International
Tables for
Crystallography
Volume H
Powder diffraction
Edited by C. J. Gilmore, J. A. Kaduk and H. Schenk

International Tables for Crystallography (2018). Vol. H, ch. 3.3, p. 266

Section 3.3.3.3. The neutron TOF powder peak profile

R. B. Von Dreelea*

aAdvanced Photon Source, Argonne National Laboratory, 9700 S. Cass Avenue, Argonne, IL 60439–4814, USA
Correspondence e-mail: vondreele@anl.gov

3.3.3.3. The neutron TOF powder peak profile

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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[link]). 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[link]; Von Dreele et al., 1982[link]) to give[\eqalignno{P(\Delta )&= {{\alpha \beta }\over{\alpha +\beta }}\Biggl\{\exp\left[{{\alpha }\over{2}}(\alpha {\sigma }^{2}+2\Delta )\right]{\rm erfc}\left[{{\alpha {\sigma }^{2}+\Delta }\over{\sigma (2^{1/2})}}\right]&\cr &\quad +\exp\left[{{\beta }\over{2}}(\beta {\sigma }^{2}-2\Delta )\right]{\rm erfc}\left[{{\beta {\sigma }^{2}-\Delta }\over{\sigma( 2^{1/2})}}\right]\Biggr\},&\cr &&(3.3.19)}]where 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 as[\alpha ={ {\alpha _{1}}/{d}}\semi\quad \beta = {\beta }_{0}+({{\beta _{1}}/{{d}^{4}}})\semi \quad \sigma = {\sigma }_{1}d.\eqno(3.3.20)]The two terms in this function are shown in Fig. 3.3.3[link]. The junction of the two exponentials defines the peak position (shown as a vertical line in Fig. 3.3.3[link]); 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[link]) was found to suffice:[{\rm TOF} = Cd+Ad^2+Z,\eqno(3.3.21)]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 (H2O 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[link]; Robinson & Carpenter, 1990[link]) to accommodate the profiles seen from liquid CH4 or H2 moderators. A drawback of this description is that the pulse profile is defined with the peak position at the low TOF edge; convolution with GI and GS results in a function where the peak position is far below the peak top. An empirical approach by Avdeev et al. (2007[link]) 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)[link] and (3.3.21)[link]; this establishes the Gλ and GI contributions to the TOF line shape. More recently, some simple extensions (Toby & Von Dreele, 2013[link]) to the empirical functions [equations (3.3.22)[link] and (3.3.23)[link]] appear to better cover the deviations arising from the enhanced epithermal contribution to the cold moderator spectrum:[{\rm TOF} = Cd+Ad^2+B/d+Z,\eqno(3.3.22)][\alpha={{\alpha_1}\over d}\semi\quad \beta=\beta_0 +{{\beta_1}\over {d^4}}+{{\beta_2}\over {d^2}}\semi \quad \sigma=\sigma_0 +\sigma_1d^2 +\sigma_2d^4 +{{\sigma_3}\over{d^2}}.\eqno(3.3.23)]

References

Avdeev, M., Jorgensen, J., Short, S. & Von Dreele, R. B. (2007). On the numerical corrections of time-of-flight neutron powder diffraction data. J. Appl. Cryst. 40, 710–715.Google Scholar
Cole, I. & Windsor, C. G. (1980). The lineshapes in pulsed neutron powder diffraction. Nucl. Instrum. Methods, 171, 107–113.Google Scholar
Ikeda, S. & Carpenter, J. M. (1985). Wide-energy-range, high-resolution measurements of neutron pulse shapes of polyethylene moderators. Nucl. Instrum. Methods Phys. Res. Sect. A, 239, 536–544.Google Scholar
Jorgensen, J. D., Johnson, D. H., Mueller, M. H., Peterson, S. W., Worlton, J. G. & Von Dreele, R. B. (1978). Profile analysis of pulsed-source neutron powder diffraction data. Proceedings of the Conference on Diffraction Profile Analysis, Cracow 14–15 August 1978, pp. 20–22.Google Scholar
Robinson, R. A. & Carpenter, J. M. (1990). On the use of switch functions in describing pulsed moderators. Report LAUR 90-3125. Los Alamos National Laboratory, USA.Google Scholar
Toby, B. H. & Von Dreele, R. B. (2013). GSAS-II: the genesis of a modern open-source all-purpose crystallographic software package. J. Appl. Cryst. 46, 544–549.Google Scholar
Von Dreele, R. B., Jorgensen, J. D. & Windsor, C. G. (1982). Rietveld refinement with spallation neutron powder diffraction data. J. Appl. Cryst. 15, 581–589.Google Scholar








































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