New experiment on non-stationary neutron diffraction by a traveling surface acoustic wave

Abstract

The results of a new experiment on neutron diffraction by surface acoustic waves are presented. The measurements were carried out at a fixed incident angle in the time-of-flight mode, which made it possible to study the diffraction pattern in a wide range of neutron wavelengths at surface acoustic wave frequencies from 35 to 117 MHz. In a number of cases, diffracted waves of not only the first but also the second order were observed. The measurement results of both the angular distribution of diffracted waves and their amplitudes are in satisfactory agreement with the calculations. A new estimation has been obtained for the range of applicability of the dispersion law of neutron waves in matter moving with large acceleration.

Introduction

Neutron diffraction by a traveling surface acoustic wave (SAW) is a non-stationary phenomenon accompanied by the transfer of energy to the neutron $\mathrm{\Delta }E=nħ\mathrm{\Omega }$, where $\mathrm{\Omega }=2\pi f$, f is the wave frequency, $ħ$ is the Planck constant and n is the diffraction order. The existence of this phenomenon was first considered within the discussion the ultracold neutrons (UCN) storage problem [1] and was first observed experimentally in 1967 [2]. Papers [3, 4] were devoted to the theoretical consideration of inelastic scattering of UCNs by waves traveling along the surface of a liquid. In Ref. [5], neutron diffraction on acoustic wave traveling on the surface of a thin film was considered as a possible method for studying the properties of such films. In Ref. [6], it was proposed to use neutron diffraction by SAW to monochromatize neutrons in a wide range of wavelengths.

Despite the fact that non-stationary neutron diffraction by SAW was the subject of several theoretical works, experiment [2] for a long time remained the only one in which this phenomenon was observed. In a relatively recent work [7], the experiment was repeated with better accuracy. A quite detailed theory of the phenomenon is also given there, and the obtained experimental results were in quite satisfactory agreement with theoretical predictions. However, according to the authors, there were several questions, the answers to which should have been obtained in subsequent experiments.

The paper presents the results of a new experiment on neutron diffraction by a traveling surface acoustic wave. Unlike works [2, 7], in which measurements were carried out in a geometry with a fixed wavelength and a variable grazing angle, this experiment was done in the time-of-flight mode and at a fixed grazing angle. That allows to perform measurements in a relatively wide range of wavelengths. In addition, measurements were carried out for several values of acoustic wave amplitudes, and with samples designed for different SAW frequencies.

Experimental setup and data processing

The experiment was conducted on the D17 reflectometer of the Institut Laue-Langevin (Grenoble, France). The diffraction pattern was measured in the neutron wavelength range from 5 to 25 Å. Figure 1 shows a scheme of the experiment.

The measurements were carried out in the time-of-flight mode. The pulse structure of the beam was formed by two choppers [8, 9]. The time resolution was almost independent of the neutron wavelength and was 1$-$1.5%. Neutrons reflected from the sample were registered by a two-dimensional position-sensitive detector. In the direction transverse to the scattering plane (x-coordinate, see Fig. 1), all detector events were integrated. The wavelength of incident neutrons was determined from the time of flight. A two-dimensional array that described the angular and spectral dependence of the scattered waves intensity was formed as a result of the measurements.

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Fig. 1

Scheme of the experiment

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Fig. 2

The layout of sample with IDTs and the structure of the SAW formation, here $\mathrm{\Lambda }$ is a SAW wavelength

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Fig. 3

2D scattering pattern on sample 1. a In the absence of a US wave. b In the presence of a US wave. The numbers indicate the order of diffraction

Lithium niobate single crystals (${\text{LiNbO}}_3$) with dimensions of $60\times 20\times 3{\text{mm}}^3$ were used as samples. Its neutron-optical properties characterized by the value $\rho b=4.25\times {10}^{10}{\text{cm}}^{-2}$, where $\rho$ is a nuclear density and b is a coherent scattering length. On the surface of each crystal, there were two interdigital transducers (IDT) with an aperture of 5 mm, which were fabricated by photolithography (see Fig. 2). A high-frequency electrical signal from a generator goes to a power amplifier and then could be applied to one of IDTs, which made it possible to excite an acoustic wave on the sample surface. Meanwhile, the second IDT remained passive. The alternating voltage arising on this IDT was proportional to the amplitude of the SAW that reached it. During the measurements, this signal was recorded. The correspondence between the magnitude of this signal and the amplitude of the surface wave was established in a specially designed optical diffraction experiment. The distance from the IDT to the sample edges was 3–4 mm. Thus, the sample area occupied by the SAW was approximately $53\times 5{\text{mm}}^2$. To be sure that the neutron beam illuminates only this area, cadmium diaphragms were placed at the front and rear ends of the sample relative to the beam, limiting the beam height. Each of them had a 5-mm wide slit. The limitation of the length of the sample area which the beam is incident on was achieved by choosing the width of the reflectometer slits S1 and S2.

The measurements were performed with three samples, the IDTs of which were designed for frequencies of 35, 70, and 117 MHz. Hereinafter, we will call these samples: sample 1, 2, and 3. The angle of incidence on the sample was fixed and was equal to ${\mathrm{\Theta }}_0$ = 20 mrad. The angular width of the beam was $0.{02}^{\circ }$. Sample 1 was a YZ-cut crystal with the velocity of SAW propagation u = 3490 m/s. Samples 2 and 3 were ${128}^{\circ }$ Y-cut crystals with a SAW velocity u = 3980 m/s. For all samples, the measurements were carried out for two directions of SAW propagation. For sample 1, the measurements were made with several amplitude values of the excited acoustic wave.

Figure 3 shows a two-dimensional neutron scattering map, which is a visual representation of the raw experimental data. The abscissa axis shows the scattering angle; the ordinate axis shows the wavelength of the incident neutrons. The data in Fig. 3a correspond to the measurements in the absence of an ultrasound (US) wave. On the left side of the figure, the residual part of the direct beam is clearly visible. The bright vertical band in the center is the specularly reflected beam. The third inclined band is a beam caused by off-specular scattering on the surface roughness with a maximum near the critical angle of the total external reflection (Yoneda scattering [10]). Figure 3b shows the scattering map of the same sample in the case when an ultrasonic wave with an amplitude of 40 Å, propagating in the direction opposite to the incident beam, is excited on its surface. It is clearly seen that additional bands caused by the diffraction by SAWs are added to the bands visible in Fig. 3a.

The task of processing the experimental data accumulated in a 2D array was to determine the amplitudes and angles of diffracted waves. From such an array, one-dimensional arrays were formed as “cuts” of a two-dimensional pattern at constant the wavelength. Each of the resulting “cuts” was approximated by a model function describing the dependence of the intensity on the reflection angle ${\mathrm{\Theta }}_f$ at a given neutron wavelength. It was assumed that the shape of the diffraction beams was identical to the shape of the specularly reflected beam, differing only in intensity. The shape itself was not a subject of interest, and the only requirement that we imposed on the form of the model function was a good description of the angular distributions of the beams.

The following processing procedure was used. At first, the data obtained in the absence of a SAW were analyzed. From these data, we extracted the position, the amplitude of the specularly reflected beam, and the parameters describing its shape. From the same data, the parameters of the Yoneda scattering peak, the background, and the peak representing the residual contribution from the direct beam were extracted. In this case, the function used to approximate the data had the following form:

1

$$\begin{array}{c}\hfill \begin{array}{cc}& f_0(y_0,A_J,x,x_{0J},{\sigma }_J,\mu )\hfill \\ \hfill & =y_0+F_D(A_D,x,x_{0D},{\sigma }_{1D},{\sigma }_{2D})\hfill \\ \hfill & +F_R(A_R,x,x_{0R},\mu ,{\sigma }_{1R},{\sigma }_{2R})\hfill \\ \hfill & +F_Y(A_Y,x,x_{0Y},{\sigma }_Y),\hfill \end{array}\end{array}$$where

2

$$\begin{array}{cc}& \begin{array}{cc}& F_D(A_D,x,x_{0D},{\sigma }_{1D},{\sigma }_{2D})\hfill \\ \hfill & =\left\{\begin{array}{c}{A}_{D}exp\left[-,\frac{{(x-x_{0D})}^2}{2{\sigma }_{1D}^2}\right],x\le x_{0D}\hfill \\ A_{D}exp\left[-,\frac{{(x-x_{0D})}^2}{2{\sigma }_{2D}^2}\right],x>x_{0D}\hfill \end{array}\right),\hfill \end{array}\hfill \end{array}$$

3

$$\begin{array}{cc}& \begin{array}{cc}& F_R(A_R,x,x_{0R},\mu ,{\sigma }_{1R},{\sigma }_{2R})\hfill \\ \hfill & =\mu A_R\left(\frac{2}{\pi }\right)\frac{{\sigma }_{1R}}{{(x-x_{0R})}^2+{\sigma }_{1R}^2}\hfill \\ \hfill & +(1-\mu )A_{R}exp\left[-,\frac{{(x-x_{0R})}^2}{2{\sigma }_{2R}^2}\right],\hfill \end{array}\hfill \end{array}$$

4

$$\begin{array}{cc}& \begin{array}{cc}& F_Y(A_Y,x,x_{0Y},{\sigma }_Y)\hfill \\ \hfill & =A_{Y}exp\left[-,\frac{{(x-x_{0Y})}^2}{2{\sigma }_Y^2}\right].\hfill \end{array}\hfill \end{array}$$Here, symbols J = D, R, Y stand for the direct (D), specular (R) beams, and the Yoneda scattering peak (Y), respectively; $x_{0}J$ are the positions of peak maxima with amplitudes $A_J$ and widths ${\sigma }_J$. The pseudo-Voit function (3) is the sum of the Lorentz and Gaussian functions. The ratio of their contributions is determined by the coefficient $\mu$.

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Fig. 4

Count rate as a function of scattering angle for several incident wavelengths. Curves are the fitting functions explained in the main text; dots are the experimental data for the following wavelengths: a$\lambda$= 6.72 Å, b$\lambda$= 8.89 Å, c$\lambda$= 10.91 Å, d$\lambda$ = 17.43 Å

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Fig. 5

Angular distributions of diffracted beams, depending on the wavelength of the incident beam. Solid and dashed lines—calculation, dots—experiment. a Sample #1, b sample #3

When analyzing the data obtained in the presence of SAW, the approximating function was supplemented by terms $F_{n,s}$ corresponding to diffracted waves of the nth order and the direction of SAW. The last was specified by the sign of the index s. The direction of a US wave propagation along the z-component of the incident beam corresponded to the value $s=+1$. In some cases, there was an acoustic wave propagating in the direction opposite to the original one. In this case, the number of terms was doubled.

In the case of low SAW amplitudes, it was sufficient to limit the analysis to first-order diffraction waves $n=\pm 1$ [7], but when the amplitude increased, it became necessary to take into account second-order waves $n=\pm 2$. When processing the diffraction pattern, the parameters of the Yoneda scattering peak, the background and the peak coming from a direct beam were assumed to be identical to those obtained without a SAW. The peak of the specularly reflected beam in the case of a SAW was approximated with completely independent parameters. Examples of the obtained count rate versus the scattering angle are presented in Fig. 4. The measurements were made with sample #1, on which the US wave with an amplitude of 40Å, directed towards the z-component of the incident beam ($s=-1$), was excited.

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Fig. 6

Angular distributions of diffracted beams, depending on the wavelength of the incident beam measured with sample #2 for two directions of the US wave

Below, in Figs. 5, 6 and 7, the diffraction order number and the direction of wave propagation relative to the incident beam direction are shown in parentheses (n, s).

Results of the data treatment

Diffraction angles

The experimental data obtained from the processing technique described above were compared with the theoretical prediction from Ref. [7]. In all cases, the dependence of the neutron diffraction angle on the wavelength is in good agreement with the calculations. In some measurements, beams of not only ± 1, but also ± 2 diffraction orders were observed. For samples #1 and #3, beams previously called anomalous in [7] were observed (see Fig. 5). Their position corresponded to US waves propagating in a direction opposite to the one they were excited in. Now, we are confident to say that their origin is associated with the reflection of acoustic waves from the far edge of the crystal. To confirm this, facets were made on the crystal edges of one of the samples, namely #2, to eliminate the reflection of the traveling SAW. In this case, the SAW propagates through the facet and is completely attenuated at the adhesive bond between the crystal and the crystal holder body.

Using this sample, we performed measurements, where the RF signal was applied to either one or both IDT simultaneously. In the first case, there was no “anomalous wave” (see Fig. 6). The second case corresponds to the standing wave mode. In this case, four diffraction beams are clearly visible, they can be identified as diffraction beams of ±1 order caused by the waves propagating towards each other (see Fig. 7).

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Fig. 7

Angular distributions of diffracted beams, depending on the wavelength of the incident beam measured with the sample #2, when a US wave is excited simultaneously in two directions (standing wave mode)

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Fig. 8

Two-dimensional scattering map of the direct beam

Figure 5 shows the angular distributions of diffraction beams as a function of wavelength measured for samples #1 and #3. In both cases, the US wave was excited in the direction opposite to the incident beam. The SAW amplitudes were 20 Å for sample #1 and 10 Å for sample #3.

The non-stationary nature of the observed quantum effect was clearly demonstrated in the experiment with sample #3. In this measurement, the neutron beam was incident not only on the central area of the crystal, where the SAW propagated, but also on the area occupied by the IDT. In this case, the electrodes played the role of an ordinary diffraction grating with a period of 17 $\mathrm{\mu }$m. Thus, it was possible to simultaneously observe both the picture of the ordinary first- and second-order diffraction by an IDT and the non-stationary diffraction by a SAW (Fig. 5b).

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Fig. 9

Relative intensity of ± 1 diffraction orders for sample #1 (35 MHz) at SAW amplitudes of 20Å (a), 30Å (b). Solid and dashed lines—calculation, dots—experiment

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Fig. 10

Relative intensity ± 1 order of diffraction for sample #2 (70 MHz) for two directions of US wave with a SAW amplitude of 23 Å. Solid and dashed lines—calculation, dots—experiment

Intensity of diffracted waves

To determine the absolute diffraction efficiencies in one or another diffraction order, it is necessary to find the ratio of the intensity of the corresponding wave to the intensity of the direct beam. However, in order to avoid systematic errors coming from slight overillumination of the small samples and the background in the direct beam measurement (see Fig. 8) it was decided to analyze the results as the ratio of the intensity of the scattered neutrons to the intensity of the specular beam measured in the absence of a SAW, as was done earlier in Ref. [7]. Figures 9 and 10 show some of the measurement results of the relative intensity of diffraction orders determined in this way for sample #1 and sample #2.

A total of eight measurements were carried out, as a result of which the relative intensity was obtained for several SAW amplitudes and for two different directions of wave propagation. The most of results of measuring the diffraction efficiency of the first-order diffraction waves are in quite a satisfactory agreement with the theoretical predictions [7] except the result shown in Fig. 9b. For sample #1 at small wavelengths appears a discrepancy with the calculation with increase of SAW amplitude, but the reason for this discrepancy has not yet been found. The calculation took into account the contribution of the second-order diffraction waves, but it was difficult to reliably measure their intensity due to its smallness.

Conclusion

We presented an experimental study of non-stationary neutron diffraction by surface acoustic waves. The diffraction spectra were measured using a neutron reflectometer in time-the of-flight mode at a fixed incident angle of neutrons. The experiment showed that the direction of the wave vectors of all diffraction orders is in a good agreement with the theoretical prediction, and it was clearly demonstrated that the diffraction pattern by a SAW has a non-stationary nature.

For the most measurements rather satisfactory quantitative agreement was obtained between the experimentally measured and calculated intensities of waves of ±1 diffraction order.

A reliable answer has been obtained regarding the nature of the waves, which were considered abnormal in the previous work [7]. The latter arises as a result of neutrons diffraction by a SAW propagating in the direction opposite to the primary one and occurring due to the reflection of an acoustic wave from the far edge of the crystal.

Since the formulas describing the intensity of diffracted waves directly include the wave number of the refracted wave [7], the agreement between the experiment and the calculation demonstrates the absence of any noticeable discrepancies from the potential dispersion law. This circumstance is by no means trivial if we keep in mind that in the presence of a US wave, the near-surface layer of matter was moving with alternating acceleration, reaching, under the conditions of our experiment, a value of about $4\times {10}^8$ m/${\text{s}}^2$. The question of the validity of the generally accepted dispersion law in an accelerating medium apparently requires both theoretical analysis and experimental verification [11, 12–13].

Important conclusions can be drawn from the very fact of observing non-stationary effects that arise when a wave is reflected from an oscillating surface, since the condition for their occurrence is that the formation time of the reflected wave is short compared to the period of surface oscillation [14, 15]. Our measurements with sample #3 (see Fig. 5b) indicate that the neutron reflection time under our conditions was significantly less than 10 nsec.

Acknowledgements

The authors express their gratitude to V.A. Bushuev for fruitful discussions and to R. Ashkar for providing access to the materials of her Ph.D. thesis. The authors thank the ILL for allocation of beam time on D17 (http://doi.ill.fr/10.5291/ILL-DATA.3-15-93).

Funding

The work was carried out in accordance with topic 03-4-1128-2017/2022 of the JINR topical plan. D.V. Roshchupkin was supported by Ministry of Science and Higher Education of the Russian Federation (Grant Number 075-00296-24-00).

Data availability statement

Data sets generated during the current study are available from the corresponding author on reasonable request.

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