International
Tables for Crystallography Volume H Powder diffraction Edited by C. J. Gilmore, J. A. Kaduk and H. Schenk © International Union of Crystallography 2018 |
International Tables for Crystallography (2018). Vol. H, ch. 3.3, pp. 265-266
Section 3.3.3. Peak profiles for neutron time-of-flight experiments^{a}Advanced Photon Source, Argonne National Laboratory, 9700 S. Cass Avenue, Argonne, IL 60439–4814, USA |
The neutron source in a time-of-flight (TOF) powder diffraction experiment produces pulses of polychromatic neutrons; these travel over the distance from the source to the sample and then to the detectors which are placed at fixed scattering angles about the sample position; the travel times are of the order of 1–100 ms. This has been briefly described in Volume C of International Tables for Crystallography (Jorgensen et al., 1992). Because neutrons of differing velocities (v) have differing wavelengths (λ) according to the de Broglie relationship (λ = h/mv) given Planck's constant (h) and the neutron mass (m), they will sort themselves out in their time of arrival at the detector. The powder pattern appears as a function of TOF via Bragg's law (λ = 2d sin θ) in which the wavelength is varied and θ is fixed. The approximate relationship between TOF, wavelength and d-spacing observed in a particular detector can be derived from the de Broglie relationship and Bragg's law to giveThe constants are such that given λ in ångströms and the total neutron flight path length L in metres, then the TOF will be in µs. An analysis of the possible variances in these components then gives an estimate of the powder diffraction peak widths:where Δd, Δt, Δθ and ΔL are, respectively, the uncertainties in d-spacing, TOF, scattering angle θ and total flight path L (Jorgensen & Rotella, 1982). Consequently, these three terms also determine the instrumental contribution to the neutron TOF powder peak profile.
The neutron pulse shape depends on the mode of production. Early studies (Buras & Holas, 1968; Turberfield, 1970) used one or more choppers to define a polychromatic pulse from a reactor source, resulting in essentially Gaussian powder peak profiles whose FWHM (Γ_{G}) is nearly constant (B ≃ 0):so that the Rietveld technique can easily be used (e.g. Worlton et al., 1976). Unfortunately, this approach gave very low intensities and relatively low resolution powder patterns.
A more useful approach uses a spallation source to produce the pulsed neutron beam. Neutrons are produced when a high-energy proton beam (>500 MeV) strikes a heavy metal target (usually W, U or liquid Hg) via a spallation process (Carpenter et al., 1984). These very high energy neutrons strike small containers of moderating material (usually H_{2}O, liquid CH_{4} or liquid H_{2}) which then comprise the neutron source seen by the powder diffraction instrument. The entire target/moderator system is encased in a neutron-reflective material (usually Be) to enhance the neutron flux and then further encased in a biological shield. Each moderator may be encased on the sides away from the instrument (e.g. powder diffractometer) in a thin neutron absorber (e.g. Cd or Gd) and may also contain an inner absorber layer (`poison') to sharpen the resulting pulse of thermal neutrons. These sources produce a polychromatic neutron beam that is rich in both thermal (<300 meV) and epithermal (>300 meV) neutrons. The proton pulses can have a very short duration (∼200 ns) (from a `short-pulse' source, e.g. ISIS, Rutherford Laboratory, UK or LANSCE, Los Alamos National Laboratory, USA) or a much longer duration (>500 ns) (a `long-pulse' source, e.g. SNS, Oak Ridge National Laboratory, USA or ESS, European Spallation Source, Sweden); the pulse repetition rate at these sources is 10–60 Hz. These characteristics are largely dictated by the proton accelerator and neutron source design. The resulting neutron pulse results from complex down-scattering and thermalization processes in the whole target/moderator assembly; it may be further shaped by choppers, particularly for long-pulse sources, to give what is seen at the powder diffractometer.
Consequently, the neutron pulse structure from these sources has a complex and asymmetric shape, usually characterized by a very sharp rise and a slower decay, both of which are dependent on the neutron wavelength. The resulting powder diffraction peak profile (Fig. 3.3.3) is then the convolution [equation (3.3.2)] of this pulse shape (G_{λ}) with symmetric functions (G_{I}) arising from beamline components (e.g. slits and choppers) and the sample characteristics (G_{S}).
An early attempt at representing the TOF peak profile used a piecewise approach combining a leading-edge Gaussian, a peak-top Gaussian and an exponential decay for the tail (Cole & Windsor, 1980). Although single peaks could be fitted well with this function, the variation with TOF was complex and required many arbitrary coefficients.
A more successful approach empirically represented the pulse shape by a pair of back-to-back exponentials which were then convoluted with a Gaussian (Jorgensen et al., 1978; Von Dreele et al., 1982) to givewhere α and β are, respectively, the coefficients for the exponential rise and decay functions; erfc is the complementary error function. Analysis of the data that were available then gave empirical relations for α, β and σ asThe two terms in this function are shown in Fig. 3.3.3. The junction of the two exponentials defines the peak position (shown as a vertical line in Fig. 3.3.3); it is offset to the low side of the peak maximum. This arbitrary choice of peak position then affects the relationship between the TOF and reflection d-spacing; an empirical relationship (Von Dreele et al., 1982) was found to suffice:with three adjustable coefficients (C, A, Z) established via fitting to the pattern from a standard reference material.
Although this profile description was adequate for room-temperature moderators (H_{2}O or polyethylene) at low-power spallation sources, it does not describe well the wavelength dependence for cold moderators feeding neutron guides used at higher-power sources. An alternative description, employing a switch function to account for the fundamental change in the neutron leakage profile from the moderator between epithermal and thermal neutrons, was proposed (Ikeda & Carpenter, 1985; Robinson & Carpenter, 1990) to accommodate the profiles seen from liquid CH_{4} or H_{2} moderators. A drawback of this description is that the pulse profile is defined with the peak position at the low TOF edge; convolution with G_{I} and G_{S} results in a function where the peak position is far below the peak top. An empirical approach by Avdeev et al. (2007) simply requires tables to be established from individual peak fits to a standard material powder pattern for the values of α, β and TOF in place of the expressions given in equations (3.3.20) and (3.3.21); this establishes the G_{λ} and G_{I} contributions to the TOF line shape. More recently, some simple extensions (Toby & Von Dreele, 2013) to the empirical functions [equations (3.3.22) and (3.3.23)] appear to better cover the deviations arising from the enhanced epithermal contribution to the cold moderator spectrum:
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