Scattering of ultracold neutrons from rough surfaces of metal foils
Stefan Döge, Jürgen Hingerl, Egor V. Lychagin, Christoph Morkel
SScattering of ultracold neutrons from rough surfaces of metal foils
Stefan Döge,
1, 2, 3, ∗ Jürgen Hingerl,
1, 2
Egor V. Lychagin,
3, 4, 5 and Christoph Morkel Technische Universität München, Department of Physics E18,James-Franck-Strasse 1, D-85748 Garching, Germany Institut Laue–Langevin, 71 avenue des Martyrs, F-38042 Grenoble Cedex 9, France Frank Laboratory of Neutron Physics, Joint Institute of Nuclear Research (JINR),6 Joliot-Curie Street, Dubna, Moscow Region, Ru-141980, Russia Lomonosov Moscow State University, GSP-1, Leninskie Gory, Moscow, Ru-119991, Russia Dubna State University, Universitetskaya Street 19, Dubna, Moscow region, Ru-141982, Russia Technische Universität München, Department of Physics E21,James-Franck-Strasse 1, D-85748 Garching, Germany
The transparency of metal foils for ultracold neutrons (UCNs) plays an important role in thedesign of future high-density UCN sources, which will feed a number of fundamental physicsexperiments. In this work, we describe and discuss the measured transmission of a collimatedbeam of very slow neutrons (UCNs and very cold neutrons) through foils of Al, Cu, and Zr ofvarious thicknesses at room temperature. Our goal was to separate scattering and absorption inthe sample bulk from surface scattering, and to quantify the contribution of the surface. We wereable to demonstrate that the surface roughness of these foils caused a significant fraction of UCNscattering. The surface roughness parameter b extracted from UCN measurements was shown to beof the same order of magnitude as the surface parameter determined by atomic-force microscopy.They lie in the order of several hundreds of angstroms. Using the formalism developed here,transmission data from previous neutron-optical experiments were re-analyzed and their surfaceroughness parameter b was extracted.Published online on 9 December 2020: https://doi.org/10.1103/PhysRevC.102.064607 S. Döge et al., J. Hingerl, E. V. Lychagin, C. Morkel, Physical Review C 102 (6), 064607 (2020)© 2020. This manuscript version is made available under the CC-BY 4.0 license.
PACS numbers: 03.75.Be, 25.40.Dn, 28.20.Cz, 83.85.Hf
I. INTRODUCTION
Metal foils are frequently used in experiments whereultracold neutrons (UCNs) need to pass but the vacuumof different volumes must be separate, e.g., the vacuumat the PF2 instrument “Turbine” at the Institut Laue–Langevin in Grenoble, France, [1, 2], and the neutronbeam guide vacuum. They also play an important roleas neutron exit windows in the exploitation of other UCNsources [3–5], such as those based on solid deuterium orliquid helium. To extract a maximum UCN flux fromthese sources, UCN losses on these exit windows need tobe as low as possible.The fact that the surface roughness of foils has atremendous effect on UCN transmission through mediahas been shown by Steyerl [6] and Roth [7] but has, ap-parently, found no attention so far.Studies of exit window materials for the UCN sources“Mini-D2” [8] in Mainz, Germany, and the one at thePaul Scherrer Institut [9], Switzerland, investigated thebest choice of materials for this application. Aluminumand zirconium were identified as the best candidates dueto their low neutron absorption cross section. Thesestudies used thin metal foils in the range of a few hun-dred micrometers thickness. They noticed a significantly ∗ Corresponding author Stefan Doege: [email protected] lower transmission of UCNs than expected. Besides neu-tron absorption by strongly absorbing isotopes presentas trace impurities in the samples, surface scattering wasconjectured to be one of the causes but no quantitativeexplanation or estimate was provided.As Lavelle et al. [10] demonstrated experimentally,the rough aluminum windows of their sample containercaused UCN losses of up to 3/4. Alas, this effect wasnot further studied and has not been taken into accountquantitatively in the literature on UCN transmission ex-periments, in which rough-surface sample containers wereused. Having recognized this, we developed and used alow-roughness sample container [11]. The present papergives an impression of the magnitude of measurementuncertainty for UCN cross sections when rough samplecontainers are used instead of low-roughness containers.The following sections deal with the purpose of thispaper (Sec. II), UCN losses in the sample bulk and theircalculation (Sec. III), and UCN losses on the sample sur-face (Sec. IV), which are calculated as the difference be-tween the measured total UCN losses and the known losschannels in the sample bulk. The experimental setupsfor both the neutron and AFM measurements are given(Sec. V), followed by a discussion of the results, their ap-plication to previously performed experiments, and theirimplication for UCN transmission measurements in gen-eral (Sec. VI). a r X i v : . [ nu c l - e x ] J a n II. IDEA OF THE EXPERIMENT
The laws of neutron optics govern the transmission ofultracold neutrons through metal foils. We hypothesizethat the foil thickness has only a very limited influenceon the total transmission – and thus the loss – of UCNs.Much more important is the surface roughness of the foil.The exact interaction of rough surfaces with neutronbeams depends on the beam distribution – collimated,isotropic, or some other shape. For exit foils and UCNtransmission experiments, the attenuation of collimatedbeams is important and thus the subject of the presentpaper. In this context, a UCN is considered lost whenit is absorbed by a nucleus of the sample bulk, gainsenergy through an up-scattering event and leaves theUCN energy range, or when it is diverted from the col-limated neutron beam by elastic scattering. In UCNstorage experiments, for example, neutrons impinging onthe surface of storage vessels are not collimated and un-dergo both specular and diffuse reflection. Their lossis only due to absorption or up-scattering by the ves-sel’s walls [12, 13] or by impurities on the wall’s sur-face [14, 15]. However, surface roughness can, depend-ing on the conditions, decrease or increase the probabil-ity of UCN loss by absorption upon a collision with thewall [16].In Frei [8] and Atchison et al. [9], the different sam-ple thicknesses were achieved by layering several thinnerfoils on top of each other. By doing that, the thicknessof the bulk material was increased, but so was the num-ber of surfaces. It was, therefore, not surprising that thethickness-dependent transmission curves showed an ex-ponential decay with increasing foil thickness, or better:with an increasing number of foils.To gain a better understanding of UCN scattering onrough surfaces, UCN transmission experiments throughmetals foils of the same bulk material and surface prepa-ration but with different thicknesses needed to be carriedout. This way, the bulk thickness – with UCN loss crosssections known from the literature – could be varied whilethe number of surfaces remained constant.
III. ULTRACOLD NEUTRON LOSSES IN THESAMPLE BULKA. Sample impurities and oxidation
We chose high-purity metal foils as samples for our ex-periments to avoid elastic scattering of UCNs on bulk in-homogeneities. The impurities of the various metal foils(Cu, Al, Zr) investigated in this work were taken from thesupplier’s (Advent Research, UK) list of typical impuri-ties, which gave a good estimate of the trace impuritiesto be expected.Taking into account the abundance of these impuri-ties as well as their respective absorption cross sections,only one or two impurities per sample actually influ- enced the total absorption cross section. Table I listsfor each sample its respective most relevant impurities.Since the impurities were very dilute, it was assumedthat they had the same particle density as their host ma-terial multiplied with their respective number concentra-tion. The absorption cross sections at thermal-neutronenergies were taken from Sears [17]. The resulting sumof bulk and impurity absorption cross sections was alsocalculated for thermal-neutron energies. For all samples,the additional absorption cross sections due to impuritieswere less than 4% and they were, therefore, neglected.
Sample Element Number σ thermalabs σ thermalabs type conc. (%) (b) (b) weightedAl 99.99+% Al > 99.99 0.231 0.231Ag 3 × − . × − B 2 × −
767 1 . × − Sum 0.231
Cu 99.9+% Cu > 99.9 3.78 3.78Ag 5 × − . × − Sum 3.81
Zr 99.8% Zr 99.8 0.185 0.185Fe 8 . × − . × − Hf 4 . × −
104 4 . × − Sum 0.192
TABLE I. Sample impurities giving rise to additional absorp-tion cross sections in the metal foil samples i) Al 99.99+%,ii) Cu 99.9+%, and iii) Zr 99.8%. The “+” denotes a samplepurity even higher than indicated. All cross sections are givenfor thermal-neutron energies.
The foils used here had received the following surfacetreatments: Al – temper as rolled, Cu and Zr – temperannealed.When metals like aluminum and copper are exposedto air at ambient temperature, they form passive oxidelayers. For pure bulk copper, a layered CuO/CuO ox-ide structure of 3.3 nm thickness has been reported [18],while 2.5 to 5.2 nm were found for copper thin films [19]and 6 nm for ultrafine particles [20]. The oxide layer onthe surface of pure aluminum has been found to be 4 nmfor ultrafine particles [20] and between 3 and 4 nm forbulk aluminum [21]. On zirconium, ZrO and substoi-chiometric oxides have a thickness of about 1.5 nm [22].These oxide layers are thinner than one typical UCNwavelength, which is a few hundred angstroms. Accord-ing to the UCN reflectivity calculations by Pokotilov-ski [23], they are thin enough to have only negligibleinfluence on the UCN transmission through the sample. B. One-phonon up-scattering
The total UCN cross section of a sample is composedof bulk scattering, surface scattering, and absorption inthe bulk. From Table I, the absorption cross sectionsat thermal-neutron energies ( E kin = 25 meV, v = 2200m/s) can be taken and extrapolated to UCN energiesby using the relation σ abs × v = const., see for exampleIgnatovich [12].Table II lists the one-phonon up-scattering cross sec-tions σ IA1-ph calculated for room temperature using theIncoherent Approximation (IA) [24], the one-phonon up-scattering cross sections σ corrIA1-ph taking into account cor-rections to the Incoherent Approximation (corrIA) byPlaczek and Van Hove [25], the absorption cross sec-tions σ abs [17], and the total UCN loss cross section σ tot for all three metals under investigation – Al, Cu, andZr. The corrections to the IA for coherent scattererscan only strictly be calculated for cubic crystals. SinceZr has a hexagonal close-packed (hcp) crystal structureat room temperature, it was assumed to have a face-centered cubic (fcc) structure for the sake of calculatingthe correction. For all three metals at room temperature,one-phonon up-scattering is not significant compared toabsorption, which is the dominant UCN loss channel. Allcross sections listed in Table II can be extrapolated toother neutron energies using the relation σ × v = const. Sample σ IA1-ph (b) σ corrIA1-ph (b) σ (b) σ (b)Al 7.9 8.8 63.5 72.3Cu 23.5 21.6 1040 1062Zr 15.9 16.7 50.9 67.6TABLE II. One-phonon up-scattering calculated according tothe Incoherent Approximation (IA), IA with corrections forcoherent effects (corrIA), as well as absorption for pure Al,Cu, and Zr, taken from Table I. The right-hand column givesthe total UCN loss cross section as the sum of σ corrIA1-ph and σ abs . All values were calculated for room temperature and anin-medium velocity of v inm = 8 m/s of the UCNs. Coherent elastic scattering in the bulk can be neglecteddue to the UCNs’ large wavelengths, which go far beyondany Bragg cutoff wavelength, and due to the high samplepurity, which avoids scattering on inhomogeneities in thesample bulk. With up-scattering and absorption in thesample bulk being known quantities, the remaining lossof UCNs in the transmission experiment on metal foilscan be attributed to surface scattering only.
IV. ULTRACOLD NEUTRON LOSS ONSURFACESA. Separating bulk from surface losses
The standard transmission equation for uniformly ab-sorbing and scattering media (in optics known as theLambert–Beer law [26, 27]), T = I n I = e − Nσ tot d n , (1) can be expanded by 1 − L t to account for surface scat-tering [6], T = I n I = e − Nσ tot d n | {z } (1 − A ) × (1 − L t ) , (2)where T represents the measured absolute transmissiv-ity of sample n , I the incoming neutron beam inten-sity, I n the neutron beam intensity behind sample n , and σ tot (for in-medium neutron velocity) the total UCN losscross section of the sample bulk, as discussed in Sec. III. A is the loss of UCNs in the sample bulk (between thetwo surfaces) and L t is the integral probability of diffusescattering from rough surfaces, i.e., the UCN loss on thetwo surfaces of one foil. The particle number density N is known from the literature and the sample thickness d n can easily be measured. To determine L t , the transmis-sivity T of a foil needs to be measured. Then, Eq. 2 canbe solved for L t , L t = 1 − T e − Nσ tot d n . (3) B. Connecting surface roughness andultracold-neutron Loss
For UCNs that are incident perpendicularly on a roughsample surface (“macroroughness”) and have a wave vec-tor (out of medium) k (cid:29) | k l | , Steyerl [6] defined the totalfraction of UCNs that is scattered out of the direct beamby the two surfaces of a sample as L t = I I = 14 b k k (cid:0) − Nσ tot d n (cid:1) , (4)where k l = m n × v crit / (cid:126) , m n is the neutron’s mass, v crit isthe critical velocity of the sample, b is the surface rough-ness parameter, I is the neutron beam intensity incidenton the rough surface, and I is the intensity of UCNsscattered out of the direct beam by the two surfaces. Theterm 1 + e − Nσ tot d n (instead of 2) accounts for the lossof UCNs in the sample bulk between the two scatter-ing surfaces. The first surface receives the full incomingneutron intensity, the surface downstream from it sees abeam attenuated by the losses in the bulk – and on thefirst surface. The condition k (cid:29) | k l | is fulfilled for themetals treated here for all except the very slow UCNs of v (cid:46) b as seen by theUCNs can thus be derived by solving Eq. 4 b = s L t k k (1 + e − Nσ tot d n ) . (5)This parameter b is defined as the mean square ampli-tude of elevations above and below the reference plane ofthe surface. For surfaces with a relatively even distribu-tion of peaks and valleys and no extreme peaks, the meansquare amplitude is quite similar to the center-line aver-age roughness R a , which we measured using atomic-forcemicroscopy, see Sec. V B. V. EXPERIMENT SETUPA. Ultracold-neutron transmission experiments
For the transmission measurements with very slow neu-trons with an out-of-medium velocity v oom , 3 ≤ v oom ≤
15 m/s, we used the time-of-flight (TOF) method and acollimated beam of UCNs and very cold neutrons (VCN)at the PF2-EDM beamline [2] of the Institut Laue–Langevin. The neutron beam was strongly collimatedin the forward direction with a solid angle of the collima-tor aperture of Ω = 2 . × − sr. The TOF geometryused in these experiments is explained in detail in Dögeet al. [28]. As sample holder for the metal foils we usedthe one that was developed for UCN transmission exper-iments on liquid and solid deuterium [11]. Aluminumclamps held the metal foils in place during the transmis-sion experiments.The foil samples were cleaned with high-purity ethanoland dried immediately prior to their installation into thevacuum vessel where the measurements took place. Be-fore the UCN measurements were started, the vacuumwas stabilized in the 10 − mbar range for half an hour.The measurements themselves ran over several hours andshowed no significant fluctuation in the UCN transmis-sion over time as the vacuum continuously improved tothe lower 10 − mbar range. It can therefore be consid-ered certain that all water and volatile compounds, whichmay have been adsorbed onto the metal surface betweenthe installation of the sample foil in the vacuum chamberand the start of the evacuation, evaporated and causedno additional scattering of UCNs. B. Atomic-force microscopy measurements ofsurface roughness
The mechanical surface roughness of the metal foilswas measured at JINR Dubna under an atomic-force mi-croscope (AFM) from NT-MDT. The center-line averageroughness, i.e., the average deviation from the imaginarycenter plane of the surface [29], R a = 1 n n X i =1 | y i | (6)for each foil sample was determined from two or moretwo-dimensional scans of 5 µ m × µ m, which were car-ried out on flat sections of overview scans of 30 µ m × µ m. This way, we minimized the role that long-wavelength surface waviness plays in the calculation ofthe surface roughness according to the standards ISO 4287-1:1984 and GOST 25142-82. The AFM was cali-brated by using two different calibration samples madeof SiO with step sizes of 21.5 nm (TGZ1) and 107 nm(TGZ2). These samples were also used to verify the re-liability of the two-dimensional roughness calculation al-gorithm. The global uncertainty of the measured surfaceroughness parameters was ± Sample Foil Thickness Transmissivity Roughness R a ( µ m) at v oom = 7 m/s (nm)Al-1 50 0.967 ± ± ± ± ×
50 0.081 ± ± ± ± ± ± R a values is ± VI. RESULTS AND DISCUSSIONA. Ultracold-neutron transmission measurements
The transmission of UCNs and VCNs through metalfoils of different thicknesses but with the same surfacetreatment was carefully recorded [30], see Figure 1 forthe copper samples. It is worth noting that the UCNtransmission through a stack of two 50- µ m-thick copperfoils is lower than that through a single foil of 100 µ mthickness. This is a direct indication of UCN losses onthe two additional surfaces. Table III gives the UCNtransmissivity of each metal foil for an out-of-medium(oom) UCN velocity v oom of 7 m/s that was corrected forthe critical velocity v crit of each sample, see for exampleIgnatovich [12], v inm = q v − v . (7)This logic was chosen in keeping with Atchison et al. [9] tomake their results comparable with those presented here.The resulting in-medium (inm) velocities v inm were 6.2m/s for Al, 4.1 m/s for Cu, and 5.8 m/s for Zr. Transmissivity
N e u t r o n v e l o c i t y i n m e d i u m v in m ( m / s )
FIG. 1. Measured transmissivity of Cu foils for UCNs plottedover UCN velocity. The in-medium velocity of 4.1 m/s (equiv-alent to 7 m/s out of medium) is marked with a vertical redline. The vertical blue line marks 8.2 m/s (equivalent to 10m/s out of medium). For data treatment, a sliding averageof 16 time bins was used. The error bars account for this. Toimprove legibility, only every fourth error bar is shown.
U C N l o s s d u e t os u r f a c e s c a t t e r i n g
E x p . d a t a f o r C u 9 9 . 9 + % a t v inm = 4 . 1 m / sE x t r a p . t h e o r e t i c a l s tot = 2 0 7 2 b a r nF i t t h r o u g h e x p . d a t a , s tot = 2 0 7 2 b a r n Transmissivity
T h i c k n e s s ( m m ) FIG. 2. The measured transmissivity of Cu foils of threedifferent thicknesses is plotted for UCNs of an in-medium ve-locity of 4.1 m/s (equivalent to 7 m/s out of medium). Thered line represents a fit to the data using the transmissionequation including a surface scattering term, see Eq. 2. Thedashed green line represents the transmissivity as expectedif the sample caused no surface scattering. For both curves, σ tot = 2072 b (at 4.1 m/s in medium) was used as UCN losscross section of the sample bulk. Figure 2 shows the UCN transmissivity of copper foilsof different thicknesses. When the data points at v inm =4 . y axis shows the fractionof UCNs that is lost on both surfaces together, L t . It isobvious that a very large share of the total UCN losses isdue to surface scattering instead of bulk scattering andabsorption. In the case of the copper foils, the two sur-faces alone scatter about 37% of UCNs out of the directbeam.When the transmissivity values are taken from Ta-ble III and plugged into Eq. 5, one can calculate thefoils’ roughnesses as seen by the UCNs. These roughnessvalues are shown as parameter b in Table IV. As the Al-1sample was very transparent and the Cu-3 sample veryopaque, these extremes were omitted when calculatingthe average sample surface roughness.As a test, the roughnesses were also calculated usingthe transmissivity values at a neutron velocity of v oom =10 m/s for all three metals. They deviated only between −
18% and +2% from the values for 7 m/s, which showsthat the foil roughness as seen by neutrons is consistentover the UCN velocity range. The copper samples Cu-1aand Cu-1b, which consisted of the same type of 50 µ mfoils, yielded roughness values within a few percent ofeach other, both for 7 m/s and 10 m/s neutrons. This isanother confirmation of the reliability of our approach. B. Surface roughness
The surface roughness of the metal foil samples bothas measured by atomic force microscopy (AFM) and ex-tracted from the cross-section measurements above areshown and contrasted in Table IV.
Sample Roughness RoughnessParameter b (Å) Parameter R a (Å)Al 182 ± ± ±
12 648 ± ±
64 397 ± b ), as well as the roughness measured mechani-cally by AFM (parameter R a ). The errors for the UCN mea-surements include statistical and systematic errors. Table IV demonstrates that the magnitudes of bothroughness parameters b and R a are of the same order ofmagnitude.AFM scans of the foils used in the experiments de-scribed above yielded values of the same order of magni-tude as reported in the work of Steyerl [6], 200 to 500 Å.In his work, Steyerl noted that the roughness values ex-tracted from electron micrographs were in quantitativeagreement with the roughness parameters extracted fromneutron measurements but that it was difficult to inter-pret these micrographs.Generally, it is difficult to obtain the same rough-ness values for measurements of the same sample thatwere done using different techniques. For example, thenonzero curvature radius of the stylus used in an atomic-force microscope (AFM) leads to smaller roughness read-ings than those obtained by noncontact optical meth-ods [31]. One single roughness parameter is often notenough to describe the entire surface roughness withshort-range and long-range correlations. An overview ofroughness parameters was published by Gadelmawla etal. [32].Considering the above, it has to be concluded that,with the theories and techniques currently available, me-chanical measurements of sample roughness can at bestserve to estimate the order of magnitude of the loss ofUCNs due to scattering on rough sample surfaces. Thetransmission of each foil, used as vacuum barrier or for adifferent purpose, still has to be measured with UCNs toknow its exact transmissivity. C. Reexamination of previous experiments
Using the equations explained in Sec. IV, the experi-mental results from Atchison et al. [9] can be reexaminedto determine the surface roughness parameter b for thosefoils as seen by UCNs. Atchison et al. were not ableto confirm the exact make-up of their foil samples [33] –stacked or single foil. For aluminum, the first sample hada thickness of 10 µ m and we suspected that the otherswere very likely stacks of 100- µ m-thick foils. The zirco-nium samples were 100, 250, and 500 µ m thick, likelylayered from 50- µ m-thick foils. For multiple layers offoil, the right side of Eq. 2 needs to be raised to thepower of n , which represents the number of foils in theneutron beam. Otherwise the lower transmissivity wouldbe erroneously attributed to σ tot and suggested a falsehigher bulk cross section. Consequently, before calculat-ing the surface roughness parameter b of one foil, the n -th root needs to be taken of the surface transmissivityterm (1 − L t ) n .Table V gives the experimental transmissivities of Aland Zr foils (second column) as well as the roughness pa-rameter b extracted from those measurements by solvingEqs. 2 and 5. In the calculation of e − Nσ tot d , σ tot wastaken as presented in the paper [9]; one-phonon scat-tering was neglected. The ratio of experimental trans-missivity to theoretical transmissivity (due to absorptiononly), i.e., column 2 divided by column 3, is given in col-umn 4. From this excess loss of transmissivity due tosurface scattering, (1 − L t ) n , the roughness parameter b was calculated.The roughness parameters of single foils extracted fromthe UCN transmission measurements of Atchison et al. [9]are consistent between the individual samples and yieldaverage values of b (Al) = 195 ±
36 Å for aluminum and
Sample Thick- Exp. Theor. Exp. / Th. Roughnessness d ( µ m) Transm. exp − Nσ tot d [1 − L t ] n b (Å)Al 1 ×
10 0.943 0.994 0.949 136Al 1 ×
100 0.837 0.941 0.890 203Al 2 ×
100 0.707 0.886 0.799 202Al 3 ×
100 0.612 0.833 0.735 196Al 4 ×
100 0.540 0.784 0.689 190Al 5 ×
100 0.484 0.738 0.656 183Zr 2 ×
50 0.860 0.955 0.901 93.1Zr 5 ×
50 0.680 0.891 0.764 96.1Zr 10 ×
50 0.449 0.793 0.566 101TABLE V. Surface roughness b of individual metal foils from afoil stack as used by Atchison et al. [9] and re-analyzed apply-ing the theory explained above. Column 4 gives the ratio ofexperimental transmissivity to theoretical transmissivity (dueto absorption only), i.e., column 2 divided by column 3. Theuncertainty of b was estimated to be ±
15% due to the uncer-tainty of the original neutron transmission measurements. of b (Zr) = 97 ±
19 Å for zirconium. These are in linewith the typical roughness parameters from Steyerl [6]and prove conclusively that surface scattering is the rea-son for the measured 2.2-fold and 2.6-fold decrease of foiltransmissivity for aluminum and zirconium, respectively,compared with theory (taking into account only absorp-tion), as reported by Atchison et al. [9].
VII. CONCLUSION
In our experiments, we have demonstrated how neu-tron scattering in the sample bulk can be separated fromscattering at the sample surface. We found that ultracoldneutrons (UCN) are very susceptible to surface scattering– an effect that should be taken into account when plan-ning transmission experiments with UCNs. Our resultsshow conclusively that low-roughness sample containers,such as the one presented previously for cryogenic sam-ples [11], must be used to increase the accuracy of resultsin UCN experiments.In particular, we found that off-the-shelf high-puritymetal foils have the following roughness parameters asseen by UCNs: Al 182 ± ±
12 Å, and Zr313 ±
64 Å.Comparing the surface roughness extracted from UCNmeasurements with that measured by AFM leads to theassumption that neutrons “see” a different spectrum ofroughness amplitudes than mechanical or optical meansof measurement. The results obtained with both meth-ods are, however, of the same order of magnitude.Applying the method described here, we re-analyzedthe UCN transmission data of various metal foils fromAtchison et al. [9] and were able to calculate the surfaceroughness of their sample foils and identify surface scat-tering as the cause of the 2.2-fold and 2.6-fold decrease offoil transmissivity for aluminum and zirconium, respec-tively, compared with theory.The results for this paper were produced as part of thePh.D. thesis of Stefan Döge [34].
ACKNOWLEDGMENTS
We thank Yuliya E. Gorshkova of JINR Dubna for herpatient support during the AFM measurements of themetal foils. Support from the reactor crew, the instru- ment scientists, and technicians of the mechanical work-shops at the Institut Laue–Langevin, Grenoble, duringthe beamtime no. 3-14-380 is gratefully acknowledged.This work received funding from the Russian Founda-tion for Basic Research (RFBR) under grants no. 17-32-50024-mol-nr and 18-29-19039, from Dr.-Ing. Leonhard-Lorenz-Stiftung, Munich, under grant no. 940/17, andfrom FRM II/ Heinz Maier-Leibnitz Zentrum (MLZ),Munich, Germany. The open access publication fee waspaid for by the University Library and the Physics De-partment of the Technische Universität München, Mu-nich, Germany. [1] A. Steyerl, H. Nagel, F.-X. Schreiber, K.-A. Stein-hauser, R. Gähler, W. Gläser, P. Ageron, J. M. Astruc,W. Drexel, G. Gervais, and W. Mampe, A new source ofcold and ultracold neutrons, Physics Letters A , 347(1986).[2] S. Döge, J. Hingerl, and C. 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