium. Temperature sensors

monitor several points in the spectrometer and one point on the sample. III. SAMPLES

The following arguments determined the choice of samples: (a) as stated in the introduction to this paper, we selected liquid Fomblin Y-HVAC 18/8 oil to be the principal sample for this study and (b) studied it as a function of various parameters, including thickness, time, and temperature. (c) We also measured samples of diamond and sapphire nanoparticles under the same experimental conditions for comparison. In order to study the small heating of UCNs on a Fomblin oil surface, we prepared samples of two types: thin layers (∼1 µm) and thick layers of oil (up to a few mm), as used in Ref. 28: thin layers next to the trap walls and thick layers at the bottom. We applied thin layers to a stainless steel foil by dipping it into a bath with oil and then removing it and letting the oil flow down for >24 h in a dust-free chamber. To prepare the thick-layer sample, we filled in flat “plates” with layers of oil with the thickness 2-3 mm and coated all other plate surfaces with a thin oil layer. To increase the sample area, we installed five identical “plates” on top of each other. Figures 2–4 show the two different types of samples.

FIG. 2. The stainless steel foil and the same foil coated with oil and placed in a dust-free chamber.

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The critical energies of the materials used in this study are the following: ∼104.5 cm for Fomblin Y-HVAC 18/8 oil, ∼194.5 cm for stainless steel, ∼165 cm for copper, ∼292 cm for diamond, and ∼152 cm for sapphire.

IV. MEASURING PROCEDURE

FIG. 3. The thin oil sample in the spectrometer; the lift is down.

FIG. 4. “Plates” with oil in the spectrometer; the lift is down.

We used Fomblin Y-HVAC 18/8 oil from the company Ausimont (now Solvay Solexis) to prepare the thin-layer samples and produced the thick-layer samples using Ausimont oil as well as oil previously used in Ref. 29. The Fomblin Y-HVAC 18/8 oil vapor pressure at ambient temperature is as low as ∼4 × 10 8 mbar, making it simple to use at this temperature and below without the worry of mass transfer to spectrometer surfaces. In addition to these oil samples, we used samples of diamond nanopowder (nano-diamond produced in accordance with TC 2-037-677-94 VNIITP, Snezhinsk) and sapphire nanopowder produced by the company Goodfellow; a characteristic diameter of sapphire nano-crystals is 20 nm. Nanopowder samples are layers with a thickness of up to 1 mm on copper or single-crystal sapphire surfaces. Figure 5 shows a nano-diamond sample.

FIG. 5. Nano-diamond powder on the spectrometer surface.

The procedure used to measure small heating and the VUCN spectrum with the BGS and samples described above consists of the following stages: “filling,” “cleaning,” “effect measurement,” and “emptying.” The total duration of a measuring cycle depends on the samples. In the measurements presented, it is equal to 365 s. Figure 6 shows a typical evolution of detector 6 and monitor 4 count rates during these stages of measurements with a sample. During all these stages (except for “emptying”), the exit valve in front of the detector is kept open and the valve in front of the monitor detector is kept closed. UCNs reach the monitor via the hole in the monitor valve;5 Fig. 1. During spectrometer filling (0-60th s), the input valve is open and the absorber is in its lower position H abs = H min at a height of 32.5 cm, i.e., 2.5 cm lower than the gravitational barrier. Some UCNs from the initial spectrum with an energy higher than the gravitational barrier escape through a slit between the absorber and the walls of the internal storage volume, jump over the gravitational barrier, and reach the detector; the count rate is therefore high. Cleaning begins after the input valve closes (60th s). Neutrons with an energy higher than the absorber height are promptly lost in it or escape from the external volume to the detector. As a result, the detector count rate drops sharply to the background value. The duration of cleaning ∆t clean is 40 s; we set this duration so that the neutrons with an energy greater than H min which survived in the internal storage volume could not affect the result. The procedures for selecting values ∆t clean and H min and measuring UCN spectra are described in Appendix A.

FIG. 6. Count rate in the detector (open circles) and the monitor (solid circles) as a function of time in measurements with a sample (see comments in the text). The lines are exponents fitted to the data. The sample surface area is 0.74 m2 ; the measurement was performed at a temperature of 24 ◦ C; the upper absorber height is 140 cm.

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At the end of cleaning, at 100th s, the absorber starts to rise; at 110th s, it reaches its upper position, H abs = H max = 140 cm, and the “effect measurement” begins. Raising the absorber does not affect neutrons with an energy lower than H min and thus does not affect the monitor count rate, while the detector count rate quickly increases and, after some time, becomes proportional to the flux density of the UCNs trapped inside the gravitational barrier. The permanent production of VUCNs in the storage volume with an energy higher than the gravitational barrier explains this dependence. Due to their absorption by the absorber, the probability of reaching the detector is significantly reduced for all neutrons with an energy greater than H max . The detector counts mostly UCNs with an energy smaller than H max making it possible to evaluate the integral VUCN spectrum by comparing the results measured with different H max . At 350th s, the absorber moves to the lower position and the detector count rate falls again to the background value. The required statistics are accumulated by repeating the measuring cycle several times. The solid lines in Fig. 6 correspond to exponents which fit the data. For the monitor count rate, the fitting function is ∼e t /τ ; for the detector, it is ∼ e−t/τ · (1 − e−t/τ1 ); here τ, the UCN storage time in the spectrometer with a sample, and τ 1 , which depends on the VUCN storage time in the spectrometer, are parameters. Further data treatment uses these values. V. CALCULATION OF THE SMALL HEATING PROBABILITY

As follows from its structure, the BGS makes it possible to measure the probability of the small heating of UCNs with an initial energy below the gravitational barrier E bound to a “visible” energy range E bound < E VUCN < H max . So defined, the probability P+ is Ndet NVUCN = . (1) P+ = Ncoll Ncoll ε Here, N coll is the total number of UCN collisions with a sample surface per “effect measurement” stage, typically 103 -104 , N VUCN is the number of VUCNs generated, N det is the number of VUCNs detected, and ε is the efficiency of VUCN detection. In its turn N coll , dS Ncoll = ∫ ΦUCN (h) dh = Seff ΦUCN (0) . dh Here, S is the sample surface area, S eff is the so-called effective sample surface area, and ΦUCN (h) is the UCN fluence per measurement at a height h (h = 0 at the monitor hole height). Provided UCN angular distribution is isotropic for every energy, ΦUCN (h) changes linearly as a function of h. Thus, with known a UCN spectrum and sample geometry, we can calculate S eff . This definition of S eff does not take into account a dependence of P+ on UCN energy. Thus, for samples of the same material at different heights, we may obtain different probabilities of small heating. To lessen the related uncertainty, the height of all samples should be small enough compared to the UCN energy. Then S eff ≈ S (the dependence of P+ on UCN energy can be taken into account within particular small heating models; they are not considered here).

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Formula (1) can be represented as P+ =

Ndet Smon × K. Nmon × ε Seff

(2)

Here, N mon is the number of counts in the monitor detector during “effect measurement” and S mon is the surface of the hole in the monitor valve. The ratio Nεdet in this formula is the result of averaging over an unknown VUCN spectrum,   dNdet 1 Ndet = dE = JVUCN (E)dE, ε dE ε(E) where E is the VUCN energy and J VUCN (E) is the differential VUCN spectrum. dNdEdet and ε(E) are measured values. We can measure the dependence ε(E) with sufficiently high accuracy, see Fig. 7; however, the measurements of differential spectra dNdet dE are more complicated and are discussed below. K in formula (2) is a correcting coefficient which takes into account the procedure for evaluating the number of VUCNs generated. For K = 1, formula (2) is correct only for the infinite duration of the “effect measurement” stage. If we were able to eliminate, at some point, all the UCNs from the spectrometer without affecting the VUCNs, the detector would continue to count only residual VUCNs. In fact, counting stops simultaneously in the detector and in the monitor. Thus, some VUCNs remain uncounted; accordingly, formula (2) underestimates the probability P+ . To compensate for this effect, we introduce a coefficient K > 1, which depends on the “effect measurement” duration and the UCN storage time in the spectrometer with a sample. K is easily calculated by fitting the data with exponents (see Fig. 6). For the data in this figure, K = 1.08. Note that introducing the ratio of simultaneous detector and monitor count rates (provided they have started to be proportional to each other; see Fig. 6) to the formula (2) instead of introducing the ratio of total counts per “effect measurement,” overestimates the value of P+ . The reason is that a VUCN

FIG. 7. Efficiency ε(E) as a function of energy. The points correspond to experimental values measured with different samples: empty circles for the empty spectrometer, solid circles for a thin Fomblin sample with a surface area of 0.74 m2 , and squares for a sample of nano-diamond powder with a surface area of 150 cm2 . Solid curves indicate calculations for the absorber height Hmax = 140 cm. The dashed curve indicates calculations for the absorber height Hmax = 70 cm.

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reaches the detector a few tens of seconds after its generation, when the UCN flux is higher. VI. MEASUREMENT OF THE EFFICIENCY OF VUCN DETECTION

A generated VUCN can escape via two loss channels: counted in the detector or lost in the spectrometer. Therefore, the efficiency of VUCN detection [see Eq. (2)] is ε=

−1 τdet −1 τstor

+

−1 τdet

=

τstor . τstor + τdet

(3)

Here, τ det is the VUCN emptying time from the spectrometer to the detector and τ stor is the VUCN storage time in the spectrometer (both inside and outside the gravitational barrier, i.e., E bound < E VUCN ) with the exit valve closed. In Eq. (3), both values, τ det and τ stor , are averaged over the VUCN spectrum, i.e., not measured directly. Instead, the measured values are τ det (E) and τ stor (E), the storage time and the emptying time for VUCNs with an energy E. Accordingly, the dependence ε(E) is also measured, ε(E) =

τstor (E) . τstor (E) + τdet (E)

(4)

The procedure for measuring ε(E) is rather timeconsuming and requires much effort, wherein only part of the information obtained is used for calculating P+ . We describe it in detail in Appendix C; here, we present only the results, see Fig. 7, and note that the efficiency (4) decreases as the absorber height increases because the storage time decreases and the emptying time increases (as shown in Figs. 16 and 17, Appendix C). This figure presents the results of the measurements of efficiency ε(E) for the empty spectrometer, the spectrometer with the thin Fomblin sample and the nano-diamond sample. The characteristic time when jumping over the gravitational barrier τ bound (E) is much smaller than τ det (E) and τ stor (E) for neutrons with the energies used in the measurement. The calculations, shown in Fig. 7 with lines, are based on measured τ det (E) and τ stor (E) and calculated τ bound (E). τ bound (E) is easy to calculate analytically for the isotropic angular distribution of the neutrons, which is provided here with good accuracy. τ bound (E) is comparable to τ det (E) for E ≈ 40 cm and increases rapidly to infinity as the neutron energy decreases down to the energy of the gravitational barrier at the height of 35 cm. At this point, the neutrons cease to jump over the barrier and the efficiency drops sharply to 0. The main information that we use here is the value of ε(E) at its maximum, ε max , which is reached for all samples at the energy E = 42 cm. Even supposing that we really do not know the VUCN spectrum, we nevertheless know that for any spectrum, the value of P+ , calculated with ε = ε max , is smaller than the true value of P+ . In other words, using this value of the efficiency, we estimate the lower limit of the probability of small heating. As seen from Fig. 7, for the empty spectrometer ε max = 77%, for the sample of thin Fomblin ε max = 75% and for nanodiamonds ε max = 53%.

The calculation of efficiency for energies greater than the absorber height takes into account the properties of the absorber, as described in Appendix A. Figure 7 reveals discrepancies between the results of calculations and measurements for Fomblin at the energy E = 120 cm. The reason is that neutrons with an energy above 105 cm, the optical potential of Fomblin, penetrate deep inside the oil bulk, and thus the efficiency decreases rapidly. Meanwhile, this effect does not modify the ε max value. The upper limit of P+ can also be estimated by measuring the total loss of UCNs on the sample in the same geometry at which small heating is measured. Indeed, all VUCNs with an energy insufficient to jump over the gravitational barrier are stored in the spectrometer exactly as their “parents” are, and hence they are not detected when we measure small heating. Within this approach, we consider that VUCNs which can jump over the barrier are lost outside the barrier. This measurement is not useful for solid samples because the probability of small heating occurring on them is much smaller than the probability of total UCN loss. For powders of nanoparticles, these probabilities may be comparable, but the powder surface is unstable over time; the probabilities of small heating and loss change. By contrast, the total UCN loss on Fomblin is low and the probability of small heating is large, wherein its surface does not change over time. Therefore, such measurements with Fomblin are meaningful. The calculation of P+ is performed for ε = ε max and the VUCN energy “mono-line” E = 42 cm. For other spectra, VUCN efficiency would decrease and thus P+ would increase. We present the measurement of the VUCN spectrum below. Here, we only provide values of ε for each sample; we used them in estimations of the small heating probability in accordance with formula (2), as follows: the empty spectrometer: ε = 0.72; a sample of thin Fomblin (S = 740 cm2 ): ε = 0.7; a sample of nanodiamond (S = 150 cm2 ): ε = 0.5. For the same energy of VUCNs, τ det (E) depends only on the spectrometer geometry and the output neutron guide diameter. As these do not change, τ det (E) does not change either. In turn, τ stor (E) depends on the state of the sample surfaces and the spectrometer walls, on the sample surface area and the temperature at which the measurement occurs. Usually, τ det (E) < τ stor (E), so there is no need to measure the efficiency again if the changes in τ stor (E) are small (for a certain energy), but if there is a significant change, a new measurement of ε(E) is needed. VII. EXPERIMENTAL RESULTS A. Probability of small heating at ambient temperature

Below are the results of the measurement of the lower limit of UCN small heating probability P+min for the following samples at a temperature of 24 ◦ C: A sample of thin Fomblin (S eff = 0.97 × S = 7180 cm2 ): P+min = (1.05 ± 0.20) × 10−5 . A sample of thin Fomblin (S eff = 0.97 × S = 1598 cm2 ): P+min = (1.10 ± 0.06) × 10−5 . A sample of thick Fomblin (S eff = 0.9 × S = 3690 cm2 ): P+min = (1.11 ± 0.0.04) × 10−5 .

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As follows from these data, the probability of small heating on Fomblin does not depend on the layer thickness. We performed measurements with samples of different surface areas to validate the results. It should be noted that the results of the measurements with Fomblin are very stable over different samples and do not change over time. By contrast, the results of the measurements with nanopowders evolve over time. Thus, the value of P+min decreased smoothly by ∼20% within a day in measurements involving layers of nanoparticles with the thickness >1 mm. It decreased even faster for thinner samples. Apparently, this is due to weakly bound nanoparticles, which acquire stronger ties with neighboring nanoparticles over time so that the amplitude of their oscillations is reduced and the probability of inelastic scattering of UCNs on them decreases. For this reason, small heating probability values for the same powder of nanoparticles may differ significantly. The results of measurements of P+min for the following samples of nanoparticles: A sample of diamond nanoparticles (S eff = 1.05 × S = 158 cm2 , averaged over 10 h): P+min = (1.60 ± 0.05)×10−3 . A sample of diamond nanoparticles (S eff = 1.05 × S = 52 cm2 , averaged over 12 h): P+min = (8.4 ± 0.3) × 10−4 . These samples were placed 1.5 cm below the monitor hole; therefore, S eff > S. The internal surfaces of the spectrometer walls also generate small heating of UCNs; this is the background for measurements with samples: The empty spectrometer (S eff = 4214 cm2 ): P+min = (1.70 ± 0.09) × 10−6 . We measured repeatedly with the empty spectrometer; throughout the entire experimental program discussed in this paper, the results were stable. B. VUCN spectra det [see formula (2)] as a funcFigure 8 shows the ratio NNmon tion of the absorber upper position, H max , where it remains

N

det FIG. 8. The ratio Nmon as a function of the absorber upper position. Squares correspond to measurements with the empty spectrometer. Empty circles indicate data for Fomblin. Solid circles show results for nanoparticles.

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during the “effect measurement;” the absorber lower position, H min , was equal to 32.5 cm, as previously. All data are normalized at the height H max = 140 cm in order to facilitate comparison. The data are similar to an integral spectrum of VUCNs, but not exactly so. Neutrons with an energy above E are absent in an integral F VUCN (E) spectrum; here, a certain fraction of such neutrons are present because a neutron with energy >H max can pass the absorber and reach the detector (see Fig. 7, dashed line). Such a probability is relatively small for large heights H max , but it can make a significant contribution to the number of detected VUCNs for small heights H max . det (Hmax ), shown in Fig. 8, To convert the dependence NNmon into the spectrum P+ (E), we must get rid of excess neutrons, det (E), and account for the calculate an integral spectrum NNmon efficiency of detection ε(E). To do that, we must differentiate the integral spectrum, dividing it by ε(E) and then integrating. With no a priori information about the spectrum shape, this procedure is not precise enough. As the VUCN energy is det (Hmax ) continreduced, the derivative in the dependence NNmon ues to grow and the accuracy of the measurements decreases. Thus, the statistical uncertainty is high at energies making a large contribution to the total number of detected VUCNs. When approaching the gravitational barrier, the systematical uncertainty and the role of absorber properties and positioning increase (see Appendix A). To obtain absolute values of small heating probability, another option is possible: if we produce a model of the differential VUCN spectrum, we can multiply it by ε(E) and integrate. Then we compare the result with the data in Fig. 8 and thus obtain a model-dependent value of P+ . This method can help us to clarify the nature of small heating as it allows very precise calculations; we intend to implement it in the near future. Here, we use a different approach. Dependences Ndet (H max ) for all the nanoparticle samples measured are staNmon tistically indistinguishable. Figure 8 therefore presents the averaged result over all such measurements. Data for thin and thick Fomblin layers are also statistically indistinguishable; we therefore average them. Moreover, the data in Fig. 8 are surprisingly similar to each other for solid, nanoparticle, and liquid samples. This universality is natural within the model of inelastic UCN scattering on nanoparticles.16 Assuming the same VUCN spectrum in all cases and the proportionality of P+ to ε, we can calculate a mean efficiency ε [in formula (2) for P+ ] for one sample and thus evaluate it for other samples. For identical VUCN spectra, relative systematic errors in estimations of ε will be the same for all samples. An important conclusion follows from the shape of dependet (Hmax ): the UCN small heating observed occurs dence NNmon in one step; it is not the result of a multi-stage process. Indeed, if we assume that small heating is the result of, say, a two-step process and take into account that the probability of VUCN generation to energies above 80 cm is P+ ∼ 10 6 , then the probability of one-step energy change would be ∼10 3 , which corresponds to a much sharper VUCN spectrum than the one observed. In addition, such a probability would exceed the total UCN loss probability, which is impossible.

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C. Small heating as a function of Fomblin temperature

We measured the probability of UCN small heating on Fomblin as a function of temperature; Fig. 9 shows the probability P+ for a sample of thin Fomblin. We assume the efficiency of VUCN detection to be equal to 0.7 in the entire temperature range; as stated above, this assumption is valid when the UCN storage time τ stor does not change significantly. Thus, in our case, when the temperature decreases from 24 ◦ C to 18 ◦ C, the UCN storage time increases by ∼10%. Accordingly, the expected efficiency increases up to ∼0.72 [see formula (3)]. Such a correction is not relevant within the available statistics and does not justify dedicated measurements at each temperature. In Fig. 9, we show the results of measurements during cooling and heating of the spectrometer separately. These results are in good agreement with each other; some differences may be due to temperature gradients as only one thermometer was used. The solid line in this figure shows an approximation of the data with exponent e T /a . The temperature dependence is quite strong: at the ambient temperature, a 1◦ temperature change results in a change in small heating probability of 5%; we will use this fact below. D. The probability of total UCN loss on Fomblin: Estimation of the upper bound for the probability of small heating

As mentioned above, an upper limit for the probability of small heating is the probability of the total loss of UCNs in their interaction with the Fomblin surface, measured in the same geometry. Such a measurement is simple. A loss probability, µ, on a sample is calculated from the difference in reciprocal storage times of UCNs in the spectrometer with/without −1 −1 a sample: τsample = τ0+sample − τ0−1 , where τ sample is the partial UCN storage time associated with the sample, τ 0+sample is the UCN storage time in the spectrometer with the sample, and τ 0 is the UCN storage time in the empty spectrometer.

FIG. 9. The small heating probability on a Fomblin oil surface as a function of temperature. Black squares correspond to measurements at a temperature of 24 ◦ C; round solid points indicate data measured during cooling; round empty points show results measured during heating; the solid curve is an exponential approximation of the data.

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With the spectrometer and sample geometries and the UCN spectrum known, we can calculate the frequency of UCN collisions with the sample surface, ν. The loss probability is µ = τ sample × ν. As the storage time in the spectrometer with copper walls is smaller by an order of magnitude than that on the Fomblin sample, we covered, in particular for this measurement, the part of spectrometer inside the gravitational barrier with Fomblin. This modification does not affect the UCN spectrum but allows us to achieve the required statistical accuracy more quickly. We measured the loss probability at three temperatures; Fig. 10 shows the results. The solid line in this figure shows the temperature dependence of small heating if all losses were due to small heating. The measured total probability of UCN loss depends to only a small degree on temperature unlike small heating probability. In any case, these other factors cause the probability of loss to decrease (or at least remain constant) with decreasing temperature. Assuming that this probability is constant over temperature and subtracting it from the loss probability at the temperature 24 ◦ C, we get an upper limit P+max = (1.31±0.24)×10−5 . A lower limit for the probability of small heating calculated with the efficiency ε = ε max = 0.75 is P+min = (0.98±0.02)×10−5 . As the VUCN spectrum (see Fig. 8) is obviously not a “mono-line” with the energy E = 42 cm, where the function ε(E) reaches its maximum, one can reasonably estimate that the efficiency for calculating P+min is slightly smaller ε = 0.7. Therefore, the probability P+ of small heating on Fomblin at the temperature 24 ◦ C is found in the range (1.05 ± 0.02stat ) × 10 5 (1.31 ± 0.24stat ) × 10 5 . As the VUCN spectra are the same for all samples, the upper limit P+ is always higher by about a factor of 1.3 than the lower limit. Note that in the data used in Fig. 10, the losses associated with small heating amount to one-third of total losses. Thus, an increase of the gravitational barrier height would significantly reduce the contribution of small heating to total losses.

FIG. 10. The probability of total UCN loss on Fomblin as a function of temperature. Points correspond to the measured results. The solid curve indicates the probability of small heating as a function of temperature.

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VIII. CONCLUSION

The method presented using the BGS spectrometer allows us to perform precise and reliable measurements of the probability of the small heating of UCNs associated with their interaction with a surface. In particular, we measured the probability of the small heating of UCNs on the surface of the hydrogen-free oil Fomblin Y-HVAC 18/8 as a function of temperature. The data obtained are well reproducible, do not change as a function of the sample thickness, and do not evolve over time. The measured model-independent probability P+ of UCN small heating from the energy “mono-line” 30.2 ± 2.5 cm to the energy range 35–140 cm is found to be in the range (1.05 ± 0.02stat ) × 10 5 (1.31 ± 0.24stat ) × 10 5 at a temperature of 24 ◦ C, in agreement with results of the first experiment (1). The systematic uncertainty would disappear if the shape of the VUCN spectrum were known (for instance, within a particular model of small heating). It should be noted that the energy range of BGS sensitivity is wider than that in alternative methods used and is directly adjacent to the UCN spectrum. It should also be noted that we do not use assumptions in our analysis but instead measure all the relevant values directly. This experiment therefore provides the most precise and reliable results on small heating on Fomblin measured so far. There are at least two hypotheses that can describe the phenomenon of small heating on Fomblin: the model of inelastic UCN scattering on surface capillary waves18 and the model of inelastic UCN scattering on near-surface nanodroplets.16 Comparison of the temperature dependence of small heating, the VUCN spectrum, and the total probability calculated within these models with measured experimental results could help to reveal a dominant mechanism. In neutron lifetime experiments with such Fomblin oil coatings, it is obvious that UCN traps must be cooled down; cooling down to a temperature of 20 ◦ C decreases the small heating probability by a factor of over 10. It would be interesting to measure the probability of small heating on oils with large molecules with a correspondingly higher viscosity and lower vapor pressure. The probability of small heating on such oils might appear to be significantly lower even at ambient temperature. By contrast, the probability of small heating on the so-called low-temperature oils might appear to be significantly larger. The BGS makes it possible to measure a small heating probability as low as ∼10 8 per bounce, i.e., with a sensitivity two orders of magnitude higher than in alternative methods.

ACKNOWLEDGMENTS

The authors are grateful to S. N. Chernyavskiy and K. N. Zhernenkov for the useful discussions and help in the preparation of this paper, and also to P. Geltenbort, T. Brenner, and A. Elaazzouzi for their help in the experiment preparation.

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APPENDIX A: SELECTION OF MEASURING CYCLE PARAMETERS: ABSORBER PROPERTIES

Prior to starting measurements of P+ , the parameters of the measuring cycle in Fig. 6 must be selected, e.g., the durations of each stage of the cycle—“filling,” “cleaning,” “effect measurement,” and “emptying”—must be defined. The durations may vary depending on the UCN storage time in the spectrometer with the sample. The lower height of the absorber must also be defined for all measurements. To evaluate the duration of the “filling” stage, we set the absorber to a certain position below the gravitational barrier and count the number of UCNs accumulated in the spectrometer as a function of filling time. Then we select the duration of this stage so that the number of accumulated neutrons approaches its saturation value. The UCN storage time in the spectrometer with the sample is then measured as follows: With the same absorber position below the gravitational barrier, the spectrometer is filled with UCNs; the valve in the input neutron guide is then closed and the UCNs are stored in the spectrometer for an interval t 1 , which is sufficient to eliminate a major portion of the UCNs with energies larger than the absorber height. The absorber is then lifted and the monitor valve opened so that the UCNs which have survived can pass from the spectrometer to the monitor detector. Let the number of UCNs counted in the monitor be N 1 . The same procedure is then repeated for another duration t 2 > t 1 and another number N 2 is counted. The UCN storage time, τ UCN , satisfies the formula ! t2 − t1 . N2 = N1 exp − τUCN The duration of the “cleaning” stage is selected as follows: On the one hand, UCNs with energies above the gravitational barrier must be eliminated during the “cleaning” stage. In this case, the requirement is much more severe than when measuring UCN storage time. Indeed, a small addition to the number N 1 does not affect the storage time value very much. When measuring a very small value P+ , even a small fraction of UCNs may exceed the number of VUCNs. On the other hand, after excessive cleaning times, the UCNs in the spectrometer simply would not survive. Therefore, the cleaning time is usually set to equal 0.5 × τ UCN . The following stage is “effect measurement.” Its duration should be set so that the coefficient K in formula (2) is not too high, say 5 or equals 2 × τ UCN . The final “emptying” stage cleans the spectrometer of all residual UCNs. Emptying starts by opening the monitor valve. It is possible to skip this stage and simply wait until the absorber has moved down to the lower position. In this case, however, it is important to make sure that the next cycle starts promptly. Otherwise, the number of UCNs filling the spectrometer would vary over cycles, which is not always acceptable. The last, very important, preparatory measurement consists of selecting the lower height of the absorber in

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FIG. 12. The absorber profile; sizes are in mm.

FIG. 11. The number of neutrons counted in the detector with the empty spectrometer during “effect measurement.” Points correspond to the data. The solid line is a linear fit to the data. The dotted line indicates the results of calculations of the number of UCNs with an energy above the gravitational barrier, which survived in the spectrometer after “cleaning.”

measurements of P+ . To do this, we measure the probability of small heating, with all the intervals already chosen for the other stages, as a function of the lower height of the absorber, H min . Figure 11 presents the results of this measurement performed with the empty spectrometer. As can be seen from the figure, at heights H min below 34 cm, the data follow a slowly changing linear dependence associated with the small heating of UCNs. A further increase of H min results in a sharp increase associated with the residual UCNs, which have survived in the spectrometer during “cleaning” with energies above the gravitational barrier. The dotted line in Fig. 11 corresponds to the calculation of the number of such residual neutrons performed using the absorber properties, as described below. The height H min is set for all measurements so that the neutrons which have survived are negligible for measurements of P+ . It should be noted that the number of neutrons to survive depends on the duration of “cleaning.” Therefore, if, for some reason, the “cleaning” time changes, the measurements of H min must be repeated. An important element of the spectrometer is the absorber. The ideal material for the absorber would have zero optical potential; if this were the case, any neutron touching it would penetrate inside and be lost. In reality, this is not possible. We used polyethylene with a density of 0.86 g/cm3 , corresponding to a small optical potential equal to 8.06 neV. Although this potential is negative, UCNs with a kinetic energy at the absorber height smaller than ∼8 cm are efficiently scattered on its surface due to quantum reflection; the smaller the energy of the UCN, the greater the probability of reflection. Increasing the absorber surface area allows the neutron to try several times to penetrate into the material on a single approach to its surface. This modification improves the “absorbing” properties of the absorber significantly and reduces the invisible part of the VUCN spectrum between the upper cutoff of the UCN spectrum and the gravitational barrier. Figure 12 shows the surface profile of our absorber.

The dashed line in Fig. 11 presents the calculation made for this absorber. Despite the fact that, visually, it describes the data well, there are differences. The rate of increase of the number of surviving neutrons is greater in the calculation than in the measurement; this is probably due to the finite precision of measuring the absorber height. We calculate this from the angle of rotation of a step motor connected to a drum, with steel cables winding around it, from which the absorber hangs. This mechanism provides high relative accuracy and reproducibility over cycles, but lower absolute accuracy ±1 mm, measured at several points, with the horizontality ±1.5 mm. The absorber weight is >10 kg. When its internal part moves below the gravitational barrier, its external part lies on the barrier edge. When this happens, the load decreases and the cables shorten, and the absorber height is therefore underestimated. Conversely, when the absorber is raised, the cables elongate at heights >34 cm, as both parts start moving together. We estimate that the associated uncertainty is equal to ±5 mm. It should be noted that although this effect does have certain drawbacks and in particular affects measurements of the VUCN spectra, it does not affect P+ . Indeed, we use virtually none of the model calculations associated with the absorber height and properties in measurements of small heating probability. All the model results presented serve as an illustration. APPENDIX B: UCN SPECTRUM

As mentioned above, the probability of small heating and the frequency of UCN collisions with a sample surface depend on the UCN spectrum. We therefore formed the narrowest UCN spectrum to minimize its effects. The edge of the input neutron guide at the height of 25 cm from the bottom of the spectrometer defines the lower cutoff of the UCN spectrum. The absorber at the height of 32.5 cm defines the upper cutoff in all measurements of P+ . Figure 13 presents the results of measurements of the number of UCNs, which survived in the spectrometer, with a Fomblin oil sample with an area of 0.74 m2 , after “cleaning,” as a function of the absorber height below 32.5 cm. These measurements are performed as follows: the duration of “filling” and “cleaning” is the same as in measurements of P+ , but the absorber is below 32.5 cm. Then the absorber is moved up, the monitor valve is opened and the UCNs, which have survived in the monitor detector, are counted. From such measurements with different absorber heights, it is possible to

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FIG. 13. The number of UCNs counted as a function of the absorber height, H. The line is the result of fitting with a function (H 25 cm)k .

obtain a dependence close to the integral UCN spectrum; this dependence closely follows a power function (H 25 cm)1.7 (line in Fig. 13). The derivative of this function provides the “differential spectrum” of UCNs shown with a solid line in Fig. 14 below 32.5 cm. Such a simplified approach to estimating the differential UCN spectrum does not take into account UCNs with an energy above the gravitational barrier (which survived in the spectrometer after cleaning) at each point of the “integral spectrum” in Fig. 13. It also does not take into account the energy dependence of the UCN emptying time to the detector and the energy dependence of the spectrometer filling time. Nevertheless, the result is good. Figure 14 compares the differential spectrum evaluated using this simplified method with the calculation, which takes into account precisely the spectrometer geometry. In this calculation, we assume isotropic angular distributions

FIG. 14. The differential spectrum of UCNs in the spectrometer with a sample of Fomblin with the surface area 0.74 m2 . The solid line below 32.5 cm corresponds to the function (H 25 cm)0.7 . The thick dashed line presents the calculation of the UCN spectrum below 32.5 cm in the spectrometer after “filling” it for 60 s. The thick solid line above 32.5 cm indicates the calculation of the spectrum of neutrons which survived after “cleaning” with energies above the absorber height 32.5 cm. The thin dotted lines designate the confidence interval for the absorber height ±1.5 mm.

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of UCNs in the spectrometer and uniform filling of the phase space √ for UCNs at the entrance to the spectrometer, dN ∼ (H − 25 cm). We assume the UCN storage time in i.e. dH the spectrometer to be equal for all energies to the value measured with this sample: 148 s; this inaccuracy is minor because of the narrow UCN spectrum. The calculation of the neutron spectrum for energies above the absorber height takes into account the properties of the absorber (see Appendix A). The positions of the thin dotted lines, indicating the confidence interval, are not associated with the instability of the absorber height but with the uncertainty of the height measurement, equal here to ±1.5 mm. High precision is not required for UCN spectrum measurements. We only use information on UCN energy in calculations of the probability of small heating to evaluate the frequency of UCN collisions with the sample surface, the probability of total loss, and for calculating S eff . Thus, a 10% change in the UCN energy changes the frequency of UCN collisions with the surface of a sample placed at the bottom of the spectrometer by 3%. Simulation showed that, for all the samples considered in this work, P+ calculated with a mean UCN energy in Fig. 14 differs from the precisely calculated value by E bound = 35 cm) in the spectrometer with the sample, and τ det (E), the time to empty UCNs with the same energy to the detector. As mentioned above, τ det (E) depends only on the spectrometer geometry and the area of the exit to the detector. These parameters are always the same. Nevertheless, it is virtually impossible to calculate them precisely because of the complexity of UCN transport to the detector (involving scattering on rough surfaces of the detector valve, output guide, etc.). τ stor (E) depends on the state of the sample surfaces and spectrometer walls and on the sample surface area and temperature. We therefore measure both these values. The procedure for measuring τ stor (E) is the following: The absorber is raised to a height H above the gravitational barrier and the “filling” and “cleaning” stages are performed with the detector valve closed. The absorber is then lifted upwards. After a delay t 1 , the detector valve is opened and neutrons with an energy higher than the gravitational barrier are counted in the detector; the number of neutrons counted is denoted as N 1 (H). This procedure is repeated with the absorber installed at a different height H + ∆H; the number of neutrons counted is N1 (H + ∆H). The difference ∆N1 = N1 (H + ∆H) − N1 (H) is the number of neutrons in the UCN spectrum with energies in the range from H to H + ∆H. The whole procedure is repeated with the same heights H and H + ∆H but with a different delay t 2 ; the number of

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FIG. 15. The detector count rate in the measurement of ∆N1 for heights H = 70 cm (open triangles) and H + ∆H = 90 cm (open circles). Black points correspond to the difference between these count rates. The line shows the approximation of the data with exponent.

neutrons counted is ∆N2 . Using the formula ! t2 − t1 ∆N2 = ∆N1 exp − , τstor (E) we obtain τ stor (E) assuming E = H + 21 ∆H. This choice of E is associated with a systematic shift related to the non-linearity of the UCN spectrum and an admixture of a fraction of the neutrons with an energy above H to numbers N(H) and N(H + ∆H). For heights below 70 cm, we usually set the interval ∆H = 10 cm; for heights above 70 cm, the interval is ∆H = 20 cm. The duration of cleaning in all cases is not shorter than 50 s. Simulations show that corrections, accounting for both these effects, are below 1 cm for the height H = 50 cm and below 1.5 cm for the height H = 100 cm, thus negligible. It should be noted that the duration of all stages of the measuring cycle for measurements of τ stor (E) may be

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FIG. 17. UCN emptying time to the detector, τ det , as a function of energy E for E > 35 cm. The line approximates the data with a smooth function.

different from the duration of these stages when measuring P+ . However, when measuring ∆N1 and ∆N2 they must match exactly. The value τ det (E) can be estimated from an analysis of the detector count rate in measurements of τ stor (E). Figure 15 shows the detector count rate in the measurement of ∆N1 for heights H = 70 cm and H+∆H = 90 cm. Here, the detector valve is opened at 180th s; before this, leakage of UCNs through slits in the detector valve defines the detector count rate. The dN1 1 black points in this figure show the difference ∆ dN dt = dt (H + dN1 −t/τ ∆H) − dt (H) as a function of time. Exponent e s+d can fit 1 the data ∆ dN dt (t); here, τ s+d is the UCN storage time for the spectrometer with the detector valve open, i.e., −1 −1 −1 τs+d = τstor + τdet .

Thus, with a known τ stor , we obtain τ det . Figure 16 presents the results of measurements of τ stor (E) for two samples and the empty spectrometer, performed at a temperature of 24 ◦ C. Figure 17 presents the results of measurements of τ det (E). As this function does not change and is a property of our spectrometer, we average the data over a few tens of measurements. A function in the form c × E k can fit the data. In calculations of ε(E) in formula (4), we use the results of this fit: τ det (E) = 4.7 × E 0.62 . We do not show the confidence interval for the fitting function in Fig. 17, as its effect on the uncertainty of ε(E) is negligible. 1 V.

FIG. 16. UCN storage time, τ stor , as a function of energy E for E > 35 cm. Open circles indicate data measured with the empty spectrometer, black circles show data measured with a thin Fomblin sample with the area 740 cm2 , and black squares correspond to measurements with a sample of nanodiamond with the area 150 cm2 .

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