Modelling of wavelength cut-off filters and polarising mirrors in a neutron guide
Introduction
Neutron guides are used to efficiently transport neutrons, sometimes over long distances up to ∼100 m, from source to instrument at almost all neutron scattering centres [1], [2], [3]. The inclusion of various neutron optical elements such as monochromators, collimators, choppers and polarisers allow the characteristics of the neutron beam to be tailored for a particular instrument in terms of the incident neutron wavelength or energy, beam divergence, time modulation or as filters to produce polarised neutron beams [4]. Elastic scattering time-of-flight (TOF) neutron diffractometers at both reactor and spallation neutron sources typically use chopper systems to shape pulses having a broad or continuous wavelength distribution or ‘white’ neutron beam such that final wavelength discrimination can be made by the time of arrival of neutron events at the detector. These instruments usually require a long wavelength cut-off in the spectrum such that the pulse repeat frequency can be optimised to maximise neutron flux while avoiding frame-overlap at the detector. A simple mirror in the neutron optical path can be used as an effective long wavelength filter based on the property that the critical angle, θc, for total reflection for neutrons is proportional to the incident neutron wavelength [5], [6], [7]. Thus, wavelengths greater than some critical wavelength, λc, can be reflected out of the neutron beam and absorbed while wavelengths less than λc are transmitted through the mirror. Similarly, polarising mirrors in neutron guides use the difference in critical angle for reflection from a magnetised mirror for neutrons polarised parallel or anti-parallel to the magnetisation to separate the two neutron spin states and provide polarised neutron beams [8]. In order to optimise the transmitted neutron flux and remove background due to scattering from the mirror such devices are best installed inside the neutron guide system before the final beam collimation for the scattering instrument.
In neutron optics it is common to define the (wavelength dependent) critical angle of reflecting surfaces relative to that of Nickel, which has the highest scattering length density and hence largest critical angle of any element. The critical angle for neutron reflection can be approximated by the relation:where m is the ratio of the critical angles of the mirror and Nickel surface, k=0.1 [degrees/Å] and λ is the neutron wavelength [Å]. The value m=1 corresponds to a Nickel reflecting surface and, for example, m=0.5 for Silicon and m=0.65 for borofloat glass often used for neutron guides. Multilayer devices can be fabricated to enhance the critical angle beyond that of elemental or compositional surfaces to values of m>6, albeit with reflectivities <1 by using the diffraction properties of bi-layers of different spacing and varying neutron contrast in the ‘supermirror’ multilayer structure [9].
Here we consider neutron optical components consisting of a neutron guide of coating mg containing a mirror of coating mm that either spans the neutron guide as a single blade or arranged in a v-geometry with mirror angle ϕ relative to the walls of the neutron guide, as shown schematically in Fig. 1. It is demonstrated how to make use of, and how to construct, phase-space diagrams to make a simple analytical approach to the appropriate design of such mirror-in-guide wavelength and polarising filter systems. This geometric and analytical approach is compared with the results of neutron Monte-Carlo (MC) simulations. This work forms the basis of the specifications of the two wavelength cut-off and two polarising mirror filters that form some of the optical elements of the new small-angle neutron scattering (SANS) instrument, D33, at the Institut Laue Langevin [10].
Section snippets
Reflection processes for a mirror in a neutron guide
Fig. 2 shows schematically the introduction of a single-blade mirror into a neutron guide, at an angle ϕ to the parallel guide walls, and depicts some characteristic neutron trajectories that might be expected, depending on the specific choice of m-values. Fig. 2(a) shows possible trajectories for incoming neutrons with zero-divergence, i.e. parallel to the guide walls and for different wavelengths. Since the critical angle of the mirror depends on the neutron wavelength, short wavelength ‘hot’
Phase-space diagrams for a mirror-in-guide system
Neutron trajectories allowed through reflecting guide and mirror systems can be visualised using so-called ‘phase-space’ or ‘acceptance’ diagrams [11], [12], [13], [14], [15], [16], [17], [18], [19], [20], [21], [22]. In general, neutron trajectories and properties can be characterised by as many as eight dimensions with three in space (x,y,z), three in divergence (θx, θy, θz), one in wavelength or energy (λ) and one in polarisation, (P). Such diagrams depict the regions of phase-space where
Guide and mirror relative critical angles
Following the principles described above, Fig. 5 depicts phase-space diagrams constructed for different ratios of the mirror and guide critical angles, mm and mg. Fig. 5(a) shows the case mm=mg, and is the same as that in Fig. 4 (iteration 7). Fig. 5(b) shows the case where mm=1.25 mg. Here it can be seen that the re-entrance of the long wavelength neutron band moves to longer wavelengths and covers a narrower band width close to zero divergence. Fig. 5(c) depicts the case where mm=1.5 mg where
Analytic approach to mirror design
Inspection of the phase space diagrams in Fig. 4, Fig. 5 lend themselves to easy parameterisation with some simple mathematics of intersecting straight lines. Using Eq. (1) we can immediately write down the angle of the mirror surface, ϕ, relative to the mean neutron beam direction aswhere λc(0) is the nominal desired cut-off wavelength at zero divergence. Inspection of Fig. 4, Fig. 5 allows us to write equations for the various boundaries on the phase diagram of gradient 1/(km),
Monte-Carlo simulation of wavelength cut-off mirrors
Monte-Carlo (MC) simulations were performed using the authors own simulation code written in Matlab and developed mostly for the simulation of small-angle neutron scattering instruments and associated optical components. Briefly, neutron trajectories are ray-traced one-by-one through a series of optical components such as source, guides, collimators, apertures and in this case mirror-in-guide optical devices, using trivial Newtonian equations of motion for a particle, including the influence of
Application of mirror-in-guide neutron optic devices
The ILL's new SANS instrument D33 will have a neutron guide of 30×30 mm and minimum collimation length of 2.8 m [10]. Even supposing large samples of the order 15 mm, the maximum divergence that can be accepted by such an instrument is ±0.46°. Setting θdiv=0.46° in Eq. (5) sets the minimum condition for the m-coatings of the D33 wavelength cut-off filters as
Substituting Eq. (2) for the mirror angle, ϕ, in terms of mm and λc gives the quadratic equation:
Conclusions
Phase-space diagrams can be a useful and rapid tool in determining the specification of neutron optical devices. Here we have demonstrated the construction of wavelength vs. divergence phase-space diagrams that describe mirror-in-guide neutron optical devices that typically find application as wavelength cut-off filters and beam polarising devices. The iterative construction of phase-space diagrams demonstrated in this article has allowed us to parametrise in sufficient detail the complexity
Acknowledgements
The author is sincerely grateful to Anna Sokolova for critical analysis of the D33 instrument specification and for pointing out that the original D33 wavelength cut-off mirrors ‘would not work’ (mm=mg) without long wavelength contamination. Bob Cubitt, Ken Andersen and Phillip Bentley are thanked for their help and general advice on neutron optics. John Barker is thanked for his critical reading of this manuscript.
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