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, p. 266
Section 3.3.3.3. The neutron TOF powder peak profile^{a}Advanced Photon Source, Argonne National Laboratory, 9700 S. Cass Avenue, Argonne, IL 60439–4814, USA |
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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