Countering the effects of gravity on a small angle neutron scattering instrument

Abstract

Small angle neutron scattering (SANS) is a powerful technique for determining chemical, biological and hard matter structures of materials over a wide range of length-scales. The larger the structure the smaller the scattering angles and the longer the collimation and detector distances need to be. In addition, in order to determine the structure of materials in the micron range long neutron wavelengths are required. The combination of the slow speed of long wavelength neutrons coupled with the large distances of the instrument results in a significant displacement of the beam with gravity. This can affect the neutron flux available, the resolution, the completeness of the signal on the detector and hence the largest length-scales that can be measured. Previous ideas of cancelling the effects of gravity have used prisms but this approach has problems working over the full height associated with typical sample sizes. This paper describes a different method using simple mirrors that can be exploited using existing components of SANS instruments and results in reducing the minimum momentum transfer by a factor of 5.

Introduction

There is a wide range of research using neutron scattering to probe structures in the micron range. This varies from the fractal nature of rocks to self organised polymer structures [1]. There have been a number of new techniques designed to reduce the minimum Q measureable on a standard SANS machine. The use of lenses reduces the size of the beam on the detector and hence minimum Q at the expense of the need for a larger sample area [2], [3]. The method is also limited by the large number of lenses often required with the associated problems of absorption and background. Prisms have been shown to eliminate the effects of gravity at a certain distance [4] but a complex stack of prisms is required for samples of more than 1–2 mm high. Work is continuing on developing a multi-aperture system for low Q but this requires changing the detector for a high resolution CCD type [5] and the effects of gravity are a problem. There exist other techniques such as the Bonse–Hart method [6] but this is based on a completely different type of machine. Particularly with delicate soft matter and biological samples there is a great advantage in measuring the widest range of Q possible at the same time on the same instrument.

On a SANS instrument the scattered intensity is measured as a function of momentum transfer Q, which is a function of the angle the beam was scattered away from the main beam, 2θ and the wavelength, λ:

$$Q=\frac{4\pi }{\lambda }\sin (\theta )$$

Q is a reciprocal length thus to measure the largest structures in real space requires a instrument capable of measuring at the lowest Q. Conventional instruments have a pinhole geometry where the pinhole is close to the sample and positioned between the source and the detector. Taking a circular source of diameter S, with the collimation distance C, equal to the detector distance and assuming the detector resolution is less than S then

$$Q_{\min }=\frac{2\mathrm{\pi S}}{\lambda C}$$

The lowest published Q [7] for the SANS instrument D11 [8] at the Institut Laue Langevin is 3.4×10⁻⁴ Å⁻¹, which corresponds to a real space size of 1.85 μm. The authors used the full source size of 30×50 mm, 40.5 m from the sample resulting in a beam of 0.042°×0.071° in 2θ on the detector. Taking the smallest value as the lowest 2θ measureable with a wavelength of 20 Å and using Eq. (2) results in a minimum Q of 2.3×10⁻⁴ Å⁻¹. Given that the fractional angular resolution is very poor at this point it is normal to use a slightly higher value of Q when fitting the data. The instrument views a cold source of neutrons and there is still a significant flux above 20 Å and SANS rises sharply at low q, so why is this the longest wavelength used on D11 at the maximum collimation distance? Neutrons must follow a parabolic path confined by the source and sample apertures due to gravity. The maximum height of the beam, h at the midpoint of the collimation distance and the drop of the beam on the detector, y are

$$h=\frac{g{\lambda }^2{C}^2}{8e^2}y=\frac{g{\lambda }^2(CD+D^2)}{2e^2}$$

where e is the product of the neutron wavelength and its velocity equal to 3956 Å m/s, g is gravitational acceleration and C and D are the collimation and detector distances, respectively Fig. 1.

The size of the apertures in the collimation limits h to 55 mm and the detector size of 0.96×0.96 m² limits y to 0.48 m (the detector has 128×128 pixels of 7.5 mm). Taking into account the finite size of the beam at the mid collimation point results in the transmission dropping to 0.5 at 20.7 Å and zero at 22.9 Å. The wavelength where the centre of the main beam hits the bottom of the detector is 21.8 Å (at the maximum detector distance D=39 m). It is a coincidence for D11 that these two wavelength limits are so similar as the limit in h is entirely due to the presence of another instruments detector arm that passes over the collimation section of D11. Both equations define a limit on the product λD and hence Q_min, which for the beam falling off the bottom of the detector is

$$Q_{\min }=\frac{2\pi S}{e}\sqrt{\frac{g}{R}}$$

where R is the distance from the centre to the bottom of the detector, assuming C=D. Thus for a gravity limited SANS machine Q_min is independent of both D and λ. The beam dropping over the detector distance requires something that cancels the effects of gravity. In theory a very dense gas with zero absorption and a high scattering length density would settle in a density gradient and could refract the beam in such a way as to completely cancel gravity i.e. neutrons would travel in straight lines. Unfortunately such a gas does not exist but refraction can be used locally to effectively cancel gravity at the detector. It has been reported previously [4] that a refracting surface, acting like an optical prism, near the sample can provide a deflection of the neutron beam proportional to λ² thus bringing all wavelengths back to the centre of the detector. Given that neutrons of different wavelengths arrive at the exit of the collimation at angles to the horizontal proportional to λ² a horizontal mirror of sufficient length will provide the same effect. In addition to the problems of beam cutting in the collimation and the beam falling off the bottom of the detector a third effect is the gravitational dispersion of the beam due to the wavelength spread (typically dλ/λ=10%) stretching the beam vertically. This reduces the resolution and the effects on the data have recently been addressed theoretically [9].

Such a mirror already exists in the collimation section at the bottom of a glass guide. A diagram of the layout is shown in Fig. 2. Normally a continuous section of neutron guide is translated in or out to vary the sample to source length. In this case all guides would be removed (to make the collimation distance 40.5 m) but a single section near the sample can be added. On D11 the sections of guide up to 5.5 m away from the sample are focusing and therefore cannot be used [10]. The first non-focusing section is between 5.5 and 8 m from the sample and has a cross-section of 45×50 mm. The detector distance where the beam is exactly in the middle of the detector, due to symmetry, is the collimation distance less twice the sample to mirror distance. Simple classical mechanics gives the relation:

$${\lambda }^{⁎}=e\sqrt{\frac{2[\Delta +w(1-f/(C-f))]}{gf(C-2f)}}$$

where λ^⁎ is the characteristic wavelength, in Å, which can bounce off the mirror and pass through the sample aperture at a height Δ above the central line, C is the collimation distance, w is half the guide height and f is the distance from the sample aperture to the middle of the mirror. Taking the first available guide section at 5.5–8 m from the sample, f=6.75 m, C=40.5 m and w is 25 mm. With Δ=0 we find λ^⁎=18.7 Å. Differentiating Eq. (5) with respect to f shows that this 2.5 m section is sufficient to reflect the 10% wavelength spread of the incoming beam. Reflections from the guide sides could only transmit to the sample if the guide was midway between the source and the sample.