New opportunities for quasielastic and inelastic neutron scattering at steady-state sources using mechanical selection of the incident and final neutron energy

1. Introduction

A recently proposed [3] neutron wide-angle velocity selector (WAVES) device makes possible inelastic neutron scattering experiments in the inverted geometry (with a fixed final energy) by means of mechanical velocity selection of the neutrons scattered at the sample position. Even though mechanical velocity selectors are not as good as crystal analyzers when it comes to the relative energy resolution, Δ E / E , they can still achieve a high absolute energy resolution, Δ E , at the low energies. This is an important advantage, because, unlike crystal analyzers, for which a few suitable low-E (large d-spacing) reflections are available, mechanical velocity selectors can continuously choose the final neutron energy, limited only by the sufficient incident flux of the long-wavelength neutrons needed to ensure the presence of the elastic line in the spectrum. In principle, it should be possible to build a WAVES-based spectrometer, such as a broad-range inelastic neutron velocity selector (BRAINWAVES) proposed for the Spallation Neutron Source second target station, that features a resolution on the µeV scale and a maximum neutron energy loss on the hundreds of meV scale, if the long final wavelength is chosen. One the other hand, direct geometry neutron spectrometers (with a fixed incident energy) can also achieve a resolution on the µeV scale using long incident wavelengths, but are then limited in the energy transfer to ∼ k B T ( k B is the Boltzmann’s constant) attainable only with the neutron energy gain.

Most of the modern high-resolution spectrometers suitable for quasielastic neutron scattering (QENS) rely on the timing of the detected neutrons to determine the neutron energy transfer. Oftentimes it is the neutron time-of-flight over the final flight path (sample-to-detector) that yields the energy transfer on direct geometry spectrometers installed at both steady and pulsed neutron sources. Less frequently, inverted geometry backscattering spectrometers [1,2,4,5] at pulsed neutron sources use the neutron time-of-flight over the initial flight path (source-to-sample) to determine the energy transfer. On the other hand, backscattering spectrometers at steady neutron sources, which rely on Doppler-driven monochromators, use timing to determine the instantaneous velocity of the monochromator and, thus, the incident neutron velocity. Here we propose a modification of WAVES that does not rely on timing at all. Instead, both the incident and final neutron energy (actually, a range of energies for the latter) are defined by the geometry and rotation frequency of the WAVES. This makes a highly efficient, easy to build and operate inelastic (in particular, quasielastic) spectrometer for a steady neutron source. The independence of timing for the proposed setup makes it somewhat similar to triple-axis spectrometers at steady neutron sources, but with simultaneous coverage of a broad angular range.

2. Design and discussion

WAVES is essentially a collimator rotated with a frequency f about the vertical axis passing through the sample position to transmit neutrons of a desired velocity v f scattered by the sample [3]. When optimized for two-dimensional scattering in the equatorial plane, the collimator blade shape mimics the trajectory of a scattered neutron with a velocity v f , ( r , φ ) = ( v f t , 360 ° f t ) , which is an Archimedean spiral, r = a f φ , a f = ( v f / 360 ° f ) , or φ = A f r , A f = ( 360 ° f / v f ) . Besides the rotation frequency, f, and the blade curvature, A f , the resolution of WAVES depends on the number of blades (per 360 ° ), n b , and the blade length, Δ R [3]: Δ E f = 2 E f v f n b f Δ R = 2 E f 360 ° n b A f Δ R .(1) While Δ E f / E f is a constant determined by the WAVES design, independent of the rotation frequency (and, thus, the velocity of the transmitted neutrons, which scales with f), Δ E f ∼ E f , that is, Δ E f ∼ v f 2 , or Δ E f ∼ f 2 . For a typical set of readily attainable parameters of λ f = 20 Å , v f = 198 m/s , E f = 0.2045 meV , f = 300 Hz , Δ R = 0.2 m , n b = 360 ( 1 ° collimator), one obtains Δ E f = 3.75 µeV .

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Fig. 1.

(Courtesy of W.M. McHargue) An illustration of a sandwich-type WAVES, SANDWAVES, featuring the blades in the middle section near the equatorial plane (for selection of the incident neutron velocity) with the curvature opposite to the curvature of the rest of the blades (for selection of the final neutron velocity).

The original WAVES concept [3] was geared toward scattering techniques where the measured neutron energy transfer is due to an energy spread of incident neutrons, whereas the final velocity of the scattered neutrons is defined by the WAVES. Let us consider a WAVES device that also defines the incident neutron velocity, v i . This can be done by building a sandwich-type WAVES device (SANDWAVES) with the middle, near-equatorial section blades mimicking the trajectory of an incident neutron (for an infinitely small sample), ( r , φ ) = ( − v i t , 360 ° f t ) = ( v i t , − 360 ° f t ) . That is, the blades in the middle section of the WAVES have the curvature opposite to the curvature of the blades in the other layers, as shown in Fig. 1. Assuming that the angular spread in the horizontal plane of the incident neutron beam focused on the small sample exceeds the inter-blade angular separation, the uncertainty in the energy of the incident neutrons is determined by an equation similar to Eq. (1): Δ E i = 2 E i v i n b f Δ R = 2 E i 360 ° n b A i Δ R(2) for the blades of the middle section, the shape of which is described as r = − a i φ , a i = ( v i / 360 ° f ) , or φ = − A i r , A i = ( 360 ° f / v i ) . If the Δ E i is defined independently from the Δ E f (e.g., by the uncertainty in the incident time-of-flight, or the monochromator crystal Bragg reflection), the overall energy resolution of the WAVES-based spectrometer can be evaluated as Δ E = ( Δ E i 2 + Δ E f 2 ) 1 / 2 , with the contributions taken in quadrature. For the SANDWAVES, however, it is more appropriate to consider the overall transmission, for the incident neutrons through the middle section blades and for the scattered neutrons through the top or bottom section blades, which will define the resolution of the entire SANDWAVES-based spectrometer. Figure 2 shows ray-tracing calculations of the transmission for idealized ( E i = E f ) SANDWAVES with 360 blades in each section of the inner and outer radius of 0.4 m and 0.6 m, respectively, optimized for transmission of neutrons with v = 198 m/s at a rotation frequency f = 10 Hz . The overall SANDWAVES transmission exhibits the main and satellite peaks; the position of the latter is somewhat dependent on the neutron scattering direction. The main transmission peak is narrower compared to the transmission of either the middle or top/bottom sections considered alone. The satellite transmission peaks, which would result in the irregular-shaped resolution function for a point sample, quickly become smeared out as the radius of the realistic finite-size sample is increased. If needed, further suppression of the satellite peaks can be achieved with the number of blades in the middle section different from that in the top/bottom section.

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Fig. 2.

Transmission in the nearly forward-scattering direction as a function of the neutron velocity for idealized ( E i = E f ) SANDWAVES with 360 blades in each section of the inner and outer radius of 0.4 m and 0.6 m, respectively, optimized for transmission of neutrons with v = 198 m/s at a rotation frequency f = 10 Hz . Open symbols: transmission through either the middle section considered alone or the top/bottom section considered alone. Filled symbols: overall SANDWAVES transmission. (a) Point sample. (b) Annular sample of 5 mm radius.

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Fig. 3.

A schematic illustration of the measurement of quasielastic and inelastic signal (as shown on the bottom) with a sandwich-type WAVES, SANDWAVES. The energy transfer is a function of the component of the scattering angle out of the equatorial plane. In particular, the elastic signal is measured at the angle of arccos ( A i / A f ) , where φ = − A i r and φ = A f r describe the shape of the blades for the middle and the other sections of the SANDWAVES, respectively.

At a first glance, in the case of the identical blade curvature, A i = A f , only the elastic scattering will be measured by SANDWAVES. However, let us now take into account the actual three-dimensional character of the scattering process and the WAVES device. If α and β are, respectively, the out-of-plane and in-plane components of the scattering angle, θ, then cos θ = cos α cos β . While only the neutrons of one selected incident velocity could be transmitted to the sample through the central layer of the SANDWAVES, a range of the final neutron velocities can be detected simultaneously as a function of the out-of-plane component of the scattering angle, as shown in Fig. 3. This is because for a neutron scattered by the sample with a velocity v f the velocity projection onto a horizontal plane is proportional to v f cos α . That is, the velocity of the scattered neutrons transmitted by a WAVES device with the constant, as a function of the height, blade curvature, is proportional to ( 1 / cos α ). Therefore, if one aims at detecting only the neutrons with a certain final velocity, irrespective of the scattering angle, then the appropriate three-dimensional optimization of the blades shape is needed [3]. On the other hand, the natural variation of the transmitted final neutron velocity allows for a simultaneous detection of neutrons with a range of v f . In this case, the energy transfer E = E i − E f = 1 2 m ( v i 2 − v f 2 ) = 1 2 m ( 360 ° f ) 2 ( 1 A i 2 − 1 A f 2 1 cos 2 α ) ,(3) where m is the neutron mass. For QENS, where it is often desirable to measure the signal as a function of both positive and negative energy transfer, the curvature of the blades for different sections can be chosen so that A i = A f cos α 0 for a certain out-of-plane scattering angle component, α 0 . Then v f = v i ( cos α 0 / cos α ) , and the energy transfer E = 1 2 m ( 360 ° f ) 2 1 A i 2 ( 1 − cos 2 α 0 cos 2 α ) = E i ( 1 − cos 2 α 0 cos 2 α )(4) is positive for α < α 0 , zero for α = α 0 , and negative for α > α 0 . The dynamic range, defined as the difference between the maximum and minimum attainable energy transfer, E max − E min = E i cos 2 α 0 ( 1 cos 2 α max − 1 cos 2 α min ) .(5) For example, with E f = 0.2045 meV , α min = 1 ° , α 0 = 35 ° and α max = 45 ° , the dynamic range is about ± 0.068 meV . The dynamic range and the energy resolution are decoupled, even though they are both proportional to E i . At a given neutron energy and SANDWAVES rotation frequency, the former can be extended by detecting neutrons up to the higher out-of-plane angles, α max , whereas the latter can be reduced by increasing the number of blades and extending their length.

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Fig. 4.

A schematic illustration of the scattering geometry (a) and the contour intensity plots of the scattering momentum transfer values in the units of Å⁻¹, powder-averaged Q (b), and its components, Q x (c), Q y (d), Q z (e) as a function of the scattering angle’s (θ) in-plane (β) and out-of-plane (α) components for E i = 0.2045 meV and α 0 = 35 ° .

Not only a range of the energy transfers is measured simultaneously by SANDWAVES, but also, at any given energy transfer associated with the angle α, a range of the scattering momentum transfers, Q = ( ( 2 π / λ i ) 2 + ( 2 π / λ f ) 2 − 2 ( 2 π / λ i ) ( 2 π / λ f ) cos θ ) 1 / 2 , is simultaneously accessible because of the dependence of θ on the in-plane scattering angle component, β. This is illustrated by Fig. 4, which shows the contour intensity plots for the powder-averaged scattering momentum transfer Q = m ℏ v i 1 − 2 cos α cos β cos α 0 cos α + cos 2 α 0 cos 2 α ,(6) where m is the neutron mass and ℏ is the reduced Planck constant, and its components Q x = − m ℏ v i cos α sin β cos α 0 cos α , (7) Q y = − m ℏ v i sin α cos α 0 cos α , (8) Q z = m ℏ v i [ 1 − cos α cos β cos α 0 cos α ] . (9) With a SANDWAVES device described above, measurements at a steady source can be conducted in a very effective manner. For any pixel of a two-dimensional detector array, both the energy transfer, E, and the scattering momentum transfer, Q, are uniquely defined irrespectively of the neutron detection time. Therefore, the scattering intensity from the sample is simply mapped onto the fixed (Q, E) grid. This is an ideal situation for QENS measurements, which typically concentrate on the fixed range of energy transfers near the elastic line. In a general inelastic neutron scattering experiment, on the other hand, there is often a need to concentrate on the region near the a priori unknown point in the (Q, E) space. Reducing the rotation frequency will reduce the energy transfer range, which is proportional to f 2 according to Eqs (3)–(5), and the accessible Q range, because the v i and v f scale with f, while at the same time improving the energy resolution, according to Eqs (1) and (2). With a SANDWAVES, the v i and v f are coupled at any given f through the blade curvature parameters and the position of the elastic line determined by the choice of the angle α 0 . A complete decoupling between the v i and v f is attainable if, instead of a SANDWAVES, the middle layer of the WAVES is transparent to neutrons (e.g., devoid of blades), and the selection of v i is done independently of the v f selection. A crystal monochromator, while suitable for the v i selection, may provide a limited choice for the long incident wavelengths, at which WAVES is the most effective. A pair of independently rotated co-axial WAVES devices, one to define v i , another to define v f , might be possible to implement, if so desired despite the increasing complexity and the reduced space around the sample. Alternatively, a traditional neutron velocity selector, which is quite effective at long wavelengths, can be used to independently select v i . These could be the options to consider for a setup for general inelastic experiments, whereas specifically QENS experiments would be invariably very effective using a SANDWAVES-based spectrometer.

In principle, such a spectrometer could be constructed fairly inexpensively, since it requires essentially a SANDWAVES device and an array of detectors and can be positioned at any distance from the source without affecting the resolution. The most important practical limitation may come from the sample size. Effective operation of WAVES for v f selection calls for a small sample size (ideally, no larger than a few mm in each direction) [3], or else the degradation of the resolution and the peak transmission will occur. Likewise, SANDWAVES selection of v i poses similar and even somewhat more stringent requirements on the incident beam size, commensurate with the sample size. Many single-crystal samples are naturally small in size and thus present no problem for experiments using WAVES. On the other hand, many time-of-flight inelastic neutron spectrometers employ the incident beam cross-section (and the typical sample size) equal to or exceeding 10 cm² for experiments with powders and liquids. The effective WAVES-based spectrometers may thus require the use of strong neutron focusing optics, such as, e.g., elliptic guides. On the positive side, a small incident beam size for a small sample on a WAVES-based spectrometer, if attainable, would be ideally suited for experiments using extreme sample environments, such as a very high pressure, or for the samples available in limited quantities, such as many biological systems. Compared to traditional reactor-based triple-axis and backscattering spectrometers, SANDWAVES offers simultaneous detection of neutrons over the full solid angle with energy resolution comparable to backscattering spectrometers, but without their reduced duty cycle.

The device that we have proposed, a sandwich-type neutron wide-angle velocity selector (SANDWAVES), uniquely defines the energy and scattering momentum transfer at any scattering angle (that is, at any detector pixel) and simply maps the scattering intensity from a particular sample onto the fixed (Q, E) grid. Not relying on the neutron time-of-flight, this device is well suited for a steady state source. There a sandwich-type WAVES can be used for inelastic experiments, similar to a triple-axis spectrometer, but with a wide angular coverage and higher energy resolution, and can be particularly effective for QENS measurements at steady-state neutron sources.