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. 2.3, pp. 66-101
https://doi.org/10.1107/97809553602060000938

Chapter 2.3. Neutron powder diffraction

C. J. Howarda* and E. H. Kisia

aSchool of Engineering, University of Newcastle, Callaghan, NSW 2308, Australia
Correspondence e-mail:  chris.howard@newcastle.edu.au

An introduction is given to neutrons and their scattering from assemblies of atoms. This is linked with how neutrons may be exploited in the study of a variety of phenomena in the fields of crystallography, magnetism, solid-state reaction chemistry, engineering and materials science. Neutron-scattering instrumentation such as neutron sources, guides, collimators, choppers and monochromators, and detectors are described individually as well as how they are combined to make distinct diffractometers for high-resolution, high-intensity or specialized studies. Examples of data recorded on selected diffractometers are given and guidance is provided on the selection of diffractometer type, experimental arrangement and the ancillary equipment required for non-ambient studies.

2.3.1. Introduction to the diffraction of thermal neutrons

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Diffraction of neutrons occurs by virtue of their wave character, the de Broglie wavelength λ being[\lambda ={{h}\over{mv}}={{h}\over{(2mE)^{1/2}}}, \eqno(2.3.1)]where m, v and E are the mass, speed and energy of the neutron, respectively, and h is Planck's constant. It may be convenient to express the neutron energy in meV, in which case the wavelength in ångströms is given by[\lambda \ ({\rm \AA}) =9.045/(E)^{1/2}\ (\rm meV).\eqno(2.3.2)]Thermal neutrons produced by a fission reactor have a representative energy of 25 meV, and accordingly a wavelength of 1.809 Å, which is well suited to the study of condensed matter since it is of the order of the interatomic spacings therein.

Neutrons have a number of distinctive properties making neutron diffraction uniquely powerful in several applications. They may be scattered by nuclei or by magnetic entities in the sample under study.

  • (a) Scattering by nuclei: The atomic nucleus is tiny compared with the atomic electron cloud, which is the entity that scatters X-rays and electrons. The scattering cross section for a particular nucleus is written as[\sigma =4\pi {b}^{2}, \eqno(2.3.3)]where σ is typically of the order of 10−28 m2 (1 × 10−28 m2 = 1 barn) and b, which is termed the scattering length, is of the order of femtometres. The small size of the nucleus relative to the wavelength of interest means that the scattering is isotropic – there is no angle-dependent form factor, as occurs in the X-ray case (cf. Section 1.1.3.1[link] ). This confers advantages in studies aimed at determining atomic displacement parameters (ADPs),1 and indeed for the total-scattering studies requiring data over a large Q range ([Q=4\pi \sin\theta /\lambda ]) that are described in Chapter 5.7[link] . Importantly, scattering lengths vary somewhat erratically with atomic number Z; this is in marked contrast to the X-ray case in which the form factor increases monotonically with Z (see Figs. 2.3.1[link] and 2.3.2[link]). This can make it much easier to detect the scattering from light (low-Z) elements in the presence of much heavier ones; it also makes it easier to distinguish scattering from elements adjacent in the periodic table, e.g. Cu with Z = 29, b = 7.718 fm and Zn with Z = 30, b = 5.680 fm. The scattering length is also different for different isotopes of the same element,2 e.g. for 1H b = −3.741 fm, whereas for 2H b = 6.671 fm, so that sometimes isotopic substitution can be employed to obtain contrast as desired.

    [Figure 2.3.1]

    Figure 2.3.1 | top | pdf |

    Representations of the scattering of X-rays and neutrons by selected elements. The scattering cross sections are proportional to the areas of the circles shown. For the neutron case, separate entries appear for the different isotopes and negative scattering lengths are indicated by shading. The figure is not intended to imply a relationship between the X-ray and neutron cross sections.

    [Figure 2.3.2]

    Figure 2.3.2 | top | pdf |

    Comparison of X-ray and neutron powder-diffraction patterns from rutile, TiO2. The patterns were recorded at the same wavelength, 1.377 Å. The differences between form factors and scattering lengths give rise to large differences in the relative intensities of the different peaks; note also that the fall off in the form factor evident in the X-ray case does not occur for neutrons.

  • (b) Scattering by magnetic entities: The neutron carries a magnetic moment of −1.913 μN (where μN is the nuclear magneton) and accordingly it interacts with magnetic entities in the sample. These may be nuclei, with magnetic moments of the order of the nuclear magneton, or atoms with much larger magnetic moments, of the order of the Bohr magneton (μB). If the magnetic entities are disordered, then the result is magnetic diffuse scattering, but if they are in some way ordered then the magnetic structure can be studied via the magnetic Bragg reflections that arise. (These may not be so obvious if they coincide with the nuclear Bragg reflections.) The magnetic moment of the neutron interacts with atomic magnetic moments, attributable to unpaired electrons in the atoms. These electrons tend to be the outer electrons, spread over dimensions comparable with atomic spacings and hence with the wavelengths used for diffraction; a consequence is that magnetic scattering is characterized by a magnetic form factor which falls off with Q more rapidly than does the form factor for the X-ray case (Fig. 2.3.3[link]). The confirmation of the antiferromagnetic ordering in MnO below its ordering (Néel) temperature of 120 K (Fig. 2.3.4[link]; Shull et al., 1951[link]) was the first of numerous studies of magnetic structure by neutron powder diffraction that have continued to the present day (Izyumov & Ozerov, 1970[link]; Chatterji, 2006[link]; Chapter 7 in Kisi & Howard, 2008[link]). Investigations of nuclear moments are more challenging largely because the smaller moments mean extremely low ordering temperatures; nevertheless neutron diffraction has been used, for example, to study the ordering of nuclear moments in metallic copper (65Cu) at temperatures below 60 nK (Hakonen et al., 1991[link]).3

    [Figure 2.3.3]

    Figure 2.3.3 | top | pdf |

    The magnetic form factor for Mn2+ compared with the normalized X-ray form factor and the normalized neutron nuclear scattering length.

    [Figure 2.3.4]

    Figure 2.3.4 | top | pdf |

    Magnetic structure for MnO proposed by Shull et al. (1951[link]). The figure shows only the Mn atoms, and indeed only those Mn atoms located on the visible faces of the cubic cell. [From Shull et al. (1951[link]), redrawn using ATOMS (Dowty, 1999[link]).]

  • (c) Low attenuation: The combination of the small scattering cross sections and generally low cross sections for absorption (notable exceptions are B, Cd and Gd) gives thermal neutrons the ability to penetrate quite deeply into most materials. Indeed, the linear attenuation coefficient for thermal (25 meV) neutrons in Fe is 110 m−1, and for neutrons in Al it is only about 9.8 m−1; the implication is that it takes about 10 cm of Al to reduce the intensity by a factor 1/e. The fact that neutrons are so little attenuated by these materials makes it easier to design large and complex sample-environment chambers which may be used for in situ studies at high temperature, under pressure or stress, in magnetic fields, and in reaction cells (Chapters 2.6[link] –2.9[link] ; Chapter 3 in Kisi & Howard, 2008[link]). Neutron powder diffraction is well suited to quantitative phase analysis (QPA, see Chapter 3.9[link] and Chapter 8 in Kisi & Howard, 2008[link]); as pointed out in Chapter 8, Section 8 of Kisi & Howard (2008[link]), neutron QPA provides a better sampling ability and is less prone to microabsorption errors than the X-ray technique; indeed, neutron diffraction was the method employed in one of the earliest and most convincing demonstrations of the Rietveld method in QPA (Hill & Howard, 1987[link]). Another advantage conferred by the deep penetration of neutrons is the ability to probe below the surface of samples to measure such aspects as structure, phase composition and stress; a particular example is the application to the analysis of zirconia ceramics (Kisi et al., 1989[link]) where the surface composition (as would be measured by X-rays) is unrepresentative of the bulk. A downside of the small scattering cross sections (along with neutron sources of limited `brightness') is that relatively large samples may be required.

  • (d) Low energy: We note from equation (2.3.1)[link] that, for a specified wavelength, the energy of the neutron is much less than that for lighter probes, such as electrons or photons. This is critically important for studying inelastic processes (e.g. measurement of phonon dispersion curves), but is usually not a factor in neutron powder diffraction.4

Neutron sources, in common with synchrotrons, are large national or international facilities, set up to cater for scientists from external laboratories. There are usually well defined access procedures, involving the submission and peer review of research proposals. Visiting users are usually assisted in their experiments by in-house staff. In some cases external users can mail in their samples for collection of diffraction data by the resident staff.

2.3.2. Neutrons and neutron diffraction – pertinent details

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2.3.2.1. Properties of the neutron

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The basic properties of the neutron are summarized in Table 2.3.1[link].

Table 2.3.1| top | pdf |
Properties of the neutron (adapted from Kisi & Howard, 2008[link])

Mass (m) 1.675 × 10−27 kg
Charge 0
Spin ½
Magnetic moment (μn) −1.913 μN
Wavelength (λ) h/mv
Wavevector (k) Magnitude 2π/λ
Momentum (p) [\hbar{\bf k}]
Energy (E) [({{1}/{2}})m{v}^{2}={{{h}^{2}}/{2m{\lambda }^{2}}}]

2.3.2.2. Neutron scattering lengths

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The scattering lengths of most interest in neutron powder diffraction are those for coherent elastic scattering, bcoh, often abbreviated to b. As already mentioned, there is no angle (Q) dependence, since the scattering from the nucleus is isotropic. A selection of scattering lengths for different isotopes and different elements is given in Table 2.3.2[link].

Table 2.3.2| top | pdf |
Coherent scattering lengths and absorption cross sections (for 25 meV neutrons) for selected isotopes

Data are taken from Section 4.4.4[link] of Volume C (Sears, 2006[link]). Where not stated, the values are for the natural isotopic mix. The X-ray atomic form factors, f, evaluated at Q = 1.2π Å−1, are included for comparison.

ElementIsotopebcoh (fm)σs(tot) (10−24cm2)σa (10−24cm2)fIsotopic abundance (%)
H   −3.7390 (11) 82.02 (6) 0.3326 (7) 0.25  
  1 −3.7406 (11) 82.03 (6) 0.3326 (7)   99.985
  2 6.671 (4) 7.64 (3) 0.000519 (7)   0.015
  3 4.792 (27) 3.03 (5) 0  
             
B   5.30 (4) − 0.213 (2)i 5.24 (11) 767 (8) 1.99  
  10 −0.1 (3) − 1.066 (3)i 3.1 (4) 3835 (9)   20.0
  11 6.65 (4) 5.78 (9) 0.0055 (33)   80.0
             
C   6.6460 (12) 5.551 (3) 0.00350 (7) 2.50  
  12 6.6511 (16) 5.559 (3) 0.00353 (7)   98.90
  13 6.19 (9) 4.84 (14) 0.00137 (4)   1.10
             
O   5.803 (4) 4.232 (6) 0.00019 (2) 4.09  
             
Ti   −3.370 (13) 4.06 (3) 6.43 (6) 13.2  
  46 4.725 (5) 2.80 (6) 0.59 (18)   8.2
  47 3.53 (7) 3.1 (2) 1.7 (2)   7.4
  48 −5.86 (2) 4.32 (3) 8.30 (9)   73.8
  49 0.98 (5) 3.4 (3) 2.2 (3)   5.4
  50 5.88 (10) 4.34 (15) 0.179 (3)   5.2
             
V   −0.3824 (12) 5.10 (6) 5.08 (2) 14.0  
             
Ni   10.3 (1) 18.5 (3) 4.49 (16) 18.7  
  58 14.4 (1) 26.1 (4) 4.6 (3)   68.27
  60 2.8 (1) 0.99 (7) 2.9 (2)   26.10
  61 7.60 (6) 9.2 (3) 2.5 (8)   1.13
  62 −8.7 (2) 9.5 (4) 14.5 (3)   3.59
  64 −0.37 (7) 0.017 (7) 1.52 (3)   0.91
             
Cu   7.718 (4) 8.03 (3) 3.78 (2) 19.9  
  63 6.43 (15) 5.2 (2) 4.50 (2)   69.17
  65 10.61 (19) 14.5 (5) 2.17 (3)   30.83
             
Zn   5.680 (5) 4.131 (10) 1.11 (2) 20.8  
             
Zr   7.16 (3) 6.46 (14) 0.185 (3) 27.0  
             
Gd   6.5 (5) 180 (2) 49700 (125) 45.9  
  155 6.0 (1) − 17.0 (1)i 66 (6) 61100 (400)   14.8
  157 −1.14 (2) − 71.9 (2)i 1044 (8) 259000 (700)   15.7
             
Pb   9.405 (3) 11.118 (7) 0.171 (2) 60.9  

The first thing to note is the variation in scattering length from element to element and indeed from isotope to isotope. The scattering lengths are in most cases positive real numbers, in which case there is a phase reversal of the neutron on scattering, but for some isotopes the scattering lengths are negative, so there is no change in phase on scattering. The scattering lengths are determined by the details of the neutron–nucleus interaction (Squires, 1978[link]).5 In the event that the neutron–nucleus system is close to a resonance, such as it is for 10B, 155Gd and 157Gd, scattering lengths will be complex quantities and the scattered neutron will have some different phase relationship with the incident one. The imaginary components imply absorption, which is reflected in the very high absorption cross sections, σa, for these isotopes.

The total scattering cross section, σs, is given by [{\sigma }_{s}=4\pi {b}_{\rm coh}^{2}] when only coherent scattering from a single isotope is involved, which is very nearly the case for oxygen since 99.76% of naturally occurring oxygen is zero-spin 16O. In most cases there is a more substantial contribution from incoherent scattering, which may be either spin or isotope incoherent scattering. Spin incoherent scattering arises because the scattering length depends on the relative orientation of the neutron and nuclear spins, parallel and antiparallel arrangements giving rise to scattering lengths [{b}_{+}] and [{b}_{-}], respectively. Isotope incoherent scattering arises because of the different scattering of neutrons from different isotopes of the same element. In almost all circumstances (except, for example, at the extraordinarily low temperatures mentioned in Section 2.3.1[link]) the distributions of spins and isotopes are truly random, which means that there is no angle dependence in this scattering: this is sometimes described as Laue monotonic scattering.

When b varies from nucleus to nucleus (even considering just a single element), the coherent scattering is determined by the average value of b, that is [{b}_{\rm coh}= \overline{b}], [{\sigma }_{\rm coh}=4\pi {(\overline{b})}^{2}], and the average incoherent cross section is given by [{\sigma }_{\rm inc}=4\pi [\overline{{b}^{2}}-{(\overline{b})}^{2}]]. The total scattering cross section σs is the sum of the two cross sections (Squires, 1978[link]; see also Section 2.3.2 in Kisi & Howard, 2008[link]). For the particular case of a nucleus with spin I, the states I + 1/2 and I − 1/2 give scattering determined by [{b}_{+}] and [{b}_{-}], respectively, and have multiplicities 2I + 2 and 2I, respectively, from which it follows that[\displaylines{{b}_{\rm coh}= \overline{b}={{I+1}\over{2I+1}}{b}_{+}+{{I}\over{2I+1}}{b}_{-},\cr {b}_{\rm inc}^{2}=\left [\overline{{b}^{2}}-{(\overline{b})}^{2}\right]={{I(I+1)}\over{{(2I+1)}^{2}}}{({b}_{+}-{b}_{-})}^{2}.}]

More information, including a comprehensive listing of scattering lengths, can be found in Section 4.4.4[link] of International Tables for Crystallography Volume C (Sears, 2006[link]). This listing presents the spin-dependent scattering lengths via bcoh and binc as just defined. Other compilations can be found in the Neutron Data Booklet (Rauch & Waschkowski, 2003[link]), and online through the Atominstitut der Österreichischen Universitäten, Vienna, at http://www.ati.ac.at/~neutropt/scattering/table.html . In addition, the majority of computer programs used for the analysis of data from neutron diffraction incorporate, for convenience, a list of bcoh values for the elements.

2.3.2.3. Refractive index for neutrons

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The coherent scattering lengths of the nuclei determine the refractive index for neutrons through the relationship (Squires, 1978[link])[n=1-{{1}\over{2\pi }}{\lambda }^{2}N{b}_{\rm coh}, \eqno(2.3.4)]where N is the number of nuclei per unit volume. For elements with positive values of the coherent scattering length the refractive index is slightly less than one, and that leads to the possibility of total external reflection of the neutrons by the element in question. In fact, when the coherent scattering length is positive, neutrons will undergo total external reflection for glancing angles less than a critical angle γc given by[\cos{\gamma }_{c}= n=1-{{1}\over{2\pi }}{\lambda }^{2}N{b}_{\rm coh}, \eqno(2.3.5)]which, since γc is small, reduces to[{\gamma }_{c}=\lambda \left({{{N{b}_{\rm coh}}\over{\pi }}}\right)^{1/2}.\eqno(2.3.6)]

It can be seen that the pertinent material quantity is Nbcoh, the `coherent scattering length density'; for materials comprising more than one element this is the quantity that would be computed. Since the critical angle for total external reflection is proportional to the neutron wavelength, it is convenient to express this as degrees per ångstrom of neutron wavelength. These are important considerations in the design and development of neutron guides (Section 2.3.3.4[link]).

2.3.2.4. Neutron attenuation

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Neutron beams are attenuated by coherent scattering, incoherent scattering and true absorption. The cross sections for all these processes are included in the tables cited above. For powder diffraction, the coherent scattering is usually small because it takes place only in that small fraction of crystallites correctly oriented for Bragg reflection; the other processes, however, take place throughout the sample.

If a particular scattering entity i with scattering cross sections (σi)inc and (σi)abs is present at a number density Ni, then the contribution it makes to the linear attenuation coefficient μ is [{\mu }_{i}={N}_{i}[{{(\sigma }_{i})}_{\rm inc}+{{(\sigma }_{i})}_{\rm abs}]]. If the mass is Mi, then the density is simply [{\rho }_{i}={N}_{i}{M}_{i}], so we have the means to evaluate the mass absorption coefficient [(\mu /\rho)_{i}]. The calculation of absorption for elements, compounds and mixtures commonly proceeds by the manipulation of mass absorption coefficients, in the same manner as is employed for X-rays (see Section 2.4.2 in Kisi & Howard, 2008[link]).

2.3.2.5. Magnetic form factors and magnetic scattering lengths

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For a complete treatment of the magnetic interaction between the neutron and an atom carrying a magnetic moment, and the resulting scattering, the reader is referred elsewhere [Marshall & Lovesey, 1971[link]; Squires, 1978[link]; Section 6.1.2[link] of Volume C (Brown, 2006[link] a[link])]. The magnetic moment of an atom is associated with unpaired electrons, but may comprise both spin and orbital contributions. The magnetic interaction between the neutron and the atom depends on the directions of the scattering vector and the magnetic moment vector according to a triple vector product. The direction of polarization of the neutron must also be taken into account. For an unpolarized incident beam, the usual case in neutron powder diffraction, it is a useful consequence of the triple vector product that the magnetic scattering depends on the sine of the angle that the scattering vector makes with the magnetic moment on the scattering atom (see Section 2.3.4 and Chapter 7 in Kisi & Howard, 2008[link]). The extent of the unpaired electron distribution (usually outer electrons) implies that the scattering diminishes as a function of Q, an effect that can be described by a magnetic form factor. For a well defined direction for the magnetic moment M, and with a distribution of moment that can be described by a normalized scalar m(r), the form factor as a function of the scattering vector h [defined in equation (1.1.17)[link] in Chapter 1.1]6 is the Fourier transform of m(r),[f\left({\bf h}\right)=\textstyle\int m({\bf r})\exp\left(2\pi i{\bf h}\cdot{\bf r}\right)\,{\rm d}{\bf r},]where m(r) can comprise both spin and orbital contributions [Section 6.1.2[link] of Volume C (Brown, 2006a[link])]. The tabulated form factors are based on the assumption that the electron distributions are spherically symmetric, so that [m({\bf r})=m(r)={U}^{2}(r)], where U(r) is the radial part of the wave function for the unpaired electron. In the expansion of the plane-wave function [\exp(2\pi i{\bf h}\cdot{\bf r})] in terms of spherical Bessel functions, we find that the leading term is just the zeroth-order spherical Bessel function [{j}_{0}(2\pi hr)] with a Fourier transform[\langle {j}_{0}(h)\rangle =4\pi \textstyle\int\limits_{0}^{\infty }{U}^{2}\left(r\right){j}_{0}\left(2\pi hr\right){r}^{2}\, {\rm d}r.]

This quantity is inherently normalized to unity at h = 0, and may suffice to describe the form factor for spherical spin-only cases. In other cases it may be necessary to include additional terms in the expansion, and these have Fourier transforms of the form[\langle {j}_{l}(h)\rangle =4\pi \textstyle\int\limits_{0}^{\infty }{U}^{2}(r){j}_{l}(2\pi hr){r}^{2}\, {\rm d}r]with l even; these terms are zero at h = 0 (Brown, 2006[link] a[link]). In practice these quantities are evaluated using theoretical calculations of the radial distribution functions for the unpaired electrons [Section 4.4.5[link] of Volume C (Brown, 2006b[link])].

Form factors can be obtained from data tabulated in Section 4.4.5[link] of Volume C (Brown, 2006b[link]). Data are available for elements and ions in the 3d- and 4d-block transition series, for rare-earth ions and for actinide ions. These data are provided by way of the coefficients of analytical approximations to [\langle {j}_{l}(h)\rangle ], the analytical approximations being[\langle {j}_{0}(s)\rangle =A\exp\left(-a{s}^{2}\right)+B\exp\left(-b{s}^{2}\right)+C\exp\left(-c{s}^{2}\right)+D]and for l ≠ 0[\langle {j}_{l}(s)\rangle ={s}^{2}\left[A\exp\left(-a{s}^{2}\right)+B\exp\left(-b{s}^{2}\right)+C\exp\left(-c{s}^{2}\right)+D\right],]where s = h/2 in Å−1. These approximations, with the appropriate coefficients, are expected to be coded in to any computer program purporting to analyse magnetic structures. Although the tabulated form factors are based on theoretical wave functions, it is worth noting that the incoherent scattering from an ideally disordered (i.e., paramagnetic) magnetic system will display the magnetic form factor directly.

It is often convenient to define a (Q-dependent) magnetic scattering length[p=\left({{{e}^{2}\gamma }\over{2{m}_{e}{c}^{2}}}\right)gJf,]where me and e are the mass and charge of the electron, γ (= μn) is the magnetic moment of the neutron, c is the speed of light, J is the total angular momentum quantum number, and g is the Landé splitting factor given in terms of the spin S, orbital angular momentum L, and total angular momentum quantum numbers by[g=1+{{J\left(J+1\right)+S\left(S+1\right)-L(L+1)}\over{2J(J+1)}}.]

For the spin-only case, L = 0, J = S, so g = 2. The differential magnetic scattering cross section per atom is then given by [{q}^{2}{p}^{2}] where [|q|=\sin\alpha ], α being the angle between the scattering vector and the direction of the magnetic moment. This geometrical factor is very important, since it can help in the determination of the orientation of the moment of interest; there is no signal, for example, when the moment is parallel to the scattering vector. Further discussion appears in Chapters 2 (Section 2.3.4) and 7 in Kisi & Howard (2008[link]).

2.3.2.6. Structure factors

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The locations of the Bragg peaks for neutrons are calculated as they are for X-rays7 (Section 1.1.2[link] ), and the intensities of these peaks are determined by a structure factor, which in the nuclear case is [cf. Chapter 1.1, equation (1.1.56)[link] ][{ F}_{hkl}^{\rm nuc}=\textstyle \sum \limits_{i=1}^{m}{b}_{i}{T}_{i}\exp(2\pi i{\bf h}\cdot{{\bf u}}_{i}), \eqno(2.3.7)]where bi here denotes the coherent scattering length, Ti has been introduced to represent the effect of atomic displacements (thermal or otherwise, see Section 2.4.1 in Kisi & Howard, 2008[link]), h is the scattering vector for the hkl reflection, and the vectors ui represent the positions of the m atoms in the unit cell.

For coherent magnetic scattering, the structure factor reads[{ F}_{hkl}^{\rm mag}=\textstyle \sum \limits_{i=1}^{m}{p}_{i}{{\bf q}}_{i}{T}_{i}\exp(2\pi i{\bf h}\cdot{{\bf u}}_{i}), \eqno(2.3.8)]where pi is the magnetic scattering length. The vector qi is the `magnetic interaction vector' and is defined by a triple vector product (Section 2.3.4 in Kisi & Howard, 2008[link]), and has modulus sin α as already mentioned. In this case the sum needs to be taken over the magnetic atoms only.

As expected by analogy with the X-ray case, the intensity of purely nuclear scattering is proportional to the square of the modulus of the structure factor [|{F}_{hkl}^{\rm nuc}|^{2}]. In the simplest case of a collinear magnetic structure and an unpolarized incident neutron beam, the intensity contributed by the magnetic scattering is proportional to [{|{F}_{hkl}^{\rm mag}|}^{2}], and the nuclear and magnetic contributions are additive.

2.3.3. Neutron sources

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2.3.3.1. The earliest neutron sources

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The earliest neutron source appears to have been beryllium irradiated with α-particles (helium nuclei), as emitted for example by polonium or radon. First described as `beryllium radiation', the radiation from a Po/Be source was identified by Chadwick (1932[link]) as comprising neutrons:[_{2}^{4}{\rm He} + {}_{4}^{9}{\rm Be}\to{} _{{\phantom 1}6}^{12}{\rm C} + {}_{0}^{1}{\rm n}.]

It was soon found (Szilard & Chalmers, 1934[link]) that the disintegration of beryllium under irradiation by the γ-rays from radium also led to the release of neutrons; this represented an alternative neutron source. The first demonstrations of the diffraction of neutrons (Mitchell & Powers, 1936[link]; von Halban & Preiswerk, 1936[link]) made use of Rn/Be sources, analogous to Chadwick's Po/Be source. These were now surrounded by paraffin to reduce the energy (`moderate') and hence increase the de Broglie wavelength of the neutrons, and so provide a reasonable match to the atomic spacings in the crystalline samples; in the Mitchell & Powers' demonstration the reflection of neutrons of estimated wavelength 1.6 Å from (100) planes in large single crystals of MgO, the separation of these planes being 4.2 Å, showed a dependence on crystal orientation that was indicative of Bragg reflection. The intensities available from these sources, however, were not sufficient to allow the observation of diffraction from polycrystalline (powder) samples.

A source based on the bombardment of Be by cyclotron-accelerated MeV deuterons (nuclei of deuterium)[{}_{1}^{2}{\rm H}+{}_{4}^{9}{\rm Be}\to {}_{\phantom{1}5}^{10}{\rm B}+{}_{0}^{1}{\rm n}]was also employed in early work, notably by Alvarez & Bloch (1940[link]) in their determination of the neutron magnetic moment.

The further development of neutron diffraction, and indeed the first observation of neutron powder diffraction, awaited the development of much more intense neutron sources; the first suitably intense neutron sources were nuclear reactors. The neutron-induced fission of uranium isotope [_{92}^{235}{\rm U}] was observed in 1938 and reported early in 1939 (Hahn & Strassmann, 1939[link]; Meitner & Frisch, 1939[link]; Anderson et al., 1939[link]). By this time Fermi and his co-workers (Fermi, Amaldi, D'Agostino et al., 1934[link]; Fermi, Amaldi, Pontecorvo et al., 1934[link]) had already carried out studies on neutron activation, in the course of which they found that neutrons could be moderated by hydrogenous materials, providing `slow' neutrons for which the activation cross sections were enhanced. Once it was established that the neutron-induced fission of a [_{92}^{235}{\rm U}] nucleus also led to the release of ~2–3 `fast' neutrons plus energy (von Halban et al., 1939[link]; Zinn & Szilard, 1939[link]), then a self-sustaining `chain reaction' based on the fission of [_{92}^{235}{\rm U}] by a slow neutron, the slowing in a moderator of the several fast neutrons released, followed by the slow-neutron-induced fission of additional [_{92}^{235}{\rm U}] nuclei, became a realistic possibility. The translation of this possibility into reality was given great impetus by the military potential of the chain reaction; the reader is referred to Mason et al. (2013[link]) for the history of this development. The first self-sustaining chain reaction took place in Chicago Pile 1 (CP-1) on 2 December 1942. CP-1 made use of uranium oxide mixed with some metallic uranium as fuel, high-purity graphite as the neutron moderator and rods of neutron-absorbing cadmium for control. CP-1 was located on a squash court under the spectator stand at a sports field at the University of Chicago; remarkably, its construction took less than a month. In November 1943, an essentially scaled up version of this reactor, the X-10 pile (also known as the Oak Ridge Graphite Reactor) achieved criticality. The fuel was now metallic uranium, and the greater power (1 MW as compared with the 200 W of CP-1) necessitated an air cooling system; the neutron flux8 was a creditable 1012 n cm−2 s−1 and the main purpose was the production of plutonium. May 1944 saw the completion of yet another reactor, Chicago Pile 3 (CP-3), outside Chicago at the site of the present Argonne laboratories. This was a 300 kW reactor, using natural uranium fuel, with heavy water serving as both moderator and coolant; this also provided a flux of 1012 n cm−2 s−1.

Early diffraction experiments using reactor neutrons were carried out `in the wings of the Manhattan project' (Mason et al., 2013[link]). Evidently, Wollan & Borst (1945[link]) obtained rocking curves when collimated thermal neutrons from X-10 were beamed onto single crystals of gypsum and rocksalt, while Zinn was able to reflect neutrons from a calcite crystal [see, for example, the post-war publication by Zinn (1947[link])]; much of the wartime interest was in using these crystals for neutron spectrometry. However, the potential use of these copious sources of neutrons was recognized, so by the early months of 1946 (according to Shull, 1995[link]) the first neutron powder-diffraction patterns, from polycrystalline NaCl and from light and heavy water, had been recorded. Wollan and co-workers (Wollan & Shull, 1948[link]; Shull et al., 1948[link]) published a number of these early diffraction patterns, along with a schematic of the diffractometer employed.

Although accelerator-based neutron sources had been around as early as 1940 (see above), the development of such sources, at least for diffraction applications, proceeded at a relatively slower pace. Indeed, it was not until 1968 that the first reports of neutron powder diffraction using accelerator-based sources appeared in the literature (Moore et al., 1968[link]; Kimura et al., 1969[link]; Day & Sinclair, 1969[link]). All this work involved the use of linear electron accelerators (LINACs) delivering pulses of ~150 MeV electrons onto a heavy-metal target; the deceleration results in Bremsstrahlung radiation (photons) of sufficient energy to bring about the release of neutrons from the target. These fast neutrons were moderated, and the result was a pulsed source of thermal neutrons. Diffraction patterns were recorded by time-of-flight methods which had already been developed on reactor sources (Buras & Leciejewicz, 1964[link]).

It may be helpful to describe one of these experiments in more detail (Kimura et al., 1969[link]). A tungsten target immersed in water was bombarded by 2.5 µs pulses of 250 MeV electrons from the Tohoku LINAC; the water, which served as a moderator, was also `poisoned' by the addition of neutron-absorbing boric acid. The thermal neutron pulses were of 30–50 µs duration. It is a fundamental problem that the time taken to moderate the fast neutrons produced at an accelerator-based source degrades the time structure, and the addition of boron here was one method to counteract this effect. Kimura et al. presented a selection of time-of-flight diffraction patterns, from Al at different temperatures, as well as from Si, Ni, ZnO, CaFe2O4 and α-Fe2O3.

The next generation of accelerator-based sources were spallation sources, based on the breaking up of heavy target elements by bombardment with 10–1000 MeV protons; up to ~30 neutrons are ejected in each spallation event; such sources can be operated in either a pulsed mode or continuously. The first spallation sources were ZING-P (100 nA of 300 MeV protons, pulsed at 30 Hz, target Pb, moderator polyethylene) and ZING-P′ (3 µA of 500 MeV protons, 30 Hz, target W/natural U, moderator polyethylene/liquid hydrogen), both at the Argonne National Laboratory (Carpenter, 1977[link]), and at the TRIUMF laboratory (400 µA of 500 MeV protons, steady, target liquid Pb/Bi, moderator light/heavy water) in Vancouver. The KENS facility (operational from 1980 to 2005, 9 µA of 500 MeV protons, 20 Hz, target W, moderator solid methane/ice) in Tsukuba, Japan, and the Intense Pulsed Neutron Source (IPNS) at the Argonne National Laboratory (operational from 1981 to 2008, 15 µA of 450 MeV protons, 30 Hz, target depleted U, moderator solid/liquid methane) are both worthy of mention for their work on techniques and applications at pulsed neutron sources; notable are contributions from IPNS on the subjects of high-temperature superconductors (Jorgensen et al., 1987[link]) and colossal magneto-resistance (Radaelli et al., 1997[link]).

The specifications and performance of modern currently operating spallation neutron sources will be presented in Section 2.3.3.3[link].

2.3.3.2. Fission reactors for neutron-beam research

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Reactors used for neutron-beam research all rely on the fissile uranium isotope9 [_{92}^{235}{\rm U}]. This constitutes only about 0.7% of natural uranium; however, enrichment in this isotope is possible. A representative fission event would be[_{0}^{1}{\rm n} +{} _{\phantom{2}92}^{235}{\rm U}\to{} _{\phantom{2}92}^{236}{\rm U}\to {}_{\phantom{1}56}^{141}{\rm Ba} + {}_{36}^{92}{\rm Kr} +{}3_{0}^{1}{\rm n} +170\ {\rm MeV}.]This equation indicates that a neutron of thermal energy is captured by 235U to form 236U in an unstable state, and in the majority of cases (88%) this breaks up almost instantly to yield fission products of intermediate mass, fast neutrons and energy. The unstable 236U can break up in many different ways – there are usually products of intermediate but unequal masses, with masses distributed around 95 and 135 (Burcham, 1979[link]), with the release of usually 2 or 3 neutrons (average 2.5; one of these neutrons is needed to initiate the next fission event), and of different amounts of energy (average around 200 MeV). As explained in Section 2.3.3.1[link], a chain reaction becomes possible if the fast neutrons released in the fission process are moderated to thermal energies so that they can be captured by another 235U nucleus. Neutrons will lose energy most rapidly through collisions with nuclei of mass equal to the neutron mass, namely nuclei of hydrogen atoms, but collisions with other light nuclei are also quite effective. Hydrogenous substances are evidently useful, and water would seem ideal; however, there is some absorption of neutrons in water, so in some reactors, heavy water (D2O, where D is [_{1}^{2}{\rm H}]) is used since, as can be seen from the absorption cross sections (Table 2.3.2[link]), thermal neutron capture in D is orders of magnitude less than for H. It has not been possible to achieve a self-sustaining chain reaction using natural uranium and light water as a moderator – for this reason uranium fuel enriched in 235U and/or heavy-water moderators are in use. Adjacent to the reactor core is a so-called reflector, which is simply in place to moderate neutrons and prevent their premature escape. The energy released in the fission process ends up as heat, which must be dissipated (or used), so cooling is required – where light or heavy water is used as the moderator it can also serve as the coolant. Control rods are also essential – these are rods containing highly neutron absorbing materials, such as boron, cadmium or hafnium, which can be inserted into or withdrawn from the reactor to increase, maintain or reduce the thermal neutron flux as required. These control rods provide the means for reactor shutdown.

The neutrons in a reactor core range from the fast neutrons (∼1 MeV) released in the fission process, through epithermal neutrons (in the range eV to keV), which are neutrons in the process of slowing down, to thermal neutrons (∼25 meV), which are neutrons in equilibrium with the moderator (see Carlile, 2003[link]). Evidently, for sustaining the chain reaction and for providing neutrons for diffraction instruments, the thermal neutrons are of the greatest interest. Neutrons in thermal equilibrium with the moderator have a Maxwellian distribution of energies, such that the number of neutrons with energies between E and E + dE is given by N(E) dE, where[N\left(E\right)={{2\pi {N}_{0}}\over{{\left(\pi {k}_{B}T\right)}^{3/2}}}(E)^{1/2}\exp(-{{E}/{{k}_{B}T}}). \eqno(2.3.9)]Here N0 is the total number of neutrons, T is the temperature (in kelvin) of the moderator, and kB is Boltzmann's constant. The neutron flux is the product of the neutron density with the neutron speed, so the energy dependence of the flux distribution takes the form[ \varphi (E)={\varphi }_{0}{{E}\over{({{k}_{B}T)}^{3/2}}}\exp(-{{E}/{{k}_{B}T}}). \eqno(2.3.10)]This distribution takes its peak value at E = kBT; for a temperature of 293 K, this leads to a peak in the flux distribution at 25.2 meV (cf. Section 2.3.1[link]). In the diffraction context the wavelength dependence of the flux is of more interest. Making use of the relationships [E={{{h}^{2}}/{2m{\lambda }^{2}}}] and [{{{\rm d}E}/{{\rm d}\lambda }}=-{{{h}^{2}}/{m{\lambda }^{3}}}], we find that the variation of flux with wavelength can be described by [\varphi (\lambda)\,{\rm d}\lambda ], where[ \varphi (\lambda)\propto {\lambda }^{-5}\exp(-{{{h}^{2}}/{2m{\lambda }^{2}{k}_{B}T}}) .\eqno(2.3.11)]This distribution peaks at [\lambda =h/({5m{k}_{B}T})^{1/2}]; at 293 K the peak in this wavelength distribution is at 1.15 Å. For some applications of neutron diffraction it may be desirable to have a greater neutron flux at shorter or longer wavelengths; as indicated in Fig. 2.3.5[link] this can be achieved by cooling or heating strategically placed special moderators.

[Figure 2.3.5]

Figure 2.3.5 | top | pdf |

The Maxwellian distribution of neutron wavelengths produced within moderators at different temperatures. Reproduced from Kisi & Howard (2008[link]) by permission of Oxford University Press .

As one specific example of a research reactor, we consider the NBSR located at the National Institute of Standards and Technology, Gaithersburg, USA. This reactor uses highly enriched (93% 235U) uranium in U3O8-Al as fuel, and heavy water as moderator and coolant. The thermal neutron flux in this reactor is 4 × 1014 n cm−2 s 1. It uses four cadmium control blades. An early plan view of this reactor and a cutaway view of the core assembly are shown in Fig. 2.3.6[link]. Note the presence of numerous beam tubes that allow neutrons to be taken out from the vicinity of the reactor core. This view of the NBSR (Fig. 2.3.6[link]a) shows provision for a cold neutron source, and for beam tubes to transport cold neutrons to experiments, but it was years before any cold neutron source was installed. The first cold source, installed in 1987, was frozen heavy water; this was replaced in 1995 by a liquid-hydrogen cold source, and that was upgraded in turn in 2003. The NBSR first went critical in December 1967; the history of its subsequent development and use in neutron-beam research has been recounted by Rush & Cappelletti (2011[link]).

[Figure 2.3.6]

Figure 2.3.6 | top | pdf |

The NBSR at the National Institute of Standards and Technology Center for Neutron Research. Part (a) is a plan view (reproduced from Rush & Cappelletti, 2011[link]) while (b) is a recent cutaway view of the reactor core showing the liquid-hydrogen cold source on the right-hand side.

The HFR at the Institut Laue–Langevin (ILL), considered to be the premier source for reactor-based neutron-beam research, serves as our second example. It too uses highly enriched uranium, here in a single centrally located U3Alx-Al fuel element, and it relies on heavy water for moderator and coolant. It operates at 58 MW and the thermal neutron flux is 1.5 × 1015 n cm−2 s−1. The reactor incorporates two liquid-deuterium cold sources, operating at 20 K, and a graphite hot source operating at 2000 K. In the HFR, being of modern design and purpose-built for neutron-beam research, the beam tubes do not view the core directly, but are `tangential' to it (Fig. 2.3.7[link]); this reduces the unwelcome fast-neutron component of the emerging beams. The HFR achieved criticality in July 1971. More details on this reactor can be found in the `Yellow Book' which is maintained on the ILL web site, https://www.ill.eu .

[Figure 2.3.7]

Figure 2.3.7 | top | pdf |

Schematic diagram of the HFR operated by the Institut Laue–Langevin in Grenoble, France. It has a compact core – the beam tubes avoid viewing the central core in favour of the surrounding moderator. This reactor also features hot (red) and cold (blue) sources. (Diagram reproduced with permission from the ILL from The Yellow Book 2008, https://www.ill.eu/fileadmin/users_files/Other_Sites/YellowBook2008CDRom/index.htm .)

From the opening paragraph of this section, it might be concluded that the more heavy water deployed, and the more highly is the uranium enriched in the fissile isotope 235U, the greater the neutron fluxes that can be obtained. This conclusion would be correct, but concerns about nuclear proliferation have brought a shift to the use of low-enrichment uranium (LEU) in which the 235U is enriched to less than 20%; however, in some reactors highly enriched uranium (HEU) with enrichment levels greater than 90% remains in use. Table 2.3.3[link] gives pertinent details on a number of research reactors important for neutron diffraction. Additional reactors are listed by Kisi & Howard (2008[link]) in their Table 3.1, and a complete listing is available from the International Atomic Energy Agency Research Reactor Database (IAEA RRDB, https://nucleus.iaea.org/RRDB/RR/ReactorSearch.aspx ).

Table 2.3.3| top | pdf |
Details on selected research reactors

The primary source of data is the IAEA Research Reactor Database (RRDB). The publicly accessible RRDB does not include information on fuel: limited information on this has been found from other internet sources.

Reactor (type)Power (MW)LocationFuel (see text)Moderator/coolantReflectorThermal flux (n cm−2 s−1)Cold/hot neutron sources
CARR (tank in pool) 60 CIAE, Beijing, China U3Si2-Al, LEU 19.75% Light water Heavy water 8 × 1014 1 cold
FRM-II (pool) 20 TUM, Garching, Germany U3Si2-Al, HEU Light water Heavy water 8 × 1014 1 cold, 1 hot
HANARO (pool) 30 KAERI, Daejeon, Korea U3Si, LEU 19.75% Light water Heavy water 4.5 × 1014 1 cold
HFIR (tank) 85 ORNL, Oak Ridge, USA U3O8-Al, HEU 93% Light water Beryllium 2.5 × 1015 1 cold
HFR (heavy water) 58.3 ILL, Grenoble, France U3Alx-Al, HEU Heavy water Heavy water 1.5 × 1015 2 cold, 1 hot
JRR-3M (pool) 20 JAEA, Tokai, Japan U3O8-Al, U3Si2-Al, LEU Light water Light water, heavy water, beryllium 2.7 × 1014 1 cold
NBSR (heavy water) 20 NIST, Gaithersburg, USA U3O8-Al HEU 93% Heavy water Heavy water 4 × 1014 1 cold
OPAL (pool) 20 ANSTO, Sydney, Australia U3Si2-Al, LEU 19.75% Light water Heavy water 2 × 1014 1 cold
This reactor has been temporarily shut down.

2.3.3.3. Spallation neutron sources

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The bombardment of heavy-element nuclei by high-energy protons, i.e. protons in the energy range 100 MeV to GeV, causes the nuclei to break up with the release of large numbers of neutrons. The word `spallation' might suggest that neutrons are simply being chipped off the target nucleus, and indeed neutrons can be ejected by protons in a direct collision process with transfer of the full proton energy, but such simple events are relatively rare. In most cases there is a sequence involving incorporation of the bombarding proton into the nucleus, intra- and internuclear cascades accompanied by the ejection of assorted high-energy particles, including neutrons, and then an `evaporation' process releasing neutrons from excited nuclei with energies comparable to those released in the fission process (Carpenter, 1977[link]; Carlile, 2003[link]; Arai & Crawford, 2009[link]). The numbers of neutrons released in these various processes depend on the proton energies and the target materials employed; for 1 GeV protons on a Pb target, around 25 neutrons are released per bombarding proton (Arai & Crawford, 2009[link]). Target materials in use include Hg, Pb, W, Ta and 238U (depleted uranium). The yield of neutrons per proton for non-fissionable target materials is approximated by [0.1(E-0.12)(A+20)] where E is the proton energy in GeV and A is the atomic number of the target nucleus; for a target such as 238U that is fissionable under bombardment by high-energy neutrons the yield is almost double that. Generally, the energy to be dissipated as heat in the spallation process will be no more than the energy of the bombarding proton, so for the example of 1 GeV protons on Pb it should not exceed 40 MeV per neutron produced. Nevertheless, cooling requires attention. The use of liquid targets such as Hg, and Pb either in pure form or in a Pb-Bi eutectic alloy, facilitates the dissipation of heat. Solid targets are usually water cooled. The fast neutrons from spallation need to be moderated, not in this case for sustaining the process, but simply to make them useful for diffraction and other applications. Moderators in common use include water, heavy water, liquid or solid methane (CH4), and liquid hydrogen (H2). The volumes of moderator are usually small, for reasons that will be explained below.

Most spallation neutron sources, though by no means all, operate in `short-pulse mode', then employ time-of-flight methods in their instrumentation. The duration of the neutron pulse is critical in determining the time-of-flight resolution. Short-pulse operation depends first of all on a short-pulse structure of the bombarding protons. This is inherent in proton-accelerating systems that incorporate synchrotron accelerators or accumulator rings, since the protons become bunched10 while travelling around these rings, and pulses of duration <1 µs are delivered. The frequency of these pulses is modest, say 50 Hz, in part to reduce power requirements, but also to avoid the situation in which the desired thermal (or cold) neutrons from one pulse are overtaken by fast neutrons from the next. For short-pulse operation, the proton pulse must be translated into a still-short pulse of moderated neutrons; this has significant implications for moderator design (Tamura et al., 2003[link]; Arai, 2008[link]; Arai & Crawford, 2009[link]; Batkov et al., 2013[link]; Zhao et al., 2013[link]; Thomsen, 2014[link]). The normal processes of moderation – neutrons giving up energy in collisions with nuclei in the moderator until thermal equilibrium is achieved – need to be to some extent curtailed. One means to curtail these processes is to use only a small volume of moderator, so neutrons escape before spending excessive time in it. Another is to place neutron absorbers – cadmium or gadolinium – around the moderator, or indeed incorporate these absorbing materials into it, so that the slow neutrons remaining in the moderator are absorbed before the pulse length becomes excessive; in this case the moderator is said to be `decoupled' from the target. For cold-neutron moderators on short-pulse spallation sources the use of an ambient-temperature `pre-moderator' may be advantageous. Whatever the means to limit the dwell time in the moderator, the emerging neutrons will be under-moderated, hence their spectrum will contain more epithermal neutrons (i.e. neutrons with energies of the order of eV to keV) than fully moderated neutrons from a continuous source. Fig. 2.3.8[link] shows the results for energy spectra and pulse length, from Monte Carlo calculations, for different cryogenic moderators for the J-PARC spallation neutron source, Tokai, Japan. The neutron dwell time and therefore the pulse length are calculated to be smaller in the decoupled moderators (Fig. 2.3.8[link]b), but comparison with the coupled moderator (Fig. 2.3.8[link]a) shows that intensity is sacrificed. The pulse length in the high-energy region, and at lower energies for the poisoned moderators, varies as roughly [1/(E)^{1/2}]; from equation (2.3.1)[link] this makes the pulse length [\Delta t] proportional to the wavelength λ. In a time-of-flight analysis we measure the flight time t over a length L; noting that [v=L/t] and using that same equation we find that t is also proportional to λ, viz. [t=({{mL}/{h}})\lambda]. The result is that the time resolution [\Delta t/t] is independent of flight time (or wavelength), which is a very satisfactory state of affairs (see Section 2.3.4.2.1).

[Figure 2.3.8]

Figure 2.3.8 | top | pdf |

(a) The neutron energy distribution (flux) of the J-PARC neutron source for coupled, decoupled and poisoned decoupled moderators. The flux consists of a Maxwell distribution at low energies and a 1/E region at higher energies. (b) Pulse duration as a function of energy calculated for the same moderators. For the decoupled moderators, the peak widths vary approximately as 1/E1/2. Reproduced from Tamura et al. (2003[link]).

As mentioned earlier, there is the problem for time-of-flight analysis that the slower neutrons from one pulse might be overtaken by the first arrivals from the next – a problem known as `frame overlap'. Taking the example of 25 meV thermal neutrons, at a speed of 2190 m s−1 and a 50 Hz pulse repetition frequency, the neutrons from one pulse will have travelled 44 m when the next pulse occurs. If instrument flight paths are longer than this, or indeed if slower neutrons are involved, then the frame-overlap problem is encountered. A conceptually simple approach is to reduce the pulse frequency, and this has been implemented at the UK's ISIS neutron facility where Target Station 2 takes just one pulse in five from the proton-acceleration system, reducing the effective pulse frequency to 10 Hz; the other four pulses are directed to Target Station 1. Neutron choppers provide an alternative means to address this problem. The simplest kind of chopper is a disc (Fig. 2.3.9[link]), usually of aluminium, nickel alloy or carbon fibre, coated in part with neutron-absorbing material such as boron, cadmium or gadolinium, rotating in a synchronous relationship with the source. A chopper located near to the source can be adjusted to block the fast neutrons and γ-rays that emerge immediately, but allow through neutrons in a restricted time window, from T0 to T0 + ΔT, measured from the time of the pulse. Evidently, time T0 + ΔT cannot exceed the time for a single rotation of the disc; when the disc is rotating at the pulse-repetition frequency this is the time between pulses. If the disc-rotation frequency is a submultiple of the pulse frequency, i.e. the rotation frequency is the pulse frequency divided by n, then the time window ΔT can be set to select only every nth pulse from the source. A two-chopper arrangement is used, for example, in the 96 m flight path of the High Resolution Powder Diffractometer (HRPD) at the ISIS facility; the first chopper at 6 m from the source runs at the pulse frequency and the second at 9 m from the source runs at one-fifth or one-tenth of that frequency, so that only every fifth or tenth pulse is used (HRPD user manual, http://www.isis.stfc.ac.uk/Pages/hrpd-manual.pdf ).

[Figure 2.3.9]

Figure 2.3.9 | top | pdf |

One of the disc choppers in use at the ISIS neutron facility. This is an aluminium (2014A) alloy disc, and the neutron-absorbing coating (the darker region) is boron carbide in a resin. The cut-out on the right-hand side provides the aperture for neutrons. (Credit: STFC.)

Although we have introduced neutron choppers in the context of spallation sources, we should acknowledge that mechanical choppers and velocity selectors have a long history, dating back long before the advent of spallation sources. In fact, the first report on a velocity selector (Dunning et al., 1935[link]) pre-dates even the earliest demonstrations of neutron diffraction. Mechanical systems have long been used at continuous neutron sources to act as velocity (wavelength) selectors, and/or to tailor pulses of neutrons suitable for time-of-flight studies. Two disc choppers can be arranged to serve both purposes – the first chopper has a limited aperture transmitting a short pulse of neutrons, and the second chopper, with a similar aperture and located at some distance from the first, is phased so as to allow through only those neutrons with a particular velocity. This arrangement can provide short pulses of more-or-less monochromatic neutrons to an experiment. The helical velocity selector (Friedrich et al., 1989[link]) is conceptually somewhat similar. This takes the form of a cylinder, or indeed a stack of discs, rotating around an axis parallel to the neutron beam, with helical slits such that exits are offset from the entrance apertures in much the same manner as described above; the difference from the two-chopper arrangement is that there are apertures located all around the cylinder, giving closely spaced pulses unsuitable for time-of-flight studies. The purpose of mechanical wavelength selection at a continuous source is to select longer wavelengths and a broader range of wavelengths than a crystal monochromator (Section 2.3.4.1.2[link]) could provide. Also worthy of mention is the Fermi chopper (Fermi et al., 1947[link]), comprising a package of neutron-transmitting slits set into a cylinder that rotates at rates of some hundreds of hertz around an axis in the plane of the slits, coincident with the cylinder axis, and perpendicular to the neutron beam. Neutrons above a threshold velocity are transmitted for the brief periods in which the slits are suitably aligned, so short (µs) but frequent pulses of neutrons are delivered. In a variation of the Fermi chopper (Marseguerra & Pauli, 1959[link]), the transmitting slits are curved, providing for the transmission of rather slower neutrons while preventing the transmission of faster ones; in this variant the chopper not only delivers short pulses of neutrons but acts as a velocity selector as well. Neutron choppers are used in various combinations at both continuous and pulsed neutron sources; the Fermi chopper in particular can be used for `shaping' the pulses at long-pulse spallation neutron sources (Peters et al., 2006[link]).

As an initial case study, we consider the ISIS neutron facility, located in Oxfordshire, England, at the Rutherford Appleton Laboratory. This is a well established neutron spallation source supporting a strong programme of research using neutron beams. Of particular note are the excellent facilities for powder diffraction. First neutrons were delivered in 1984, but there have been upgrades since then, including the commissioning of a second target station in 2009. Fig. 2.3.10[link] is a schematic showing the layout of this facility. Some details about its operation are available on the ISIS web site, at https://www.isis.stfc.ac.uk/Pages/How-ISIS-works.aspx . Briefly, an ion source and radio-frequency quadrupole accelerator (not shown) inject bunches of negative hydrogen ions, H, into the linear accelerator where they are accelerated to 70 MeV. These are passed through aluminium foil, which strips them of their electrons, so they become protons, H+, which are then accelerated to 800 MeV in the proton synchrotron. The protons, then travelling in two 100 ns bunches 230 ns apart, are kicked out of their synchrotron orbits and directed toward the targets. The whole process is repeated at a frequency of 50 Hz; the kickers are arranged to send one pulse in five to Target Station 2 (so that the pulse frequency there is just 10 Hz), and the remainder to Target Station 1. Both targets are made of tantalum-coated tungsten, as a stack of water-cooled plates in Target Station 1 and as a heavy-water surface-cooled cylinder in Target Station 2. As explained earlier, the fast neutrons produced in the spallation process must be moderated, and for this purpose moderators are located adjacent to the targets: two water moderators at 300 K, one liquid-methane moderator at 100 K and one liquid-hydrogen moderator at 20 K at Target Station 1; and one decoupled solid-methane moderator at 26 K and one coupled liquid-hydrogen/methane moderator at 26 K at Target Station 2. The widths of the pulses of the moderated neutrons are typically 30–50 µs, but 300 µs for the coupled moderator at Target Station 2. The target/moderator assemblies are surrounded, apart from beam exit ports, by beryllium reflectors. The schematic of Fig. 2.3.10[link] indicates the placement of the various neutron-beam instruments around the target stations.

[Figure 2.3.10]

Figure 2.3.10 | top | pdf |

Layout of the ISIS spallation neutron source. (Credit: STFC.)

The Swiss neutron spallation source, SINQ, located at the Paul Scherrer Institute in Villigen, is the only spallation source operating in continuous mode. SINQ reached full power in 1997. Since there is no time structure to be preserved, more generous quantities of moderator can be used; in fact the target, which becomes the source of neutrons, is located centrally in a moderator tank. The situation here is not very different from that in a medium-flux research reactor. The target comprises lead rods in Zircaloy tubes, the moderator is heavy water and there is a light-water reflector outside the moderator tank. Protons accelerated first by a Cockroft–Walton accelerator, then to 72 MeV by an injector cyclotron, and finally to 590 MeV in a proton ring cyclotron are directed onto the target from below (Fig. 2.3.11[link]). The proton current is initially 2.4 mA, but this is reduced in muon production, so that only about 1.65 mA reaches the spallation target. The power is thus close to 1.0 MW. A horizontal insert in the moderator tank houses a liquid-deuterium cold source at 25 K.

[Figure 2.3.11]

Figure 2.3.11 | top | pdf |

Layout at the SINQ neutron source. (a) Elevation: the target is located in the moderator tank, the high-energy protons being delivered from below. (b) Plan: showing the location of guide tubes relative to this central target. (Courtesy: Dr Bertrand Blau, Paul Scherrer Insitut.)

As a final example we describe the 5 MW long-pulse European Spallation Source, now under construction in Lund, Sweden (see Fig. 2.3.12[link]). A more detailed description is available at the ESS web site, https://europeanspallationsource.se/technology . The proton-acceleration system, although comprising a number of different components, will be linear. The protons from the ion source will be accelerated through a radio-frequency quadrupole and drift tube LINAC up to 90 MeV, then through a series of superconducting cavities up to the final energy of 2 GeV. This system will deliver proton pulses of 2.86 ms duration at a 14 Hz repetition rate; the average current will be 6.26 mA and hence the total power 5 MW. The target material will be helium-cooled tungsten encased in stainless steel, in the form of a 2.5 m-diameter rotating wheel. Such an arrangement assists in dissipation of the heat deposited in the target. Coupled liquid-hydrogen moderators will be located above and below the rotating wheel, and this assembly will be partially surrounded by a water pre-moderator and beryllium reflector. Neutron choppers will be used to shape the neutron pulses as required, and neutron optical systems will deliver neutrons to the experiments. First beam on target is expected in 2019.

[Figure 2.3.12]

Figure 2.3.12 | top | pdf |

Schematic diagram of the ESS facility. The proton beam enters at the right, strikes the target and liberates neutrons for instruments in the three neutron experiment halls. (Image courtesy of the ESS.)

Characteristics of these and other neutron spallation sources are recorded in Table 2.3.4[link]. The information included there has been taken from the respective facility web sites.

Table 2.3.4| top | pdf |
Details of selected spallation neutron sources

SourceTypeLocationProton energyCurrentAverage powerTarget(s)Repetition rate (Hz)Moderator(s)
CSNS Short pulse Institute of High Energy Physics, Guangdong, China 1.6 GeV 62.5 µA 100 kW Tungsten 25 Water, 2 × liquid hydrogen
ESS Long pulse European Spallation Source, Lund, Sweden 2 GeV 2.5 mA 5 MW Tungsten wheel (helium cooled) 14 2 × Liquid hydrogen (pancake geometry)
ISIS Short pulse Rutherford Appleton Laboratory, Oxfordshire, UK 800 MeV 200 µA 160 kW 2 × Tungsten 50 2 × Water, liquid methane, liquid hydrogen
              10 Hydrogen/methane, solid methane at 26 K
JSNS Short pulse J-Parc Centre, Tokai-mura, Japan 3 GeV 333 µA 1 MW Liquid mercury 25 Supercritical hydrogen
LANSCE Long pulse Los Alamos National Laboratory, Los Alamos, USA 800 MeV 125 µA 100 kW Tungsten 20 Water, 2 × liquid hydrogen
SINQ Continuous Paul Scherrer Institute, Villigen, Switzerland 590 MeV 1.64 mA§ 0.97 MW Lead Heavy water; cold source: liquid deuterium at 20 K
SNS Short pulse Oak Ridge National Laboratory, Oak Ridge, USA 1 GeV 1.4 mA 1.4 MW Liquid mercury 60 2 × Water, 2 × liquid hydrogen
Under construction.
Currently operating at <0.5 MW.
§Current reaching spallation target after attenuation in muon source.

2.3.3.4. Neutron beam tubes and guides

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Ideally, neutron diffractometers should be designed following a holistic approach, designing the source of moderated neutrons, through the delivery system, to the instrument itself. This is not often possible in practice; for example the source must often be taken as a given, and in some cases the delivery of the neutrons as well. The holistic approach is commonly a very large Monte Carlo simulation, not suitable for purposes of description; in this chapter, therefore, we provide separate descriptions of these different components.

The simplest delivery system is a neutron beam tube or collimator. A collimator could comprise just two pinholes of diameters a1 and a2 cut into neutron-absorbing material, and placed at a distance L apart; this limits the divergence of the beam to (full angle) [2\alpha =({a}_{1}+{a}_{2})/L]. It is of course possible to use apertures of different cross section, for example rectangular slits, if the divergence must be smaller in one direction than another.

Neutron guides are now widely used at both reactor and spallation neutron sources. These are able to transport neutrons over distances ranging to 100 m or more. They are evacuated tubes, normally of rectangular cross section, and transmission depends on the reflection of glancing-angle neutrons from the walls of the guide. The guides are constructed from glass plates with a reflective coating deposited on the internal surfaces.

Initially, total external reflection (Section 2.3.2.3[link]) provided the basis for reflection; the coating was nickel, or preferably 58Ni. Given that nickel has a face-centred cubic structure (4 atoms per unit cell) with lattice parameter 3.524 Å, and taking the scattering lengths from Table 2.3.2[link], we find from equation (2.3.6)[link] that the critical glancing angles per unit wavelength for total external reflection are 0.10° Å−1 and 0.12° Å−1 for nickel and 58Ni, respectively. Taking wavelengths of 0.4, 1.2 and 5 Å as representative of hot, thermal and cold neutrons, respectively (cf. Fig. 2.3.5[link]), these angles for a nickel mirror are just 0.04, 0.12 and 0.5°. Consequently, these guides are most useful for transmitting cold neutrons and are moderately useful for thermal neutrons, but are not used for hot neutrons. The small glancing angles are demanding, not only on the precision of manufacture, but also because it is highly desirable to use a curved guide tube so there is no direct line of sight to the source (as in Fig. 2.3.13[link]); this is a way of preventing fast neutrons and γ-radiation from impacting on the experiment. The guide tube still transmits a range of wavelengths, although only the longest wavelengths can travel by the zig-zag path indicated in Fig. 2.3.13[link]. If the guide width is a, and its radius of curvature ρ (see Fig. 2.3.13[link]), then the minimum length to avoid direct transmission is [(8a\rho )^{1/2}]. Critical to the transmission of a guide tube is the angle [{\theta }^{*}], which is the minimum glancing angle of incidence onto the outer surface that permits subsequent reflection from the inner surface, and is given by [{\theta }^{*}=(2a/\rho )^{1/2}]. The shortest wavelength, then, that can be transmitted involving reflection from the inner surface is given by [cf. equation (2.3.6)[link]][{\lambda }^{*}={\theta }^{*}\left({{{\pi }\over{N{b}_{\rm coh}}}}\right)^{1/2}. \eqno(2.3.12)]This is known as the `characteristic' wavelength of the guide [see Section 4.4.2[link] of Volume C by Anderson & Schärpf (2006[link])]; the majority of transmitted neutrons will have longer wavelengths than this.

[Figure 2.3.13]

Figure 2.3.13 | top | pdf |

Plan of a curved neutron guide, indicating different possible neutron paths, labelled `garland' and `zig-zag'. Only the longer-wavelength neutrons can travel the zig-zag path because the glancing angles on this path (which must be less than the critical angle) are greater. In this schematic, the glancing angles, the width and the curvature have all been exaggerated. [From Section 4.4.2[link] of Volume C (Anderson & Schärpf, 2006[link]).]

The desire to use guides for shorter (e.g. thermal-neutron) wavelengths, and for retaining more neutrons at a given wavelength, has motivated the development of mirrors capable of reflecting neutrons incident at greater glancing angle. The earliest such mirrors were in fact monochromating mirrors obtained by laying down alternate layers of metals with contrasting coherent-scattering-length densities (Fig. 2.3.14[link]). For a bilayer thickness d and angle of incidence θ these would select wavelengths according to Bragg's law [equation (1.1.3)[link] ],[\lambda =2d \sin(\theta).]

[Figure 2.3.14]

Figure 2.3.14 | top | pdf |

Schematic diagrams of (a) a multilayer monochromator and (b) a neutron supermirror.

In an early implementation (Schoenborn et al., 1974[link]), the metals were Ge and Mn (which have coherent scattering lengths opposite in sign) and the bilayer thickness was of the order of 100 Å; this is a larger d-spacing giving access to longer wavelengths than would be accessible with the usual crystal monochromator (Section 2.3.4.1.2[link]). The idea of supermirrors, comprising bilayers of graduated thickness, and in effect increasing the critical angle, was suggested by Turchin (1967[link]) and Mezei (1976[link]). For a perhaps simplistic explanation, we note first that since the bilayer dimension d is large compared with the neutron wavelength, we can approximate the above equation for reflection as[\theta \simeq \lambda {{1}\over{2d}},]in which form it is reminiscent of equation (2.3.6)[link]. If we take dmin to be the thickness of the thinnest bilayer, then we can propose that the critical angle for reflection by the supermirror should be[{ \theta }_{c}^{\rm SM}\simeq \lambda {{1}\over{2{d}_{\rm min}}}. \eqno(2.3.13)]In order to ensure that all neutrons incident at angles less than this critical angle should be reflected, we need to incorporate a more-or-less continuous range of thicker bilayers into the supermirror (Fig. 2.3.14[link]b). A more rigorous treatment (Hayter & Mook, 1989[link]; Masalovich, 2013[link]) takes account of the transmission and reflection at each interface, and lays down a prescription as to how the thicknesses should be varied. The most common pairing for the bilayer is now Ni with Ti; the coherent scattering cross sections are of opposite sign (see Table 2.3.2[link]). The performance of a supermirror is normally quoted as the ratio m of the critical angle for the supermirror, [{\theta }_{c}^{\rm SM}], to that for natural nickel, [{\theta }_{c}^{\rm Ni}]; a high value for reflectivity is also important. Supermirrors to m of 2 or 3 are in quite common use, while now Ni/Ti supermirrors with m up to 7 are offered for purchase (Swiss Neutronics AG; see also Maruyama et al., 2007[link]).

Consideration is currently being given to the variation of the cross section of the guide along its length. There is some loss on reflection by supermirrors, so these studies aim to reduce the number of reflections involved in transmission along the guide. One suggestion (also attributable to Mezei, 1997[link]) is to use a `ballistic guide', in which neutrons from the source travel through a taper of widening cross section into a length of larger guide, then through a taper of narrowing cross section to restore the original cross section at the exit. This is said to reduce the number of reflections suffered by the neutron by a factor of [{({w}_{0}/w)}^{2}], where w0 is the width at entrance and exit and w the larger width along the main part of the guide (Häse et al., 2002[link]). Such a guide has been installed and is operating successfully on the vertical cold source at the Institut Laue–Langevin (Abele et al., 2006[link]). An extension of this idea is based on the well known property of ellipses that a ray emanating from one focus is reflected (just one bounce) to pass through the other; so if the guide cross section could be varied to give a very long ellipse, a source of neutrons placed at one focus, and the target point at the other, then perhaps the neutrons could be transmitted along the guide with just a single reflection (Schanzer et al., 2004[link]; Rodriguez et al., 2011[link]). Accordingly a number of neutron facilities have installed elliptical guides, and indeed a number of neutron powder diffractometers now are located on elliptical guides; these include diffractometer POWTEX at FRM-II, the high-resolution diffractometers HRPD and WISH at ISIS, and Super-HRPD at JSNS. Computer simulation by Cussen et al. (2013[link]), however, questions whether, given the practicalities of finite source sizes and the approximation of elliptical variation by a number of linear segments, the theoretical improvement is fully realized.

2.3.4. Diffractometers

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Put simply, the diffracted neutron beams associated with the different d-spacings in the sample under study satisfy Bragg's law,[\lambda =2d \sin(\theta). \eqno(2.3.14)]As always, λ is the wavelength of the incident neutrons, and these neutrons are scattered through an angle 2θ.

There are basically two ways of exploiting this relationship. The first is to use a single wavelength for the investigation, in which case diffracted neutrons are observed at different angles 2θ corresponding to different d-spacings in the sample. A neutron powder diffractometer designed to carry out an investigation by this means we choose to call a `constant wavelength' (CW) diffractometer. The other means is to fix the angle 2θ, illuminate the specimen with a range of wavelengths, and note the different wavelengths that are diffracted. In this case, we determine the wavelengths of the diffracted neutrons via their speed [\lambda =h/(mv)] [equation (2.3.1)[link]], and that in turn is measured by their flight time t over a path of length L, [v=L/t]; this leads to[\lambda ={{ht}\over{mL}}. \eqno(2.3.15)]A diffractometer designed to carry out such an analysis of wavelengths we call a `time-of-flight (TOF) diffractometer'.

The distinction between these two modes of operation can also be indicated via the Ewald construction in reciprocal space (Section 1.1.2.4[link] ). In this, the ideal powder is represented by concentric spheres in reciprocal space. In the constant-wavelength situation, the primary beam is fixed in direction and the Ewald sphere has a fixed radius; diffracted (reflected) beams are observed at any angle at which the surface of the Ewald sphere intersects one of the concentric spheres mentioned just above. In the wavelength-analysis (time-of-flight) situation, the directions of the primary and diffracted beams are fixed, but the radius of the Ewald sphere (1/λ) is variable through a range; diffracted beams are observed whenever the wavelength is such that the tip of the vector representing the reflected beam lies on one of the concentric spheres.

2.3.4.1. Constant-wavelength neutron diffractometers

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The salient features of a constant-wavelength diffractometer are perhaps most easily explained by reference to a particular example; for this purpose we consider the High Resolution Powder diffractometer for Thermal neutrons (HRPT) installed at the SINQ continuous spallation source (Fischer et al., 2000[link]). Neutrons from the source travel through a guide tube to the crystal monochromator, which directs neutrons of a selected wavelength toward the sample. The diffracted neutrons are registered in a detector or detectors that cover a range of angles of scattering from the sample. Collimation is used to better define the directions of the neutron beams; in this instance a primary collimator is included in the guide tube and additional collimation is included between the sample and the position-sensitive detector. The various components will be described in more detail below.

2.3.4.1.1. Collimation

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There need to be restrictions on the angular divergences of the neutron beams. The divergence of the beam impinging upon the crystal monochromator must be limited to better define the wavelength of the neutrons directed to the sample, whereas the divergences of the beams incident upon and diffracted from the sample will control the precision with which the scattering angle 2θ can be determined. For a diffractometer detecting neutrons and measuring scattering angles in the horizontal plane (as shown in Fig. 2.3.15[link]) the horizontal divergences are critical, the vertical divergences less so.11 Indeed, the horizontal divergences are key parameters in the determination of resolution and intensity (Section 2.3.4.1.4[link]); for this reason we denote by α1, α2 and α3 the (half-angle) angular divergences of the primary beam (i.e. the beam onto the monochromator), the monochromatic beam (from monochromator to sample) and the diffracted beam (from sample to detector), respectively.

[Figure 2.3.15]

Figure 2.3.15 | top | pdf |

A constant-wavelength neutron powder diffractometer. This figure shows (a) a layout diagram and (b) the physical appearance (dominated by the monochromator and detector shieldings) for the HRPT diffractometer installed at the SINQ continuous spallation source. (Figures from https://www.psi.ch/sinq/hrpt/ .)

The divergences are limited by various forms of collimation. The divergence of the primary beam will be limited in the first instance by the delivery system. For delivery through a simple beam tube of length L, with entrance and exit apertures of dimensions a1 and a2, respectively, the angular divergence (half-angle) is given by (as already noted in Section 2.3.3.4[link])[{\alpha }_{1}={{{a}_{1}+{a}_{2}}\over{2L}}. \eqno(2.3.16)]Neutrons emerging from a guide tube would have divergence equal to the critical angle of the guide, [{\alpha }_{1}={\theta }_{c}]. Soller collimators (see below) can be used if there is a need to further reduce the horizontal divergence of the primary beam. The divergence of the monochromatic beam may be limited by slits, or a beam tube. The divergence of the diffracted beam, α3, is often defined using another Soller collimator. Sometimes this divergence is limited just by the dimensions of the sample and the detecting elements; equation (2.3.16)[link] gives α3 if it now references the sample and detector element dimensions and the distance between them. Even in this circumstance (as in HRPT), Soller collimators may be used in front of the detector to reduce scattering from ancillary equipment and other background contributions.

Soller collimators (Soller, 1924[link]) are used to transmit beams of large cross section while limiting (for example) horizontal divergence. They are in effect narrow but tall rectangular collimators stacked side by side; in practice they comprise thin neutron-absorbing blades equally spaced in a mounting box. It should be evident from equation (2.3.16)[link] that if the length of the collimator is L and the separation between the blades is a, then the (half-angle) horizontal divergence is a/L. The transmission function for a Soller collimator is ideally triangular. It is technologically challenging to make compact Soller collimators, since, for a given collimation, a shorter collimator needs a smaller blade spacing. One very successful approach, due to Carlile et al. (1977[link]), has been to make the neutron-absorbing blades from Mylar, stretched on thin steel or aluminium alloy frames, and subsequently coated with gadolinium oxide paint; these blades are stacked and connected via the frames which become the spacers in the final product. The collimators made by Carlile et al. were 34 cm long, and the blade spacing was 1 mm, giving a horizontal divergence of 0.17°. Compact Soller collimators of this type (Fig. 2.3.16[link]) are now commercially available, with blade spacings down to 0.5 mm.

[Figure 2.3.16]

Figure 2.3.16 | top | pdf |

Commercially available compact Soller collimators. (Reproduced with permission from Eurocollimators Ltd, UK.)

Even more compact collimators can be produced by eliminating the gaps in favour of solid layers of neutron-transmitting material; for example, a collimator only 2.75 cm long made by stacking 0.16 mm thick gadolinium-coated silicon wafers gave a divergence of 0.33° (Cussen et al., 2001[link]). Microchannel plates (Wilkins et al., 1989[link]) may offer additional possibilities for collimation and focusing.

2.3.4.1.2. Monochromators

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The wavelength in a constant-wavelength powder diffractometer is almost invariably selected by a single-crystal monochromator. If the primary beam is incident onto the monochromator in such a way as to make an angle θM with a chosen set of planes in the crystal, then the wavelength that will be reflected from these planes is given by Bragg's law,[\lambda =2d \sin({\theta }_{M}),]where d is the spacing of the chosen planes. A spread of angles of incidence represented by ΔθM will result in the selection of a band of wavelengths Δλ given by[{{\Delta \lambda }\over{\lambda }}=\cot{\theta }_{M}{\Delta \theta }_{M}. \eqno(2.3.17)]For high-resolution performance we need a rather precisely defined wavelength, so Δλ should be small; if, on the other hand, intensity is an issue then a wider band of wavelengths needs to be accepted. It should be evident from equation (2.3.17)[link] that a high-resolution diffractometer will operate with a take-off angle from the monochromator, 2θM, as high (i.e. as close to 180°) as practicable, and with tight primary collimation α1.

It might be noticed that the integer n appearing on the right-hand side of equation (1.1.3)[link] has been omitted from our formulation of Bragg's law. If the Miller indices of the chosen planes are hkl, if the spacing of these planes is dhkl, and if we introduce dnh,nk,nl = dhkl/n [cf. equation (1.1.23)[link] ], then the factor n is effectively restored. This means that, as well as reflecting the selected wavelength through the hkl reflection, the monochromator has the potential to reflect unwanted harmonics [\lambda /n] of the desired wavelength through the nh,nk,nl reflections. This problem can be largely overcome using the hkl planes with h, k, l all odd in crystals with the diamond structure, such as silicon and germanium; for this structure the structure factors [equation (2.3.7)[link]] for the 2h,2k,2l reflections are zero so that there is no contamination by [\lambda /2], and at the shorter wavelengths, [\lambda /3] and so on, there are very few neutrons in the thermal neutron spectrum (Fig. 2.3.5[link]).

Since `perfect' crystals (of silicon and germanium, for example) have low reflectivity, for monochromator applications imperfect or `mosaic' crystals are usually preferred. A mosaic crystal can be pictured as comprising small blocks of crystal with slightly differing orientations, the distribution in angle of these blocks being characterized by a full-width at half-maximum angle, β, known as the `mosaic spread'. In addition to improving the intensity markedly,12 this `mosaic spread' will also increase the range of wavelengths obtained. Crystals intended for use as monochromators are very often deliberately deformed to achieve the desired mosaic structure. Further gains in intensity are sought by using vertically focusing monochromators, since the vertical divergence can be increased without serious detriment to the diffraction patterns. Vertically focusing monochromators usually comprise a number of separate monochromator crystals either individually adjustable (Fig. 2.3.17[link]) or in fixed mountings on a bendable plate.

[Figure 2.3.17]

Figure 2.3.17 | top | pdf |

The vertically focusing monochromator constructed at the Brookhaven National Laboratory (Vogt et al., 1994[link]) and now used by the high-resolution powder diffractometer ECHIDNA at OPAL. The 24 monochromating elements are individually adjustable, and each of these is a 30-high stack of 0.3 mm thick Ge wafers, deformed to yield a suitable mosaic structure and then brazed together. (Reproduced with permission from ANSTO.)

It is not common to find polarized neutrons being used in neutron powder diffractometers. Nevertheless, we think it appropriate to mention here that one means to obtain a polarized neutron beam is to use an appropriate polarizing crystal monochromator.13 The 111 reflection from the ferromagnetic Heusler alloy Cu2MnAl is commonly used for this purpose; the nuclear and magnetic structure factors [equations (2.3.7)[link] and (2.3.8)[link]] are of similar magnitude and they add or subtract depending on whether the neutron spin is antiparallel or parallel to the magnetization of the alloy. The beam reflected from such a monochromator can be polarized to better than 99%.

The reader is referred to Section 4.4.2[link] of Volume C (Anderson & Schärpf, 2006[link]) and to Kisi & Howard (2008[link]) Sections 3.2.1 and 12.3 for further details.

2.3.4.1.3. Neutron detectors

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Neutrons, being electrically neutral, do not themselves cause ionization and so cannot be detected directly; their detection and counting therefore depend on their capture by specific nuclei and the production of readily detectable ionizing radiation in the ensuing nuclear reaction. Only a limited number of neutron-capture reactions are useful for neutron detection [see Chapter 7.3[link] of Volume C (Convert & Chieux, 2006[link])]; they include[\eqalign{_{0}^{1}{\rm n} + {}_{2}^{3}{\rm He}&\to {}_{1}^{1}{\rm H} +{} _{1}^{3}{\rm H}+0.76 \ {\rm MeV}\cr _{0}^{1}{\rm n} + {}_{3}^{6}{\rm Li}&\to {}_{1}^{3}{\rm H}+{}_{2}^{4}{\rm He}+4.79 \ {\rm MeV}\cr _{0}^{1}{\rm n}+{}^{10}_{\phantom{1}5}{\rm B}&\to {}_{3}^{7}{\rm Li}+{}_{2}^{4}{\rm He}+2.8\ {\rm MeV}\cr _{0}^{1}{\rm n}+{}_{\phantom{1}64}^{157}{\rm Gd}&\to {}_{{\phantom 1}64}^{158}{\rm Gd}+\gamma\hbox{-rays + conversion electrons}\cr &\quad+ \le 0.18\ {\rm MeV}\cr _{0}^{1}{\rm n}+{}_{\phantom {1}92}^{235}{\rm U}&\to \hbox{fission fragments}\ +\ {\sim}200\,{\rm MeV}}](cf. Section 2.3.3.2[link]).

The attenuation of neutrons in these materials (Section 2.3.2.4[link]) will be dominated by the high absorption (by capture) cross sections (Table 2.3.2[link]), so the linear attenuation coefficient will be given by [\mu =N{\sigma }_{\rm abs}] where N is the number of absorbing nuclei per unit volume. We remark that absorption cross sections, with the exception of Gd, increase linearly with wavelength. The factor by which the neutron beam is diminished in a detector of thickness x is [\exp(-\mu x)=\exp(-N{\sigma }_{\rm abs}x)]. The detector efficiency, then, given by the fraction of neutrons absorbed (captured) in the detector, is [1-\exp(-N{\sigma }_{\rm abs}x)]. In most cases the aim is to have high detector efficiency; however, in some circumstances it is desirable to monitor an incident neutron beam, in which case attenuation should be kept to a minimum. The account of neutron detection given here will be kept relatively brief since much has been written on this subject elsewhere [Oed, 2003[link]; Chapter 7.3[link] of Volume C (Convert & Chieux, 2006[link])].

The task, following the neutron-capture reaction, is to detect the various charged particles or ionizing radiations that are produced. These are registered by the electrical signals they generate in a gas-filled proportional counter or ionization chamber, or in a semiconductor detector, recorded on film, or detected from the flashes of light they produce in a scintillator, for example ZnS. It is well worth noting that the secondary radiation carries no record of the energy of the detected neutrons; so whatever the means of detection, detectors can count neutrons but can provide no information on their energy distribution.

The gas-filled radiation detectors are essentially Geiger counters, comprising a gas-filled tube with a fine anode wire running along its centre. The anode collects the electrons released by ionization of the gas; if the anode voltage is high enough, there is a cascade of ionization providing amplification of the signal.14 Detectors filled with boron trifluoride, 10BF3, and helium-3, 3He, have high efficiencies and are in common use; in these the nucleus designated to capture neutrons is incorporated in the filling gas. Such detectors operate with pulse-height discrimination, not in any attempt to determine neutron energy, but to discriminate against lower-voltage signals from γ-rays and other unwanted background. Another approach is to have a thin solid layer15 of neutron-absorbing material, 235U for example, releasing secondary radiation, in this case fission products, into a gas proportional counter filled with a standard argon/methane mixture; this would represent a low-efficiency neutron detector suitable for use as an incident-beam neutron monitor. Neutron detection based on semiconductor particle detectors is still in the developmental stage. The main problem is that the semiconductors used for charged-particle detection do not contain neutron-absorbing isotopes. Semiconductor particle detectors could be used to register the secondary radiation from an abutting layer of neutron-absorbing solid, but that layer would need to be thin, and another low-efficiency neutron detector would result. Scintillation detectors involve the placement of neutron-absorbing materials, such as 6LiF, adjacent to a scintillator such as a ZnS screen, or perhaps the use of a Ce-doped lithium silicate glass, and counting the flashes of light that are produced. These light flashes can be recorded by photomultiplier tubes or on film. Scintillation detectors are, however, not used in constant-wavelength diffractometers because of their sensitivity to γ-radiation. They are used in time-of-flight diffractometers at spallation sources by exploiting the fact that the unwanted fast neutrons and γ-rays, and the thermal neutrons of interest, are separated in time (Section 2.3.4.2.2[link]).

Much of the preceding description refers to single neutron counters, although it should be noted that there is a position-sensitive capability inherent in a film or scintillator screen. The earliest CW diffractometers employed just a single detector set on an arm that scanned through the scattering angle 2θ; the deployment of a Soller collimator just in front of the detector was advantageous. Conceptually the simplest but not necessarily the cheapest means for improvement was to mount a number of collimator/detector pairs on the detector arms. Such an improvement was made to diffractometer D1A at the Institut Laue–Langevin (Hewat & Bailey, 1976[link]) by mounting ten sets of 10′ divergence Soller collimators/3He detectors at intervals of 6°. The BT-1 diffractometer at NBSR operates with 32 3He detectors set at 5° intervals, so a scan through 5° covers a total angular range of 160°. The ultimate level for this kind of development was reached when D2B at the Institut Laue–Langevin operated in its former mode, with 64 detectors set at 2.5° intervals, each with its own 5′ Mylar Soller collimator; this required a scan through only 2.5° to record 160° of diffraction.

The alternative to using large numbers of individual detectors is to make use of position-sensitive neutron detectors (PSDs), and these have been in use for quite some time. The technology is that of the position-sensitive detection of charged particles, the important issue for neutrons being that the charged-particle detection should be located close to the neutron-capture event so that positional information is retained. A gas-filled 3He detector with a single anode wire can serve as a linear PSD, for example by comparing the charges collected, after the capture event, at the opposite ends of the wire. The D2B diffractometer at the Institut Laue–Langevin has now been upgraded to `SuperD2B', which uses 128 linear PSDs with their axes (anode wires) vertical, at 2θ intervals of 1.25°; this operates as a quasi-two-dimensional PSD. Diffractometers SPODI at FRM-II (80 detectors) and ECHIDNA at OPAL (128 detectors) are fitted with similar detector arrays. A single gas-filled chamber containing a number of separate parallel vertically aligned anodes, termed a multi-wire proportional counter (MWPC), provides another approach; the electronics needs to register at which of the wires the capture event occurred. This technology has extended from the first multi-wire PSD with 400 wires at 5 mm (0.2°) separation, used on the D1B diffractometer at the Institut Laue–Langevin in the 1970s, to a PSD with 1600 wires at 0.1° separation now in use on HRPT at SINQ (Fig. 2.3.15[link]). A further advance is the development of the micro-strip gas chamber (MSGC) detector (Oed, 1988[link]). In this detector the anodes and the cathodes are printed circuits on glass substrates, which are then mounted into the chamber. With this arrangement, an anode separation of 1 mm is achievable and the stability is excellent. The high-intensity diffractometer D20 at the Institut Laue–Langevin has a detector assembled from plates of micro-strip detectors and achieves 1600 anodes at 0.1° angle separation.

As pointed out above, the detection systems on SuperD2B, SPODI and ECHIDNA achieve a quasi-two-dimensional position capability by using banks of linear PSDs located side by side. An MWPC detector can achieve two-dimensional capability in a very similar manner, using the anode position to locate in the horizontal direction and charge division measurements at the ends of each anode wire to find the vertical position. An MWPC detector can also be fitted with segmented cathodes, either side of the anodes, one returning positional information in the horizontal direction and the other giving the vertical position. A detector of this kind is used on the WOMBAT diffractometer at the OPAL reactor. MSGC detectors can also be adapted to provide two-dimensional positional information after printing a set of cathodes orthogonal to the primary set on the back surface of the glass.

A few general comments about detecting systems are in order. The time for a detector to recover after registering a neutron count is known as the dead time, and this may be significant when count rates are high, in which case corrections are needed [Chapter 7.3[link] of Volume C (Convert & Chieux, 2006[link])]. For banks of detectors, and also for position-sensitive detectors, calibration for position and sensitivity becomes a critical issue. In the case of a smaller bank of detectors, it may be possible to scan the detector bank so the same diffraction pattern is recorded in the different detectors, in which case the relative positions and efficiencies of the different detectors can be determined quite well (see Section 4.1 of Kisi & Howard, 2008[link]). For more extensive banks or large position-sensitive detectors, detector sensitivity calibration is performed by examining the very nearly isotropic incoherent scattering from vanadium. In this case checking for angular accuracy can be more difficult. The time taken to register a neutron count cannot be said to be a fundamental issue in CW powder diffraction, since in some applications it is scarcely relevant, although in other applications, such in the study of very fast reaction kinetics (Riley et al., 2002[link]), the constraints on time are very demanding.

2.3.4.1.4. Resolution and intensity

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The resolution and intensity of a CW powder diffractometer are strongly influenced by the divergences α1, α2 and α3 of the primary, monochromatic and diffracted beams, respectively, along with the mosaic spread β of the crystal monochromator. The situation was analysed by Caglioti et al. (1958[link]) on the basis that the triangular transmission factor of each collimator, total width 2α, could be approximated by a Gaussian with full-width at half-maximum (FWHM) α, that the mosaic distribution of the monochromator could also be described by a Gaussian with FWHM β, but that there was no sample contribution to the peak widths. On this basis the diffraction peaks were found to be Gaussian, with the FWHM of the diffraction peak occurring at scattering angle 2θ given by (Hewat, 1975[link])[{\rm FWHM}^{2}=U \tan^{2}\theta +V\tan\theta +W, \eqno(2.3.18)]where[\eqalignno{U&={{4({\alpha }_{1}^{2}{\alpha }_{2}^{2}+{\alpha }_{1}^{2}{\beta }^{2}+{\alpha }_{2}^{2}{\beta }^{2})}\over{\tan^{2}{\theta }_{M}({\alpha }_{1}^{2}+{\alpha }_{2}^{2}+4{\beta }^{2})}}, &(2.3.18a)\cr V&={{-4{\alpha }_{2}^{2}({\alpha }_{1}^{2}+2{\beta }^{2})}\over{\tan{\theta }_{M}({\alpha }_{1}^{2}+{\alpha }_{2}^{2}+4{\beta }^{2})}}, &(2.3.18b)\cr W&={{{\alpha }_{1}^{2}{\alpha }_{2}^{2}+{\alpha }_{1}^{2}{\alpha }_{3}^{2}+{\alpha }_{2}^{2}{\alpha }_{3}^{2}+4{\beta }^{2}({\alpha }_{2}^{2}+{\alpha }_{3}^{2})}\over{{\alpha }_{1}^{2}+{\alpha }_{2}^{2}+4{\beta }^{2}}} & (2.3.18c)}]and θM is the Bragg angle (2θM is the take-off angle) at the monochromator. Under these conditions the total (integrated) intensity in the diffraction peak is given by[L\propto {{{\alpha }_{1}{\alpha }_{2}{\alpha }_{3}\beta }\over{{({\alpha }_{1}^{2}+{\alpha }_{2}^{2}+4{\beta }^{2})}^{1/2}}}.\eqno(2.3.19)]

These equations have important implications and accordingly have received a good deal of attention. They return at once the well known resolution advantage in setting up the diffractometer in the parallel configuration (that seen in Fig. 2.3.15[link], in this configuration θM taken to be positive). Caglioti et al. (1958[link]) deduced that for the simple case of [\alpha_1=\alpha_2=\alpha_3=\beta =\alpha ] equations (2.3.18)[link] and (2.3.19)[link] reduce to[{\rm FWHM}=\alpha {\left({{11-12a+12{a}^{2}}\over{6}}\right)}^{1/2} \ {\rm and}\ L\propto {\alpha }^{3}/(6)^{1/2},]where [a=\tan\theta /\tan{\theta }_{M}]; they went on to record results for a number of other combinations. In his design for a high-resolution diffractometer, Hewat (1975[link]) considered the case [{\alpha }_{2}=] [2\beta \,\gt\, {\alpha }_{1}\simeq {\alpha }_{3}]. Under these conditions, the peak widths are close to their minimum around the parallel focusing condition [\theta ={\theta }_{M}], their widths there are given by[{\rm FWHM}^{2}=\left({\alpha }_{1}^{2}+{\alpha }_{3}^{2}\right)-{{{\alpha }_{1}^{4}}\over{{\alpha }_{1}^{2}+{\alpha }_{2}^{2}+4{\beta }^{2}}}\simeq {\alpha }_{1}^{2}+{\alpha }_{3}^{2}, ]and the total intensity is approximately[L\propto {\alpha }_{1}{\alpha }_{3}\beta /(2)^{1/2}. ]Hewat's conclusions, put briefly, were that good resolution could be obtained by keeping divergences α1 and α3 small, while intensity could be somewhat recovered by adopting relatively large values for the monochromator mosaic spread β and divergence α2 of the monochromatic beam. Hewat also argued for a high monochromator take-off angle 2θM, not only to reduce peak widths [through the term [\cot{\theta }_{M}] appearing in equation (2.3.17)[link] and reappearing in equations (2.3.18)[link]], but also to match the region of best resolution to that of the most closely spaced peaks in the diffraction pattern. Hewat's design was implemented in the D1A diffractometer at the Institut Laue–Langevin (Hewat & Bailey, 1976[link]), subsequently in the D2B diffractometer at the same establishment, and elsewhere. In a version installed at the (now retired) HIFAR reactor in Sydney, Howard et al. (1983[link]), using an Al2O3 (corundum) ceramic sample, reported a peak-width variation in close agreement with that calculated from equation (2.3.18)[link]. Although more sophisticated analyses are available in the literature (Cussen, 2000[link]), this result would suggest that equations (2.3.18)[link] still provide a good starting point.

The usual trade-off between intensity and resolution applies, and since neutron sources are rather less intense than X-ray sources, this is an important consideration. Intensity is sacrificed by using high monochromator take-off angles to limit the wavelength spread [equation (2.3.17)[link]], and by using tight collimation [equation (2.3.19)[link]]. Evidently intensities could be increased by relaxing these constraints. These days it is more common to build diffractometers of good-to-high resolution, and then to seek other means to improve data-collection rates. Focusing monochromators, such as described in Section 2.3.4.1.2[link], serve to increase the neutron intensity at the sample position without seriously degrading the resolution. In addition, the use of multi-detector banks and the development and deployment of position-sensitive detectors, as described in Section 2.3.4.1.3[link], has been very much driven by the desire to increase the speed of data collection. As mentioned earlier, the design and analysis of neutron powder diffractometers should be treated in a holistic fashion, and although some advanced analytical methods have been applied (Cussen, 2016[link] and references therein), Monte Carlo analyses using programs such as McStas (Willendrup et al., 2014[link]) and VITESS (Zendler et al., 2014[link]) to track large numbers of neutrons from the source right through to the neutron detectors are now widely employed.

2.3.4.2. Time-of-flight (TOF) diffractometers

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Time-of-flight (TOF) diffractometers differ substantially from CW diffractometers. Neutrons delivered to the instrument are already partially collimated and TOF instruments have no monochromator and consequently no moving parts. The full incident neutron spectrum is utilized and needs to be well characterized in order to extract meaningful intensities; in addition the wavelength dependence of detector efficiencies needs to be taken into account. In principle, measurements from an incoherently (therefore isotropic and wavelength-independent) scattering sample such as V or H2O provide the required characterization.16 In practice, however, incident spectra are usually recorded using a low-efficiency detector (beam monitor) in the incident beam. Data from V are still required to correct for the relative efficiency of individual detectors or detector elements and their wavelength dependence (Soper et al., 2000[link]).

The basic components of a TOF powder diffractometer are the flight tube from the neutron source or a neutron guide, a precisely located sample position, banks of detectors at various positions around the sample position and a neutron-absorbing beam stop. In early TOF diffractometers, detector banks were relatively localized typically in forward scattering, close to 2θ = 90° and backscattering locations. More modern diffractometers have very extensive detector arrays such as the newly upgraded POLARIS instrument at the ISIS facility, which is illustrated in Fig. 2.3.18[link]. Neutrons enter the diffractometer at the right of Fig. 2.3.18[link](a) through a number of adjustable neutron-absorbing jaws which trim the beam size to match the sample size. The beam is then incident on the sample, which is located within the chamber where the detectors, arranged in numbered banks, are housed. The entire sample/detector chamber (and flight tube) is evacuated during data collection in order to reduce absorption and scattering of the incident neutron beam by air, effects which both decrease the intensity of the neutrons incident on the sample and increase the background scattering. A human figure in Fig. 2.3.18[link](a) indicates the large scale of the device and it should be noted that the substantial neutron shielding surrounding the detector chamber (known as the blockhouse) is not shown.

[Figure 2.3.18]

Figure 2.3.18 | top | pdf |

(a) Schematic cross section of the POLARIS diffractometer at the ISIS facility, UK, and (b) a three-dimensional solid model of the detector chamber. (Credit: STFC.)

2.3.4.2.1. Instrument resolution and design

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In a TOF instrument, all of the incident spectrum of neutron wavelengths is utilized, appropriately trimmed by the chopper system as previously described. The different wavelengths (λ) are identified through their time-of-flight (t) according to equation (2.3.15)[link]. Substituting that equation into Bragg's law, we obtain[\eqalignno{{d}_{hkl}&={{ht}\over{2mL\sin\theta }}&(2.3.20)\cr &={{t}\over{505.554L\sin\theta }}}]for t in microseconds, d in ångstroms and L in metres.

The resolution of a TOF diffractometer is defined by the uncertainty in the d-spacing (Δd) relative to its absolute value d. Apparent as the width of the diffraction peaks, the resolution is given primarily by (Buras & Holas, 1968[link]; Worlton et al., 1976[link])[{{\Delta d}\over{d}}={\left[{\Delta \theta }^{2}\cot^{2}\theta +{\left({{\Delta t}\over{t}}\right)}^{2}+{\left({{\Delta L}\over{L}}\right)}^{2}\right]}^{{{1}/{2}}}. \eqno(2.3.21)]There are a number of important things to note concerning this equation:

  • (i) The terms [\Delta \theta \cot\theta ] and [{{\Delta L}/{L}}] are fixed and independent of flight time once the diffractometer is constructed; in addition, as we have already noted (Section 2.3.3.3[link]), for a spallation source with a suitably poisoned moderator the time resolution [\Delta t/t] is practically constant. Thus the resolution of a TOF diffraction pattern is virtually constant across the entire range of d-spacing explored in a given detector bank.17

  • (ii) Uncertainties in the neutron path length, ΔL, can arise due to measurement uncertainty in determining L; however, these are usually overshadowed by the uncertainty that arises because neutrons can emerge into the neutron guide from any position within the finite-sized moderator and this uncertainty constitutes the major contribution to ΔL.

  • (iii) As ΔL is a constant, a linear improvement in resolution can be achieved merely by making the instrument longer, such as HRPD at ISIS and S-HRPD at J-PARC, which are almost 100 m long.

  • (iv) The contribution of the diffraction angle 2θ to resolution is considerable. For a fixed angular uncertainty (detector positioning and finite width) the cot θ term varies from infinite at 2θ = 0 to zero at 2θ = 180°. Therefore, the higher the detector angle, the better the resolution.

With these matters considered, we can return to our example of a modern TOF diffractometer in Fig. 2.3.18[link] and in particular the arrangement of the detectors. The strategy employed is to group multiple individual detector elements into a number of discrete banks. It may be seen from equation (2.3.21)[link] that decreasing 2θ and increasing L have opposing effects on resolution. By appropriate manipulation of the equation and by expressing the overall neutron flight path as L = L1 + L2 where L1 is the moderator-to-sample distance and L2 is that from the sample to the detector, it is straightforward to obtain[{L}_{2}=\Delta L{\left[{\left({{\Delta d}\over{d}}\right)}^{2}-{\left({{\Delta \theta }\over{\tan\theta }}\right)}^{2}-{\left({{\Delta t}\over{t}}\right)}^{2}\right]}^{-{{1}/{2}}}-{L}_{1}. \eqno(2.3.22)]Therefore by adjusting 2θ and L2 correctly, it is possible to construct banks of detectors covering a range of 2θ, for which the resolution is identical. This allows neutrons recorded in the entire detector bank to be `focused' into a single diffraction pattern. The resulting curved detector arrangement is obvious in the high-resolution detector bank labelled 5 and 6 in Fig. 2.3.18[link](a). For a fixed (small) value of [{{\Delta d}/{d}}], eventually space limitations impose restrictions on L2 and a new, lower-resolution detector bank (4) commences. As the benefits of a curved arrangement become insignificant, the appropriate curve is approximated by a straight arrangement in the lower-angle banks and dispensed with altogether in the very low angle bank. In Fig. 2.3.18[link] the backscattering (5, 6), 90° (4), two separate low-angle (2 & 3) and the very low angle (1) detector banks of POLARIS are identified. These have average 2θ angles of 146.72, 92.59, 52.21, 25.99 and 10.40°, respectively.

Raw diffraction patterns recorded in the various detector banks are compared in Fig. 2.3.19[link]. Note that the curved background due to the incident spectrum is flattened when the patterns are normalized. A logarithmic scale is necessary to display the very wide range of d-spacings accessible across the whole instrument and this scale emphasises the near-constant resolution across each pattern. In keeping with equations (2.3.21)[link] and (2.3.20)[link], the effects of changing the detector angle are obviously greater resolution and access to shorter d-spacings as 2θ increases. Each detector bank can provide data for a different purpose according to its resolution and d-spacing coverage. For example, the combination of good resolution (4 × 10−3) and a wide range of d-spacing (0.2–2.7 Å) makes data from the backscattering bank (Fig. 2.3.19[link]e) ideal for the refinement of medium- to large-scale crystal structures. The 90° bank (Fig. 2.3.19[link]d) is optimized for use with complex sample environments such as high-pressure cells or reaction vessels, as this geometry combined with appropriate collimation of the incident and scattered neutron beams enables diffraction patterns to be collected that only contain Bragg reflections from the sample being studied. It can be used to obtain good-resolution data (7 × 10−3) during a variety of in situ studies. The low-angle and very low angle banks with their access to very large d-spacings up to 20 Å are invaluable in determining unknown crystal structures and complex magnetic structures by allowing the indexing of low-index reflections and determining reflection conditions.

[Figure 2.3.19]

Figure 2.3.19 | top | pdf |

Raw neutron diffraction patterns from Y3Al5O12 (YAG). Patterns from the five POLARIS detector banks, (a) very low angle, (b) low angle 1, (c) low angle 2, (d) 90° and (e) backscattering, are shown separately. Note that the very wide range of d-spacings accessible (~0.2–25 Å) necessitates the use of a log10 scale. Insets for the backscattering bank illustrate that useful data are obtained even at very small d-spacing (red) and that the resolution is very good (blue). Note the asymmetric peak shape that results from a rapid rise, followed by a slower exponential decay, in the number of neutrons emerging from the moderator after each incident proton pulse.

In order to reduce unwanted background counts and give better localization of the diffraction pattern from the sample, i.e. to better exclude sample environments such as cryostats or furnaces, the instrument is fitted with a radial collimator surrounding the sample position.18 For more common sample environments, e.g. furnaces, this collimation allows all detector banks to view the sample unimpeded. The detector banks are contained within the large vacuum vessel shown in Fig. 2.3.18[link](b). This reduces attenuation and background due to scattering by air. The detector coverage on such an instrument is very large, in the case of POLARIS up to 45% of the available solid angle is covered. A full description of this instrument may be found in Smith et al. (2018[link]).

2.3.4.2.2. Detection

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All the neutron detector types discussed in Section 2.3.4.1.3[link] are capable of detecting the scattered neutrons in a TOF pattern. Gas-filled proportional counters such as BF3 and 3He detectors have efficiencies governed by the neutron energy (or wavelength). Detectors on CW diffractometers are optimized for the narrow band of wavelengths available when using a crystal monochromator, say 1–2.4 Å. The wavelength range in TOF diffraction is generally much wider; as much as 0.2–6 Å or more, and proportional detectors need to be specifically optimized. There is of course the added complexity of tracking the arrival time of each neutron and this has worked against the use of multi-wire proportional detectors and microstrip detectors as described in Section 2.3.4.1.3[link]. Instead, there is extensive use of scintillation detectors, which are usually based on the 6Li(n;t,α) reaction (Section 2.3.4.1.3[link]). When doped into the ZnS film of a scintillator, the 6Li provides excellent detection sensitivity and energy range. Discrimination against fast neutrons and γ-ray contamination in the incident beam is easily accommodated as these have different velocities to the thermal and epithermal neutrons used for TOF diffraction and are therefore readily excluded by the chopper system and detector electronics.

The detector electronics on older instruments recorded the diffraction pattern in a fixed set of time channels or bins; typically 1024 to begin with and progressively more as electronic and computational advances occurred. More recently, the technique has shifted to recording the data to memory in a continuous stream known as event mode, where the arrival time of each neutron is recorded. The user may then bin (and re-bin) the data into time channels to suit the resolution of the diffraction pattern, which may differ significantly from the instrument resolution because of microstructural features of the sample. Such features are discussed at length in Chapters 5.1 and 5.2[link] .

In a new development, a neutron-sensitive microchannel plate detector has been developed (Tremsin, McPhate, Vallerga, Siegmund, Feller et al., 2011[link]). Microchannel plate detectors (MCPs) are divided into discrete pixels and record the arrival time of each neutron in each pixel. Initially used for high-resolution radiography at pulsed neutron sources, it was quickly realized that MCP detectors can be used for diffraction via the Bragg-edge phenomenon (Tremsin, McPhate, Vallerga, Siegmund, Kockelmann et al., 2011[link]). The resolution is typically 55 µm due to the data-acquisition electronics but can be sharpened to less than 15 µm using centroiding techniques. This type of detector opens the door to spatially resolved neutron powder diffraction in materials as well as strain-imaging applications on TOF neutron diffractometers.

2.3.4.3. Variations on a theme

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The diffractometers HRPT (Fig. 2.3.15[link]) and POLARIS (Fig. 2.3.18[link]) are general-purpose instruments suitable for solving and studying medium-sized crystal structures under a range of non-ambient conditions and in some cases the study of non-crystalline or poorly crystalline materials. There are several such diffractometers at reactors [HB-2A at Oak Ridge (https://neutrons.ornl.gov/powder ), D1B at ILL (https://www.ill.eu/instruments-support/instruments-groups/instruments/d1b/description/instrument-layout/ ), HRPD at KAERI (http://www.kaeri.re.kr:8080/english/sub/sub03_04_01_01.jsp ), C2 at CINS (http://cins.ca/get-beam-time/beamline-specs/c2/ )] and spallation sources around the world [POWGEN and NOMAD at SNS (https://neutrons.ornl.gov/powgen ; https://neutrons.ornl.gov/nomad ), GEM at ISIS (http://www.isis.stfc.ac.uk/instruments/gem/gem2467.html ), iMATERIA at J-PARC (https://j-parc.jp/researcher/MatLife/en/instrumentation/images/BL20.gif ) etc].

A more specialized type of TOF powder diffractometer is the High Resolution Powder Diffractometer (HRPD) at ISIS (https://www.isis.stfc.ac.uk/Pages/Hrpd.aspx ) and a similar instrument, Super-HRPD at J-PARC (https://j-parc.jp/researcher/MatLife/en/instrumentation/images/BL08.jpg ). Although both of these instruments have 90° and low-angle detector banks, their overall design has strongly centred on extremes of resolution, attaining [{{\Delta d}/{d}} ] values of 4 × 10−4 and 3 × 10−4, respectively. Such extremes of resolution are attained primarily through making the flight path of both instruments nearly 100 m long and placing detectors at very high Bragg angles (150–176°). Data from these can supply individual peak positions to a precision of approximately 5 parts per million and whole pattern fitting can give correspondingly precise lattice parameters. Recalling that in TOF powder diffraction the resolution is constant across the whole pattern, this makes the instruments ideal for the solution of large crystal structures in which a great many diffraction peaks need to be resolved, for tracking phase transitions, and for solving structures involving pseudo-symmetry, which even in relatively small structures (e.g. perovskites) can be a challenge for lower-resolution instruments. Example diffraction patterns are shown in Fig. 2.3.20[link] for the structural transitions in SrZrO3 (Howard et al., 2000[link]).

[Figure 2.3.20]

Figure 2.3.20 | top | pdf |

Parts of the very high resolution neutron powder-diffraction patterns recorded by the backscattering detector bank on the instrument HRPD at ISIS from SrZrO3 at (a) 1403, (b) 1153, (c) 1053, (d) 933 and (e) 293 K. Insets to the left and right show subtle changes to the reflection shapes and splitting of reflections due to phase transitions from the cubic ([Pm{\bar 3}m]) in pattern (a), to the tetragonal phase (I4/mcm) in (b), an orthorhombic phase (Imma) in (c) and a second orthorhombic phase (Pnma) in (d) and (e). Note the intensity reversal in the 002 reflection (right insets), which was pivotal in finding and solving the orthorhombic phase in Imma (Howard et al., 2000[link]).

At the other extreme of instrument design are the very high intensity diffractometers exemplified by the CW instruments D20 at ILL (https://www.ill.eu/instruments-support/instruments-groups/instruments/d20/ ) and WOMBAT at ANSTO (http://www.ansto.gov.au/ResearchHub/Bragg/Facilities/Instruments/Wombat/ ). These diffractometers use a large degree of vertical focusing to greatly increase the incident flux on the sample and are fitted with large position-sensitive detectors from which the data can be stored at 1 MHz or faster. If there is a periodic time structure to the phenomenon under study due to some driving stimulus (e.g. a periodic laser, electric or magnetic field pulse), then the data can be analysed stroboscopically by synchronizing with the driving stimulus, giving an effective time resolution in the MHz range. Even in the absence of a periodic stimulus, useful diffraction patterns on these diffractometers can in favourable circumstances be stored at rates of 2, 10 or with a large enough sample even 50 Hz (Fig. 2.3.21[link]).

[Figure 2.3.21]

Figure 2.3.21 | top | pdf |

Neutron powder-diffraction patterns during combustion synthesis of Ti3SiC2 recorded in 400 ms each on the diffractometer D20 at ILL (Riley et al., 2002[link]). Panel (a) shows an overview of the reaction process with time vertical, diffraction angle horizontal and intensity as colour/brightness. Panel (b) is a three-dimensional view of the portion enclosed by dashed lines in (a), representing 140 s of reaction, wherein the numbered reflections show (i), (ii) a phase change in Ti, (iii) SiC, (iv) formation of an intermediate phase Ti(Si,C) and (v) growth of the Ti3SiC2 product. Panel (c) illustrates via Rietveld refinement the high quality of diffraction patterns even on this short timescale.

It should be noted that for TOF diffractometers, the time structure imposed by the pulsed neutron source and chopper system places absolute limitations on the most rapid diffraction pattern that can be recorded. This is typically ~0.1 s at sources such as ISIS, J-PARC or SNS. An additional time penalty is often paid due to the time taken to save such large amounts of data (typically between 10 and 30 s). There is therefore no TOF equivalent of the very rapid stroboscopic mode of operation.

Other forms of specialized neutron powder diffractometer have also been developed. Among these are the engineering or residual stress diffractometers, exemplified by the TOF diffractometers ENGIN-X at ISIS (https://www.isis.stfc.ac.uk/Pages/Engin-X.aspx ), VULCAN at SNS (https://neutrons.ornl.gov/vulcan ), TAKUMI at J-PARC (https://j-parc.jp/researcher/MatLife/en/instrumentation/ns.html ) and the CW diffractometers SALSA at ILL (https://www.ill.eu/instruments-support/instruments-groups/instruments/salsa/description/instrument-layout ) and KOWARI at ANSTO (http://www.ansto.gov.au/ResearchHub/Bragg/Facilities/Instruments/Kowari/ ). The pur­pose of these diffractometers is to measure accurate interplanar spacing (d) within a small gauge volume defined by the intersection of incident and diffracted beams inside a larger sample, as illustrated for constant wavelength in Fig. 2.3.22[link].

[Figure 2.3.22]

Figure 2.3.22 | top | pdf |

Illustrating (a) a CW engineering diffractometer and (b) the formation of a gauge volume at the intersection of the incident and diffracted beams.

Variations in the d-spacing relative to a strain-free reference value (do) represent the average strain in the gauge volume parallel to the scattering vector (i.e. perpendicular to the diffracting planes) as is also illustrated in Fig. 2.3.22[link]. By determining strains in several directions, it is possible to reconstruct the full strain tensor within each gauge volume, and this may be converted into the stress tensor, the desired outcome for engineering purposes (Noyan & Cohen, 1987[link]; Fitzpatrick & Lodini, 2003[link]; Kisi & Howard, 2008[link]). This procedure is widely used in residual stress analysis to study stress distributions in fabricated or welded components and also to observe the internal stress distribution due to an externally imposed load. An example is illustrated in Fig. 2.3.23[link] in relation to in situ experiments and the stress distribution in granular materials.

[Figure 2.3.23]

Figure 2.3.23 | top | pdf |

Stress distribution for four stress components in an iron powder compacted within a convergent die (see also Zhang et al., 2016[link]).

The required localization of the gauge volume is achieved by shaping the incident and diffracted beams with slits/collimators and is greatly assisted by fixing the diffraction angle 2θ at ±90°. In CW instruments, the need for high resolution and good intensity is met by using a focusing (bent Si) monochromator and a small area detector to record the data. This generally limits the investigation to a single Bragg peak (reflection), the position of which is carefully mapped over the sampled area for each strain component under investigation.

TOF engineering diffractometers record a full diffraction pattern at each position. Localization of the gauge volume is achieved using symmetric detector banks and radial collimators on either side of the sample position (Fig. 2.3.24[link]). All other instrument-design criteria are generally secondary to this, as a parallelepiped-shaped gauge volume allows a seamless strain (stress) map to be obtained. These instruments are usually 40–50 m long and have moderately high resolution, which allows peak positions and hence strains to be measured to a precision of 5 × 10−5 in favourable circumstances. In common engineering materials (steels, aluminium alloys etc.) this equates to an absolute minimum stress uncertainty of 4–10 MPa. The extreme resolution that would be available using very high resolution designs like HRPD and Super-HRPD (above) is sacrificed in order to obtain data on a reasonable timescale given the generally small gauge volume (0.5–30 mm3) and the need to map the strain field piecewise over an extended region of the sample.

[Figure 2.3.24]

Figure 2.3.24 | top | pdf |

The engineering diffractometer ENGIN-X at ISIS. The incident beam enters through the flight tube at the top and the left (L) and right (R) 90° detector banks simultaneously record patterns with the scattering vector perpendicular and parallel to the sample axis, respectively. A mechanical testing machine used for in situ application of loads is also shown (https://www.isis.stfc.ac.uk ). (Credit: STFC.)

Although it is not usual for instruments to be specifically designed for the purpose, neutron diffraction is also particularly useful for studying crystallographic texture in materials, as the neutron-diffraction pattern is not distorted by surface coatings or preparation methods. In principle, any diffractometer can be used for measuring texture simply by recording a large number of diffraction patterns with the sample rastered in small angular intervals (5° is common) about two mutually perpendicular axes to form a grid over all orientations. This is extremely time consuming on a conventional CW diffractometer, although the whole pattern is captured each time, as the intensity recorded for the different reflections is subject to different corrections. This can be greatly sped up by using a CW engineering diffractometer (SALSA, KOWARI) with an intense, well collimated incident beam and fitted with an area detector. For example, on KOWARI, the detector spans 15° in both horizontal and vertical directions and so the sample needs to be re-positioned far fewer times. An added advantage is that the diffraction geometry is identical for each sample position and almost so for each reflection studied, and so a pure (i.e. model-independent) texture measurement is obtained. Texture measurements on modern TOF diffractometers (e.g. GEM, POLARIS, POWGEN, NOMAD and iMATERIA) are in principle quite straightforward. Because there are detectors in many positions all around the sample, the scattering vector and hence orientation of diffracting planes (crystal orientation) is sampled in many orientations all in one data collection. If data from the individual detectors are not `focused' into composite diffraction patterns as for crystal-structure studies, then very few re-orientations are required to record data representing the full texture. However, since each reflection in each detector bank is sampled using neutrons of different wavelength, each is recorded under different conditions for attenuation and extinction. In addition, to make full use of all the data, whole pattern or Rietveld analysis using a preferred-orientation (texture) model has to be conducted for each of the multitude of diffraction patterns recorded. As well as being time consuming, the reliability of the resultant pole figures and orientation density function is governed by the quality of all the individual models (for background, peak shape, peak width, sample centring, attenuation etc.) within the Rietveld refinement as well as the ability of the preferred-orientation model in the Rietveld program to accurately fit the real texture. A pure model-independent texture measurement can only be obtained using CW or TOF single-peak methods.

The instrument WISH at ISIS represents a departure from the normal TOF diffractometer design in that it receives long wavelength neutrons (1.5–15 Å) from a cold neutron source at Target Station 2. Ballistic supermirror neutron guides and three choppers deliver neutrons in an active bandwidth of 8 Å for a given chopper setting (https://www.isis.stfc.ac.uk/Pages/Wish.aspx ). The pixelated 3He detectors cover Bragg angles in the very wide range 10–170°. WISH is designed for the study of complex magnetic structures and large-unit-cell structures in chemistry and biology. Polarization analysis is available to assist the former.

The concept of long-wavelength neutron powder diffraction will be taken a step further in the DREAM instrument planned for the European Spallation Source (ESS, https://europeanspallationsource.se/realizing-dream-versatile-powder-diffractometer ). This instrument will receive neutrons simultaneously from thermal and cold neutron moderators. It will have a complex array of choppers to shape the incident pulse prior to arrival at the sample. Modelling has indicated that intensity gains of a factor of 10–30 are to be expected and that the instrument may be able to deliver [{{\Delta d}/{d}}] as low as 4 × 10−5, albeit at very long wavelengths. More typically the projection is that [{{\Delta d}/{d}}] as low as 1 × 10−4 could be achieved with more conventional wavelengths. Perhaps the major advantage of the instrument will not be its absolute resolution but the ability to change resolution over the full range during the experiment by simply altering the chopper settings. Therefore unexpected phenomena (phase transitions etc.) can be tracked during the initial experiment with no time lost by having to prepare a proposal for a different higher-resolution instrument.

2.3.4.4. Comparison of CW and TOF diffractometers

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The preceding discussion has demonstrated that, although not necessarily the case for other types of neutron scattering, powder diffraction can be very successfully conducted on either CW or TOF instruments. Their relative advantages for the various types of powder-diffraction experiment are embedded in the discussion above and summarized in Table 2.3.5[link].

Table 2.3.5| top | pdf |
Advantages of CW and TOF instruments (modified from Kisi & Howard, 2008[link])

CWTOF
(1) Incident beam may be essentially monochromatic, in which case the spectrum is well characterized (1) The whole incident spectrum is utilized, but it needs to be carefully characterized if intensity data are to be used
   
(2) Large d-spacings are easily accessible for study of complex magnetic and large-unit-cell structures (2) Data are collected to very large Q values (small d-spacings)
   
  (3) Few cold neutron instruments are available for study of complex magnetic and large-unit-cell structures
   
(3) Can fine tune the resolution during an experiment (4) Resolution is constant across the whole pattern
   
  (5) Very high resolution is readily attained by using long flight paths
   
(4) More common (6) Complex sample environments are very readily used if 90° detector banks are available
   
(5) Peak shapes are simpler to model  
   
(6) Absorption and extinction corrections are relatively straightforward (7) Simpler to intersect a large proportion of the Debye–Scherrer cones with large detector banks
   
(7) Data storage and reduction is simpler  
   
(8) Extremely rapid data collection and stroboscopic measurements are feasible (8) Very fast data collection is feasible
   
(9) Engineering diffractometers are very well suited for strain scanning in complex objects (9) Engineering diffractometers use an extended diffraction pattern, ideal for in situ loading and/or heating
   
(10) Texture is straightforward to measure on engineering diffractometers (10) Texture can be measured on universal instruments

Plotting and summarizing the approximate intensity and resolution of different types of neutron diffractometer may be of assistance in assessing the options (Fig. 2.3.25[link]). In the figure, resolution is shown as the inverse of the FWHM (Δd/d) and intensity is shown as the inverse of the time in seconds taken to record a single diffraction pattern, so that improvements follow the positive x and y axes.

[Figure 2.3.25]

Figure 2.3.25 | top | pdf |

Schematic showing regions of intensity–resolution space in which different diffractometer types typically operate. High-resolution TOF diffractometers operate in the green area, engineering diffractometers (TOF or CW) in the purple area, multi-purpose TOF diffractometers such as POLARIS in the orange area and very high intensity CW diffractometers in the blue area.

There are two particular cases where the distinction between CW and TOF instruments can determine the success or failure of a neutron powder-diffraction experiment. The first is where crystal structures or phase transitions involving extreme pseudosymmetry are being studied. In this case, the very high resolution available over the entire Q-range (d-spacing range) using high-resolution TOF instruments such as HRPD at the ISIS facility (UK) or SuperHRPD at J-PARC confers a particular advantage. The CW equivalent high-resolution powder diffractometers such as D2B at ILL and ECHIDNA at ANSTO can almost match the absolute resolution of the TOF instruments, D2B achieving [{{\Delta d}/{d}}] of 5.6 × 10−4; however, the resolution function for a CW diffractometer [equation (2.3.18)[link]] has a strong minimum and so this resolution can only be achieved over a restricted range of d-spacing. The reflections appearing in the highest-resolution zone can be shifted by wavelength changes, which of necessity require re-recording of the pattern.

The second extreme case is when rapid kinetic behaviours are to be studied. In this case, a small number of CW diffractometers (e.g. D20 at the Institut Laue–Langevin or WOMBAT at ANSTO) have a distinct advantage. Therefore at this time, processes that occur reproducibly and uniformly over a large sample on sub-1 s timescales are best suited to stroboscopic studies using one of the very rapid CW diffractometers available. There are nonetheless a great number of processes that can be studied on the timescales accessible using TOF, where near-constant resolution across the entire diffraction pattern lends considerable advantage.

If unaffected by extremes of resolution, intensity or highly specialized data types (stress, texture etc.), the choice between a CW or TOF instrument can be made based more casually on proximity to neutron sources and the access arrangements for national or regional neutron users.

2.3.5. Experimental considerations

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2.3.5.1. Preliminary considerations

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Neutron-diffraction studies are motivated by a desire to exploit the unique properties of neutrons as listed in Sections 2.3.1[link] and 2.3.2[link]. As access to neutron diffraction is carefully regulated through an experiment proposal system, considerable planning is required in order to write a successful proposal. Owing to the expense of operating a neutron source and pressure on instrument time, there is an onus on the experimental team to make the best use of neutron beam time. Consideration should be given to the type of instrument required, the resolution that is needed, the d-spacing range of interest, how long each pattern will take to record, the requirement (or not) for standard samples and whether a special sample environment is needed.

In the general case, there is competition between the resolution and the intensity of diffractometers, although some of the modern TOF diffractometers (e.g. POLARIS, GEM, POWGEN, NOMAD and iMATERIA) simultaneously record patterns of moderate resolution and intensity, and high-intensity patterns at low resolution, in different detector banks. For the purposes of this chapter, high resolution is defined as a minimum diffraction peak width at half maximum height corresponding to [{{\Delta d}/{d}}\le {10}^{-3}]. This is the resolution typically required to observe lattice-parameter differences [e.g. (ab)/a] of as little as 4 × 10−5 or so in the absence of sample-related peak broadening. Such a diffractometer is typically of the order of 10 to 1000 times slower than corresponding high-intensity diffractometers at the same neutron source. The decision to opt for a high-resolution diffractometer or a high-intensity diffractometer will depend critically on the nature of the problem under study. This situation was considered in Kisi & Howard (2008[link]) and their conclusions are reproduced in Table 2.3.6[link].

Table 2.3.6| top | pdf |
Suitability of problems to high-resolution or high-intensity diffractometers

Reproduced from Kisi & Howard (2008[link]) by permission of Oxford University Press .

ProblemHigh resolutionHigh intensity (medium resolution)
Solve a complex crystal or magnetic structure Essential, especially in the presence of pseudosymmetry Not usually suitable
     
Refine a complex crystal or magnetic structure Essential. Will benefit from a high Q-range if available Not usually suitable
     
Solve or refine small inorganic structures Beneficial, but not usually essential unless pseudosymmetry is present Usually adequate
     
Quantitative phase analysis Only required when peaks from the different phases are heavily overlapped Usually adequate. Allows phase quantities to be tracked in fine environmental variable steps (T, P, E, H etc.) during in situ experiments
     
Phase transitions Depends on the nature of the transition and complexity of the structures. Essential for transitions involving subtle unit-cell distortions and pseudosymmetry Often adequate for small inorganic structure transitions and order–disorder transitions. Allows fine steps in an environmental variable (T, P, E, H etc.)
     
Line-broadening analysis Essential for complex line broadening such as from a combination of strain and particle size, dislocations, stacking faults etc. Adequate for tracking changes in severe line broadening as a function of an environmental variable (T, P etc.) especially if the pure instrumental peak shape is well characterized
     
Rapid kinetic studies Not appropriate Essential
In some cases the symmetry and lattice parameters are such that the diffraction peaks are well spaced and not severely overlapped even at modest resolution.
May be necessary to supplement high-resolution data to observe weak superlattice reflections in the presence of very subtle or incomplete order–disorder transitions.

It might be expected that the total information content in a diffraction pattern correlates with the d-spacing range covered and therefore this should be maximized. However, this expectation overlooks the different purposes for which powder-diffraction patterns are used. A greater density of diffraction peaks (e.g. in a CW pattern recorded using a short neutron wavelength) makes the detailed refinement of complex crystal structures more precise; however, it makes the determination of unit cell and systematic absences more difficult as well as reducing access to information contained within the peak shapes concerning the sample microstructure. Table 2.3.7[link] summarizes these effects. It should be noted that in this context parallels exist between a short-wavelength CW diffraction pattern and a low-angle-detector-bank TOF pattern; and between a longer-wavelength CW pattern and a high-angle-detector-bank TOF pattern, subject to limitations imposed by the wavelength distribution in the incident spectrum.

Table 2.3.7| top | pdf |
Guidance on choice of wavelength/detector bank

Reproduced from Kisi & Howard (2008[link]) by permission of Oxford University Press .

ProblemChoiceReasons
Solve complex or low-symmetry structures Longer wavelength Increase d-spacing resolution to allow correct symmetry and space group to be assigned
     
Refine a large or complex crystal structure Shorter wavelength Ensure that the number of peaks greatly exceeds the number of parameters. Improve determination of site occupancies and displacement parameters
     
Solve or refine magnetic structures Longer wavelength Ensure that large d-spacing peaks are observed. Spread the magnetic form factor over the entire diffraction pattern
     
Quantitative phase analysis Usually shorter wavelength Improve the accuracy of the determination. Longer wavelengths only required if peak overlap is severe
     
Phase transitions Shorter wavelength Ensures adequate data for order–disorder or other unit-cell-enlarging transitions
  Longer wavelength Subtle unit-cell distortion or pseudosymmetric structures

A decision must be made on how long to spend recording each diffraction pattern, such that the greatest number of patterns (samples) may be studied without compromising the information content of each pattern. Since counting is governed by Poisson statistics, the statistical precision of N counts in a radiation detector (X-ray, electron or neutron) is represented by the standard deviation σ:[\sigma = N^{1/2}. \eqno(2.3.23)]This is true regardless of whether a single count is made or multiple counts are summed to give an integrated intensity or a total count from several detectors. For a relatively constant arrival rate of neutrons, the precision of each data point will increase with counting time t in proportion to t1/2, and this will be reflected in the agreement indices (e.g. Rwp; Chapter 4.7[link] ) between the observed and calculated neutron intensities during structure refinements (e.g. Rietveld refinement) as well as in the estimated standard deviation (e.s.d.) of the refined crystal structure and other parameters. It has been shown by Hill & Madsen (1984[link]) using CW X-ray powder-diffraction patterns that this is the case for small counting time; however, the agreement and e.s.d.'s quickly attain a plateau for counting times where 2000–5000 counts are recorded at the top of the largest diffraction peak. Beyond this, systematic errors in the models used for peak shapes, background etc. begin to dominate the fitting procedure. An important consequence is that since the expected values of the parameter e.s.d.'s fall in proportion to [{t^{ - 0.5}}] whereas their actual values plateau, the statistical χ2 increases for patterns recorded beyond the limit suggested by Hill & Madsen. A number of recommendations may be derived from these results:

  • (i) It is of no benefit for routine crystal structure refinements to record data beyond the point where the strongest peak has 5000 or so counts at its apex and to do so may render parameter e.s.d.'s invalid.

  • (ii) Counting for longer times is however recommended for problems that hinge upon weak superlattice or magnetic peaks. Similarly, it may be of benefit when minor phases are of interest, such as in complex engineering materials, in samples undergoing phase transitions or in multi-component geological materials.

  • (iii) An equally important result from Hill & Madsen is that respectable refined parameter estimates could be obtained using powder-diffraction patterns with only 200–500 counts at the apex of the strongest peak. This is extremely useful when assessing counting times in rapid kinetic studies where the shortest acceptable counting time is preferred. Modern data-acquisition electronics are often configured to allow very short acquisition times or `event-mode operation' (Section 2.3.4.2.2[link]) with patterns subsequently added together to obtain the required statistical and/or time resolution. In this case, the shortest time step available should be used provided sufficient data storage capacity is at hand.

In CW measurements with a detector bank scanned in small angular steps, similar arguments to those above apply to the sampling interval. This too has been investigated by Hill & Madsen (1986[link]) and again, improvements to the agreement between the calculated and observed patterns and indeed improvements to refined parameter e.s.d.'s were only observed until systematic errors begin to dominate the fit. As a general rule of thumb, once the applicable counting time has been established, the counting interval should be adjusted to give at least 2 (but typically around 5) sampling points in the top half of the diffraction peak for routine crystal structure refinements. Finer sampling intervals are however beneficial in the case of:

  • (i) subtle symmetry changes that manifest in the peak shape well before peak splitting is observable,

  • (ii) following the evolution of a minor phase during an in situ experiment, or

  • (iii) peak-shape analyses to explore the sample microstructure (crystallite size, strain distribution, dislocation density, stacking-fault probability etc.).

CW measurements using instruments with a fixed position-sensitive detector and TOF measurements both have their raw sampling interval fixed by the instrument architecture, which cannot be varied. The recorded patterns can be subsequently re-binned to a larger sampling interval, although this would usually only be considered to reduce serial correlations during profile refinement (Hill & Madsen, 1986[link]).

2.3.5.2. Sample-related factors

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Recording a neutron powder-diffraction pattern is in itself a simple operation. There are, however, a number of sample-related variables that can affect the accuracy or the precision of the resulting patterns, or the ability to analyse them. It is worth mentioning here that neutron-diffraction samples are often large, in the range 1–40 g, to compensate for the lower incident fluxes and scattering cross sections as compared with the X-ray case. Large sample size has a strong mitigating effect on many of the sample-induced problems to be discussed in Chapter 2.10[link] and below.

The absolute accuracy of the position, intensity and shape of neutron powder-diffraction peaks is primarily determined by:

  • (i) How representative the whole sample is of the whole system. Known as disproportionation, this problem results from any non-random factor during sampling. For example, within rocks there is spatial variability in the mineral content (where to sample), hardness differences (different mineral particle sizes) and differing density (settling effects). Similar considerations apply to multiphase ceramic materials and metal alloys. With highly penetrating neutrons, this can be greatly reduced by using the polycrystalline solid sample provided that the crystallite size is relatively small [see (iii)[link] below]. Disproportionation primarily influences quantitative phase analysis studies. Crystal-structure results are unaffected provided there is enough of each phase of interest to give a high-quality diffraction pattern.

  • (ii) How representative the irradiated part of the sample is of the whole system. Although ideally the entire sample is bathed in the incident beam, for highly focused neutron beams on high-intensity and/or strain-scanning diffractometers, the beam–sample interaction volume is smaller than the whole sample. In such cases, if a gradient in an experimental variable such as temperature, pressure or composition is present, then the irradiated portion of the sample can be quite unrepresentative and this needs to be addressed in the overall experiment plan.

  • (iii) How representative the diffracting part of the sample is of the whole sample. There are two circumstances in which the observed diffraction pattern may be unrepresentative of the irradiated portion of the sample. First, very large crystallite size leads to the phenomenon of granularity, which is dealt with in detail in Section 2.10.1.1[link] . Crystallites diffract only when the Bragg condition is met, so if the crystallite size is a sizable fraction of the irradiated part of the sample, only a small number of crystallites are aligned for diffraction. With only relatively few crystals diffracting, the peak shapes, intensities and apparent d-spacings are strongly distorted. Second, when there is amorphous material present, it is visible in the diffraction pattern only as structure in the background signal and is not analysed using standard crystallographic techniques.

  • (iv) How representative the recorded pattern is of the sample. There are two other factors that can affect accuracy of the diffraction pattern.

    The first is that the crystallites may have preferred (rather than random) orientations, so that some sets of atomic planes are overrepresented and others underrepresented in the diffraction pattern. This effect and the means to overcome it in X-ray diffraction measurements are covered in Section 2.10.1.2[link] . Neutron powder diffraction, by using large samples on a rotating sample holder in transmission geometry, is generally far less susceptible to preferred orientation than X-ray diffraction. In cases where preferred orientation is unavoidable, it is generally of a simple axial form due to the sample rotation. Quite good analytical means for modelling preferred orientation of this type are available in the various refinement programs described in Chapter 4.7[link] .

    The second effect is attenuation. For most materials, thermal neutrons are attenuated comparably by true absorption and scattering, the overall effect being very minor. For a small number of elements (e.g. B, Cd, Gd – see Table 2.3.2[link]) the absorption is high, and in an even smaller number of isotopes (e.g. H) the incoherent scattering is high enough to give significant attenuation. Details of these processes are dealt with in Section 2.10.2.4[link] as well as in Sections 2.4.2 and 3.5.3 of Kisi & Howard (2008[link]). In summary, when using transmission geometry and absorbing samples, diffraction peaks at low angle (CW) are attenuated more than those at higher angles. An additional linear dependence on neutron wavelength occurs in TOF patterns. Therefore the relative intensities are incorrect and during structure refinements unreasonable (often negative) displacement parameters will result. When strongly attenuating elements or isotopes are present three approaches are available; the data can be recorded in reflection geometry, the capillary-coating method can be adapted from X-ray diffraction, or the sample can be diluted with a large amount of a weakly absorbing material. The latter two methods are explained in Section 2.10.1.4.2.2[link] .

Sample-related factors that interact with the precision of various crystallographic and microstructural parameters determined from a given diffraction pattern are:

  • (i) The crystallite size within the sample. As discussed at length in Chapter 2.10[link] , the ideal size for crystallites in a powder-diffraction measurement is 2–5 µm. The upper limit is determined by onset of granularity [see (iii)[link] above]. The lower limit is set by the onset of detectable crystallite size broadening (Chapter 5.1[link] ). To first order, the broadening of diffraction peaks due to small crystallite size is well understood. It has negligible effect on the measured intensity of diffraction peaks and does not affect the numerical value of the peak positions (hence d-spacings); however, the precision or standard error of such measured positions is strongly affected. In addition, the precision (standard error) of measurements of other microstructural features such as strain distributions, dislocation density or stacking-fault probability are strongly affected. Powdered samples should be sized to lie within the range 2–5 µm with the lower limit being the more important in this case. The crystallite size within solid polycrystalline samples is an inherent part of the system. Forming a material with a fine grain size is a universal method for strengthening metals and ceramics alike. In systems undergoing phase transitions the crystallites typically subdivide into small portions during the transition. Consequently, crystallite size broadening is often an inevitable part of a powder-diffraction experiment.

  • (ii) How ideal the crystal structure is within the crystallites. The preparation of powder samples can induce several types of lattice defects (dislocations, stacking faults, twin faults etc.) into the material under study. Each of these leads to changes to the peak positions, shapes and breadths. Likewise, in solid polycrystalline samples, thermal-expansion anisotropy and mismatch between different phases cause intergranular strains which manifest themselves in broadened peaks. Each new source of broadening strongly affects the precision with which other microstructural features of the sample can be determined from peak-shape analysis. In ground powders, it is sometimes possible to relieve stresses and repair defects by annealing, but only if it is certain that no detrimental changes to the material occur under the annealing conditions.

A common prerequisite for the detailed analysis of diffraction patterns is a good understanding of the instrument's characteristic peak shapes and widths, i.e. the resolution function (Sections 2.3.4.1.4[link] and 2.3.4.2.1[link]). The parameters of the resolution function are needed to enable Rietveld (Chapter 4.7[link] ) or whole-pattern (Chapter 3.6[link] ) analysis of the diffraction patterns. A good description for the instrument resolution function is important in the study of sample microstructure (e.g. crystallite size, strain distribution or dislocation studies) and may be established using standard samples. Early versions of the NIST LaB6 lattice-parameter and peak-shape standards (SRM 660) were unsuitable because of the high neutron absorption of natural boron. More recently, NIST has developed LaB6 standards SRM 660b and 660c made with 11B that can be used for neutron diffraction (see Section 3.1.4[link] ). Suitable air- and moisture-stable alternatives with a closely regulated crystallite size and a moderate density of diffraction peaks include Al2O3, CeO2, Y2O3 and some intermetallic compounds such as Cu9Al4 and Cu5Zn8.

One's ability to successfully analyse a diffraction pattern is then strongly affected by:

  • (i) Smooth and locally monotonic peak shapes. The two primary causes of failing to meet this requirement are granularity (crystallites significantly above the preferred 2–5 µm size) and unusual sample shapes such as hollow samples. Examples of the former may be seen in Figs. 2.10.2[link] and 2.10.3[link] , where large single crystals in the sample each give a discrete diffraction peak, the composite of which looks nothing like the true powder peak shape. The case of hollow samples is rarely seen unless the `capillary-coating' technique (see Section 2.10.1.4.2.2[link] ) is adopted for a highly absorbing sample or diffraction peaks from a hollow sample container are also to be analysed. In this case, the peak shape will have a depression in the centre due to the non-uniform distribution of diffracting matter across the specimen.

  • (ii) Crystallite perfection. For crystal-structure studies, it is preferred that the crystallites in the sample be as near perfect19 as possible. However, materials of interest are often far from perfect, containing stacking faults, domain walls, antiphase boundaries, compositional gradients, strain gradients etc. Fig. 9.22 in Kisi & Howard (2008[link]) illustrates this for a ferroelectric material. Here the individual crystallites are subdivided into ferroelectric domains with different orientations defined by the symmetry relationship between the parent (cubic) and daughter (tetragonal) structures. Where differently oriented domains abut, there is a strain gradient over a finite portion of crystal. This is visible in the diffraction pattern as a plateau between twin-related pairs of peaks such as the 200/002 pair shown, because in the strain gradient all d-spacings between d200 and d002 are present.

  • (iii) Sample perfection. The major types of imperfection in sampling are described under accuracy in the preceding discussion. Our main interest here is in the preferred orientation of crystallites, which means some diffraction peaks are exaggerated and others underrepresented in the diffraction pattern. Methods for avoiding or reducing preferred orientation are dealt with in Section 2.10.1.2[link] . In addition, whole-pattern fitting and reasonably robust mathematical models for preferred orientation, principally the March–Dollase model (Dollase, 1986[link]) and models based on spherical harmonics (Ahtee et al., 1989[link]), have reduced the effect of preferred orientation on crystal-structure parameters and quantitative phase analyses derived from powder-diffraction patterns. In a small number of cases of severe and/or multi-axis preferred orientation, these models can fail and efforts to reduce the effect within the sample need to be revisited.

2.3.5.3. Sample environment and in situ experiments

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It is more often the case with neutron diffraction than with X-ray or electron diffraction that the purpose is an experimental study involving rather more than a simple room-temperature data collection.20 As such, there are a great variety and complexity of sample environments available, relating to studies: at room temperature, cryogenic temperatures, high temperature and high pressure; under magnetic fields, electric fields or applied stress; during gas–solid, liquid–solid, solid–solid or electrochemical reactions; and almost any combination of these. There are several other chapters in this volume that include descriptions of sample environments for neutron powder-diffraction experiments under high (hydrostatic) pressure (Chapter 2.7[link] ), electric and magnetic fields (Chapter 2.8[link] ) and chemical and electrochemical reactions (Chapter 2.9[link] ). Some general guidance on the mounting of samples is also given in Chapter 2.10[link] . Additional information concerning sample containers for non-ambient studies, as well as sample environments not expressly covered in these chapters, will be presented briefly below.

2.3.5.3.1. Sample containers

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Solid polycrystalline samples can be directly mounted on the diffractometer; however, powder samples require careful containment. Powder spillage must be avoided because samples may become activated in the neutron beam and spilled powders present a radiological hazard. Owing to the low neutron attenuation by most materials, neutron diffraction patterns are generally recorded in transmission (Debye–Scherrer) geometry. Therefore sample containers that do not contribute significantly to the diffraction pattern are required. Fortunately there are several materials that have essentially zero coherent neutron scattering length, i.e. they give no discernible diffraction peaks and minimal contribution to the background. Most versatile is elemental vanadium, which has a scattering length of just −0.3824 fm (Table 2.3.2[link]), making its diffraction pattern 100–750 times weaker than most other metals. Coupled with excellent room-temperature resistance to atmospheric corrosion, it is not surprising that it is the material that is used most often for neutron powder diffraction sample holders. Typical designs are discussed in Section 2.10.2.3[link] . Another useful material for room-temperature containment is Al, which has very low attenuation and few diffraction peaks of its own. This is especially useful in cases where only the large d-spacing peaks are of interest, for example with magnetic materials or large-scale structures, or where a fine radial collimator is able to exclude diffraction from the sample container.

Sample containers for specialized sample environments vary greatly. Low-temperature studies routinely use V or Al cans, as for room-temperature studies. High-temperature studies of powders can use V cans up to approximately 1073 K provided that an inert gas or vacuum environment is present. At higher temperatures, thin-walled fused silica (silica glass) can be used as it has several advantages: it is amorphous and therefore gives no sharp diffraction peaks; it is vacuum tight and relatively easy to seal to vacuum fittings via O-rings outside or graded glass–metal seals within the hot zone of the furnace; it is transparent, so the state of the sample can be viewed during loading and after the experiment; and it is immune to thermal shock. Silica can survive at temperatures up to 1473 K and for short periods can resist temperatures up to 1673 K, although some devitrification may occur. Care should be exercised since although fused silica has no sharp diffraction peaks, its short-range order does give a structured background which has to be carefully treated in subsequent analyses. Containers for still higher temperatures can be made from other ceramics such as alumina or from refractory metals such as Nb, Ta or W in increasing order of temperature resistance. Noble metals such as Pt may seem to have some advantages; however, they are extremely weak and fragile after high-temperature annealing. All high-temperature sample-container materials are able to chemically react with some samples at high temperature and great care must be taken when selecting them. If possible, a trial heating should be conducted off-line prior to the experiment.

2.3.5.3.2. Non-ambient temperature

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As neutron powder diffraction is routinely conducted in transmission geometry, non-ambient sample environments have many common features. They are typically cylindrical in shape, with the sample can loaded centrally from above on a `sample stick', which goes by various names in different fields.

An example is the liquid-helium cryostat developed at the Institut Laue–Langevin, shown in Fig. 2.3.26[link]. The internal space is evacuated and heat is removed from the sample via conduction through the sample stick to cold reservoirs in contact with the liquid-helium tank. The sample protrudes below the helium and nitrogen tanks into the `tails', which are thin-walled Al or V cylindrical sections that allow ready transmission of neutrons but preserve the vacuum and exclude radiant heat from the outside world. Liquid-helium cryostats can generally attain base temperatures of 4.2 K (He alone) or 1.9 K if pumped. Liquid-nitrogen cryostats are limited to 77 K. A second type of low-temperature device is the closed-cycle He refrigerator, commonly referred to by the trade name Displex. These are more compact than a liquid-helium cryostat and do not require refilling. Depending on the number of stages and internal design, refrigerators with base temperatures as low as 4 K are available.

[Figure 2.3.26]

Figure 2.3.26 | top | pdf |

(a) Exterior and (b) interior of the standard ILL liquid-helium cryostat for cooling samples in the range 1.8–295 K. An internal heater allows samples to be studied without interruption from 1.8–430 K. Reproduced with permission from the ILL.

Samples are typically first cooled to base temperature and then studied at the chosen sequence of increasing temperatures. This is achieved through a small electric resistance heater and control system. As heat transfer to and from the sample is deliberately poor in these devices, sufficient time should be allowed for the (often large) sample to reach thermal equilibrium before recording its neutron-diffraction pattern. It is worth noting that the attainment of thermal equilibrium does not guarantee that the sample has attained thermodynamic equilibrium. Some phase transitions are notoriously slow, for example the ordering of hydrogen (or deuterium) in Pd metal at 55 K and 75 K, which can take up to a month (Kennedy et al., 1995[link]; Wu et al., 1996[link]), or the ordering of C in TiCx (0.6 < x < 0.9) around 973 K, which can take a week to complete (Moisy-Maurice et al., 1982[link]; Tashmetov et al., 2002[link]).

Raising samples to above ambient temperature is, for X-ray diffraction, the subject of a separate chapter (Chapter 2.6[link] ); however, neutron-diffraction high-temperature devices are somewhat different. Most commonly used and most versatile is the foil element resistance furnace, in which Cu bus bars transfer electric current to a cylindrical metal foil which heats up as a result of its electrical resistance. Foil elements are typically 30–60 mm in diameter and up to 200 or 250 mm long so as to provide a long hot zone of uniform temperature within the furnace. The sample is located, via a sample stick from above or occasionally via a pedestal support from below, in the centre of the foil heating element, ensuring that it is uniformly bathed in radiant heat. Concentric metal-foil heat shields greatly reduce heat loss to the exterior by radiation, while convective losses are avoided by evacuating the interior of the furnace to ∼10−5 mbar. Metals for manufacture of the foil elements include V, which has almost no coherent diffraction pattern and can operate continuously up to 1173 K or intermittently to 1273 K. For temperatures above this, progressively more refractory metals are chosen such as Nb (<1773 K), Ta (<2473 K) or W (2773 K). These materials will contribute some small diffraction peaks to the observed patterns, which requires the recording of reference patterns from the empty furnace before commencing. Owing to the internal vacuum, some types of sample are at risk of subliming, decomposing or disproportioning during the experiment. In such cases, sample cans that extend outside the hot zone, where they can be coupled to a gas-handling system and filled with an internal atmosphere of air, an inert gas or a reactive gas of interest as required, are used.

Alternatives to foil furnaces include variations of the wire-wound laboratory furnace with a split winding and reduced insulating material in the neutron beam path, Peltier devices, hot-air blowers and induction heaters. The first three of these are discussed by Kisi & Howard (2008[link]).

Non-ambient temperature devices are usually designed for operation either below or above ambient temperature. However, there are a large number of phase transitions and other phenomena that span from below to above ambient temperature. In order to avoid transferring samples from one sample environment to another mid-experiment, a useful hybrid device is the cryo-furnace. Cryo-furnaces are based around the liquid-helium cryostat and are equipped with more powerful heaters, allowing temperatures typically in the range 4–600 K to be covered.

2.3.5.3.3. Uniaxial stress

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There are two major applications of in situ uniaxial loading. In the first, stress-induced phase transitions, ferroelasticity or simply mechanical response are studied throughout the whole sample as a function of applied stress. This may be undertaken on any powder diffractometer with a reasonable data-collection rate, depending on the resolution required. Parameters typically monitored are the relative phase proportions of parent and daughter structures, lattice parameters, individual peak shifts, which can yield the single-crystal elastic constants (Howard & Kisi, 1999[link]), peak widths, which can indicate the breadth of strain distributions, and preferred-orientation parameters, which can indicate the degree of ferroelasticity (Kisi et al., 1997[link]; Ma et al., 2001[link]; Forrester & Kisi, 2004[link]; Forrester et al., 2005[link]). The second application involves strain scanning using an engineering diffractometer as described in Section 2.3.4.3[link]; however, in this instance an external load is applied to the object under study. This technique can be used to validate finite element analysis simulations of complex components with or without internal residual stresses.

Devices for the in situ application of uniaxial stress include adaptations of laboratory universal testing machines such as the 100 kN hydraulic load frame shown in Fig. 2.3.27[link]. Devices such as this may be used in tension, compression, fatigue or even creep conditions depending on the sample and the problem under study.

[Figure 2.3.27]

Figure 2.3.27 | top | pdf |

Elements of a typical mechanical testing machine used for applying uniaxial stress (pressure) to samples on an engineering neutron diffractometer. This example of a 100 kN device is from the instrument ENGIN-X at the ISIS facility, UK. (Credit: STFC.)

For more specialized applications, it is sometimes possible to create a more compact device. A recent adaptation of strain scanning is to study the stress distribution within granular materials subjected to a variety of load cases as either the average stresses shown in Fig. 2.3.23[link] (Wensrich et al., 2012[link]; Kisi et al., 2014[link]), or the stress tensor in individual particles throughout a granular material bed. The latter provides insight into inhomogeneous stress distributions such as force chains (Wensrich et al., 2014[link]). The device that was used in these studies (Fig. 2.3.28[link]) is a self-loading die within which a granular material is compacted while diffraction studies are conducted.

[Figure 2.3.28]

Figure 2.3.28 | top | pdf |

(a) Cross section and (b) exterior of a self-loading die for the study of stresses in granular materials.

2.3.6. Concluding remarks

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Neutron powder diffraction is just one of many neutron-scattering techniques available; however, it is one that is very commonly used. In fact, the demand for this particular neutron technique is rivalled only by that for small-angle neutron scattering. The close analogy with X-ray powder diffraction makes the technique very familiar to many practitioners of that technique. The differences from X-rays are also critical (Sections 2.3.1[link] and 2.3.2[link]), since these are the means by which neutron diffraction can obtain information not otherwise accessible. In this chapter we have included descriptions of the various types of neutron source, the neutron powder diffractometers installed at these sources, and a selection of routine and more specialized applications. Demand for the technique is expected to continue, buoyed by further developments in instrumentation and the exploration of new applications.

Acknowledgements

The authors thank Judith Stalick (NIST), Masatoshi Arai (ESS), Peter Galsworthy (ISIS), Philip King (STFC), Bertrand Blau (PSI), Oliver Kirstein (ESS), Vladimir Pomjakushin (PSI) and Ron Smith (ISIS) for their assistance in organizing Figs. 2.3.6, 2.3.8, 2.3.9, 2.3.10, 2.3.11, 2.3.12, 2.3.15 and 2.3.18, respectively. Ron Smith also provided the data for Fig. 2.3.19. The authors are also grateful for selected proof reading and specialized advice from Greg Storr (ANSTO) as well as Bertrand Blau, Oliver Kirstein and Ron Smith. The authors are particularly appreciative of the efforts of Mark Senn (Oxford) in reading the entire chapter and offering useful constructive comment.

Supporting information


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