Tables for
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, pp. 265-266

Section 3.3.3. Peak profiles for neutron time-of-flight experiments

R. B. Von Dreelea*

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

3.3.3. Peak profiles for neutron time-of-flight experiments

| top | pdf | The experiment

| top | pdf |

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[link]). 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 give[{\rm TOF} = 252.7784 L\lambda = 505.5568 Ld \sin\theta.\eqno(3.3.16)]The 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:[\Delta d/d =[(\Delta t/t)^2 +(\Delta \theta \cot\theta)^2 +(\Delta L/L)^2]^{1/2},\eqno(3.3.17)]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[link]). Consequently, these three terms also determine the instrumental contribution to the neutron TOF powder peak profile. The neutron pulse shape

| top | pdf |

The neutron pulse shape depends on the mode of production. Early studies (Buras & Holas, 1968[link]; Turberfield, 1970[link]) used one or more choppers to define a polychromatic pulse from a reactor source, resulting in essentially Gaussian powder peak profiles whose FWHM (ΓG) is nearly constant (B ≃ 0):[{\Gamma }_{G}^{2}=A+Bd^2,\eqno(3.3.18)]so that the Rietveld technique can easily be used (e.g. Worlton et al., 1976[link]). 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[link]). These very high energy neutrons strike small containers of moderating material (usually H2O, liquid CH4 or liquid H2) 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[link]) is then the convolution [equation (3.3.2)[link]] of this pulse shape (Gλ) with symmetric functions (GI) arising from beamline components (e.g. slits and choppers) and the sample characteristics (GS).

[Figure 3.3.3]

Figure 3.3.3 | top | pdf |

The observed and calculated Ni 222 diffraction line profile from the Back Scattering Spectrometer, Harwell Laboratory, Chilton, UK. The curves A and B are computed from the two terms in equation (3.3.19)[link] and curve C is the sum (from Von Dreele et al., 1982[link]). The neutron TOF powder peak profile

| top | pdf |

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)]


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
Buras, B. & Holas, A. (1968). Nukleonika, 13, 591–620.Google Scholar
Carpenter, J. M., Lander, G. H. & Windsor, C. G. (1984). Instrumentation at pulsed neutron sources. Rev. Sci. Instrum. 55, 1019–1043.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., David, W. I. F. & Willis, B. T. M. (1992). White-beam and time-of-flight neutron diffraction. In International Tables for Crystallography, Vol. C, edited by A. J. C. Wilson. Dordrecht: Kluwer.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
Jorgensen, J. D. & Rotella, F. J. (1982). High-resolution time-of-flight powder diffractometer at the ZING-P′′ pulsed neutron source. J. Appl. Cryst. 15, 27–34.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
Turberfield, K. C. (1970). Time-of-flight neutron diffractometry. In Thermal Neutron Diffraction, edited by B. T. M. Willis. Oxford University Press.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
Worlton, J. G., Jorgensen, J. D., Beyerlein, R. A. & Decker, D. L. (1976). Multicomponent profile refinement of time-of-flight neutron diffraction data. Nucl. Instrum. Methods, 137, 331–337.Google Scholar

to end of page
to top of page