International
Tables for Crystallography Volume C Mathematical, physical and chemical tables Edited by E. Prince © International Union of Crystallography 2006 
International Tables for Crystallography (2006). Vol. C, ch. 2.5, pp. 8488
doi: 10.1107/97809553602060000580 Chapter 2.5. Energydispersive techniques^{a}ISIS Science Division, Rutherford Appleton Laboratory, Chilton, Didcot, Oxfordshire OX11 0QX, England,^{b}Physics Department, Technical University of Denmark, DK2800 Lyngby, Denmark,^{c}Materials Science Division, Building 223, Argonne National Laboratory, Argonne, IL 60439, USA,^{d}Chemical Crystallography Laboratory, University of Oxford, 9 Parks Road, Oxford OX1 3PD, England, and ^{e}Physics Department, Risø National Laboratory, DK4000 Roskilde, Denmark In the first part of this chapter the principles of the whitebeam, energydispersive method for Xray diffraction are described with special emphasis on the use of synchrotron radiation. Expressions are given for the integrated intensity, the polarization factor and other correction factors. Applications in highpressure Xray diffraction are mentioned. In the second part of the chapter, neutron singlecrystal Laue diffraction and neutron timeofflight powder diffraction are discussed. 
Xray energydispersive diffraction, XED, invented in the late sixties (Giessen & Gordon, 1968; Buras, Chwaszczewska, Szarras & Szmid 1968), utilizes a primary Xray beam of polychromatic (`white') radiation. XED is the analogue of whitebeam and timeofflight neutron diffraction (cf. Section 2.5.2). In the case of powdered crystals, the photon energy (or wavelength) spectrum of the Xrays scattered through a fixed optimized angle is measured using a semiconductor detector connected to a multichannel pulseheight analyser. Singlecrystal methods have also been developed.
In XED powder work, the incident and scatteredbeam directions are determined by slits (Fig. 2.5.1.1). A powder spectrum is shown in Fig. 2.5.1.2. The Bragg equation is where d is the latticeplane spacing, the Bragg angle, λ and E the wavelength and the photon energy, respectively, associated with the Bragg reflection, h is Planck's constant and c the velocity of light. In practical units, equation (2.5.1.1a) can be written The main features of the XED powder method where it differs from standard angledispersive methods can be summarized as follows:
The scattering angle is chosen to accommodate an appropriate number of Bragg reflections within the available photonenergy range and to avoid overlapping with fluorescence lines from the sample and, when using an Xray tube, characteristic lines from the anode. Overlap can often be avoided because a change in the scattering angle shifts the diffraction lines to new energy positions, whereas the fluorescence lines always appear at the same energies. Severe overlap problems may be encountered when the sample contains several heavy elements.
The detector aperture usually collects only a small fraction of the Debye–Scherrer cone of diffracted Xrays. The collection of an entire cone of radiation greatly increases the intensities. Also, it makes it possible to overcome crystallite statistics problems and preferred orientations in very small samples (Holzapfel & May, 1982; Häusermann, 1992).
(a) Bremsstrahlung from an Xray tube. Bremsstrahlung from an Xray diffraction tube provides a useful continuous spectrum for XED in the photonenergy range 2–60 keV. However, one has to avoid spectral regions close to the characteristic lines of the anode material. A tungsten anode is suitable because of its high output of white radiation having no characteristic lines in the 12–58 keV range.
A drawback of Bremsstrahlung is that its spectral distribution is difficult to measure or calculate with accuracy, which is necessary for a structure determination using integrated intensities [see equation (2.5.1.7)]. Bremsstrahlung is strongly polarized for photon energies near the highenergy limit, while the lowenergy region has a weak polarization. The direction of polarization is parallel to the direction of the electron beam from the filament to the anode in the Xray tube. Also, the polarization is difficult to measure or calculate.
(b) Synchrotron radiation. Synchrotron radiation emitted by electrons or positrons, when passing the bending magnets or insertion devices, such as wigglers, of a storage ring, provides an intense smooth spectrum for XED.
Both the spectral distribution and the polarization of the synchrotron radiation can be calculated from the parameters of the storage ring. Synchrotron radiation is almost fully polarized in the electron or positron orbit plane, i.e. the horizontal plane, and inherently collimated in the vertical plane. Full advantage of these features can be obtained using a vertical scattering plane. However, the mechanical construction of the diffractometer, the placing of furnaces, cryogenic equipment, etc. are easier to handle when the Xray scattering is recorded in the horizontal plane. Recent XED facilities at synchrotronradiation sources have been described by Besson & Weill (1992), Clark (1992), Häusermann (1992), Olsen (1992), and Otto (1997).
The momentum resolution in energydispersive diffraction is limited by the angular divergence of the incident and diffracted Xray beams and by the energy resolution of the detector system. The observed profile is a convolution of the profile due to the angular divergence and the profile due to the detector response. For resolution calculations, it is usually assumed that the profiles are Gaussian, although the real profiles might exhibit geometrical and physical aberrations (Subsection 2.5.1.5). The relative full width at halfmaximum (FWHM) of a diffraction peak in terms of energy is then given by where is the electronic noise contribution, F the Fano factor, the energy required for creating an electron–hole pair (cf. Subsection 7.1.5.1 ), and the overall angular divergence of the Xray beam, resulting from a convolution of the incident and the diffractedbeam profiles. For synchrotron radiation, can usually be replaced by the divergence of the diffracted beam because of the small divergence of the incident beam.
Fig. 2.5.1.3 shows as a function of Bragg angle . The curves have been calculated from equations (2.5.1.1) and (2.5.1.2) for two values of the latticeplane spacing and two values of , typical for Bremsstrahlung and synchrotron radiation, respectively. It is seen that in all cases decreases with decreasing angle (i.e. increasing energy) to a certain minimum and then increases rapidly. It is also seen that the minimum point of the curve is lower for the small d value and shifts towards smaller values for decreasing . Calculations of this kind are valuable for optimizing the Bragg angle for a given sample and other experimental conditions (cf. Fukamachi, Hosoya & Terasaki, 1973; Buras, Niimura & Olsen, 1978).

Relative resolution, , as function of Bragg angle, , for two values of the lattice plane spacing: (a) 1 Å and (b) 0.5 Å. The full curves have been calculated for , the broken curves for . 
The relative peak width at halfheight is typically less than 1% for energies above 30 keV. When the observed peaks can be fitted with Gaussian functions, one can determine the centroids of the profiles by a factor of 10–100 better than the value of equation (2.5.1.2) would indicate. Thus, it should be possible to achieve a relative resolution of about 10^{−4} for high energies. A resolution of this order is required for example in residualstress measurements.
The detector broadening can be eliminated using a technique where the diffraction data are obtained by means of a scanning crystal monochromator and an energysensitive detector (Bourdillon, Glazer, Hidaka & Bordas, 1978; Parrish & Hart, 1987). A lowresolution detector is sufficient because its function (besides recording) is just to discriminate the monochromator harmonics. The Bragg reflections are not measured simultaneously as in standard XED. The monochromatorscan method can be useful when both a fixed scattering angle (e.g. for samples in special environments) and a high resolution are required.
The kinematical theory of diffraction and a nonabsorbing crystal with a `frozen' lattice are assumed. Corrections for thermal vibrations, absorption, extinction, etc. are discussed in Subsection 2.5.1.5. The total diffracted power, , for a Bragg reflection of a powder sample can then be written (Buras & Gerward, 1975; Kalman, 1979) where h is the diffraction vector, the classical electron radius, the intensity per unit energy range of the incident beam evaluated at the energy of the diffraction peak, V the irradiated sample volume, the number of unit cells per unit volume, j the multiplicity factor, F the structure factor, and the polarization factor. The latter is given by where P(E) is the degree of polarization of the incident beam. The definition of P(E) is where and are the parallel and normal components of with respect to the plane defined by the incident and diffractedbeam directions.
Generally, has to be calculated from equations (2.5.1.4) and (2.5.1.5). However, the following special cases are sometimes of interest: Equation (2.5.1.6a) can often be used in connection with Bremsstrahlung from an Xray tube. The primary Xray beam can be treated as unpolarized for all photon energies when there is an angle of 45° between the plane defined by the primary and the diffracted beams and the plane defined by the primary beam and the electron beam of the Xray tube. In standard configurations, the corresponding angle is 0° or 90° and equation (2.5.1.6a) is generally not correct. However, for it is correct to within 2.5% for all photon energies (Olsen, Buras, Jensen, Alstrup, Gerward & Selsmark, 1978).
Equations (2.5.1.6b) and (2.5.1.6c) are generally acceptable approximations for synchrotron radiation. Equation (2.5.1.6b) is used when the scattering plane is horizontal and (2.5.1.6c) when the scattering plane is vertical.
The diffraction directions appear as generatrices of a circular cone of semiapex angle about the direction of incidence. Equation (2.5.1.3) represents the total power associated with this cone. Generally, only a small fraction of this power is recorded by the detector. Thus, the useful quantity is the power per unit length of the diffraction circle on the receiving surface, . At a distance r from the sample, the circumference of the diffraction circle is and one has (constants omitted) The peak areas in an XED powder spectrum are directly proportional to the of equation (2.5.1.7).
Quantitative structural analysis requires the knowledge of and P(E). As mentioned above, these quantities are not known with sufficient accuracy for Bremsstrahlung. For synchrotron radiation they can be calculated, but they will nevertheless contribute to the total uncertainty in the analysis. Accordingly, XED is used rather for identification of a known or assumed structure than for a full structure determination.
(a) Temperature effects. The effect of thermal vibrations on the integrated intensities is expressed by the Debye–Waller factor in the same way as for standard angledispersive methods. Notice that irrespective of the method used. The contribution of the thermal diffuse scattering to the measured integrated intensities can be calculated if the elastic constants of the sample are known (Uno & Ishigaki 1975).
(b) Absorption. The transmission factor ) for a small sample bathed in the incident beam and the factor for a large sample intercepting the entire incident beam are the same as for monochromatic methods (Table 6.3.3.1 ). However, when they are applied to energydispersive techniques, one has to note that the absorption corrections are strongly varying with energy. In the special case of a symmetrical reflection where the incident and diffracted beams each make angles with the face of a thick sample (powder or imperfect crystal), one has where μ(E) is the linear attenuation coefficient evaluated at the energy associated with the Bragg reflection.
(c) Extinction and dispersion. Extinction and dispersion corrections are applied in the same way as for angledispersive monochromatic methods. However, in XED, the energy dependence of the corrections has to be taken into account.
(d) Geometrical aberrations. These are distortions and displacements of the line profile by features of the geometry of the apparatus. Axial aberrations as well as equatorial divergence contribute to the angular range of the Bragg reflections. There is a predominance of positive contributions to , so that the diffraction maxima are slightly displaced to the lowenergy side, and show more tailing on the lowenergy side than the highenergy side (Wilson, 1973).
(e) Physical aberrations. Displacements due to the energydependent absorption and reflectivity of the sample tend to cancel each other if the incident intensity, , can be assumed to be constant within the energy range of Bragg reflection. With synchrotron radiation, varies rapidly with energy and its influence on the peak positions should be checked. Also, the detector response function will influence the line profile. Lowenergy line shapes are particularly sensitive to the deadlayer absorption, which may cause tailing on the lowenergy side of the peak. Integrated intensities, measured as peak areas in the diffraction spectrum, have to be corrected for detector efficiency and intensity losses due to escape peaks.
The Rietveld method (see Chapter 8.6 ) for refining structural variables has only recently been applied to energydispersive powder data. The ability to analyse diffraction patterns with overlapping Bragg peaks is particularly important for a lowresolution technique, such as XED (Glazer, Hidaka & Bordas, 1978; Buras, Gerward, Glazer, Hidaka & Olsen, 1979; Neuling & Holzapfel, 1992). In this section, it is assumed that the diffraction peaks are Gaussian in energy. It then follows from equation (2.5.1.7) that the measured profile of the reflection k at energy corresponding to the ith channel of the multichannel analyser can be written where is a constant, is evaluated at the energy , and is the full width (in energy) at halfmaximum of the diffraction peak. is a factor that accounts for the absorption in the sample and elsewhere in the beam path. The number of overlapping peaks can be determined on the basis of their position and halfwidth. The full width at halfmaximum can be expressed as a linear function of energy: where U and V are the halfwidth parameters.
Energydispersive diffraction is mainly used for powdered crystals. However, it can also be applied to singlecrystal diffraction.
A twocircle system for singlecrystal diffraction in a diamondanvil highpressure cell with a polychromatic, synchrotron Xray beam has been devised by Mao, Jephcoat, Hemley, Finger, Zha, Hazen & Cox (1988).
Formulae for singlecrystal integrated intensities are well known from the classical Laue method. Adaptations to energydispersive work have been made by Buras, Olsen, Gerward, Selsmark & LindegaardAndersen (1975).
The unique features of energydispersive diffraction make it a complement to rather than a substitute for monochromatic angledispersive diffraction. Both techniques yield quantitative structural information, although XED is seldom used for a full structure determination. Because of the fixed geometry, energydispersive methods are particularly suited to in situ studies of samples in special environments, e.g. at high or low temperature and/or high pressure. The study of anomalous scattering and forbidden reflections is facilitated by the possibility of shifting the diffraction peaks on the energy scale by changing the scattering angle. Other applications are studies of Debye–Waller factors, determinative mineralogy, attenuationcoefficient measurements, onstream measurements, particlesize and strain determination, and texture studies. These and other applications can be found in an annotated bibliography covering the period 1968–1978 (Laine & Lähteenmäki, 1980). The short counting time and the simultaneous recording of the diffraction spectrum permit the study of the kinetics of structural transformations in time frames of a few seconds or minutes.
Energydispersive powder diffraction has proved to be of great value for highpressure structural studies in conjunction with synchrotron radiation. The brightness of the radiation source and the efficiency of the detector system permit the recording of a diffraction spectrum with satisfactory counting statistics in a reasonable time (100–1000 s) in spite of the extremely small sample volume (10^{−3}–10^{−5} mm^{3}). Reviews have been given by Buras & Gerward (1989) and Häusermann (1992). Recently, XED experiments have been performed at pressures above 400 GPa, and pressures near 1 TPa may be attainable in the near future (Ruoff, 1992). At this point, it should be mentioned that XED methods have limited resolution and generally give unreliable peak intensities. The situation has been transformed recently by the introduction of the imageplate area detector, which allows angledispersive, monochromatic methods to be used with greatly improved resolution and powder averaging (Nelmes & McMahon, 1994, and references therein).
In traditional neutrondiffraction experiments, using a continuous source of neutrons from a nuclear reactor, a narrow wavelength band is selected from the wide spectrum of neutrons emerging from a moderator within the reactor. This monochromatization process is extremely inefficient in the utilization of the available neutron flux. If the requirement of discriminating between different orders of reflection is relaxed, then the entire white beam can be employed to contribute to the diffraction pattern and the countrate may increase by several orders of magnitude. Further, by recording the scattered neutrons on photographic film or with a positionsensitive detector, it is possible to probe simultaneously many points in reciprocal space.
If the experiment is performed using a pulsed neutron beam, the different orders of a given reflection may be separated from one another by timeofflight analysis. Consider a short polychromatic burst of neutrons produced within a moderator. The subsequent timesofflight, t, of neutrons with differing wavelengths, λ, measured over a total flight path, L, may be discriminated one from another through the de Broglie relationship: where m_{n} is the neutron mass and h is Planck's constant. Expressing t in microseconds, L in metres and λ in Å, equation (2.5.2.1) becomes Inserting Bragg's law, , for the nth order of a fundamental reflection with spacing d in Å gives Different orders may be measured simply by recording the time taken, following the release of the initial pulse from the moderator, for the neutron to travel to the sample and then to the detector.
The origins of pulsed neutron diffraction can be traced back to the work of Lowde (1956) and of Buras, Mikke, Lebech & Leciejewicz (1965). Later developments are described by Turberfield (1970) and Windsor (1981). Although a pulsed beam may be produced at a nuclear reactor using a chopper, the major developments in pulsed neutron diffraction have been associated with pulsed sources derived from particle accelerators. Spallation neutron sources, which are based on proton synchrotrons, allow optimal use of the Laue method because the pulse duration and pulse repetition rate can be matched to the experimental requirements. The neutron Laue method is particularly useful for examining crystals in special environments, where the incident and scattered radiations must penetrate heat shields or other window materials. [A good example is the study of the incommensurate structure of αuranium at low temperature (Marmeggi & Delapalme, 1980).]
A typical timeofflight singlecrystal instrument has a large area detector. For a given setting of detector and sample, a threedimensional region is viewed in reciprocal space, as shown in Fig. 2.5.2.1. Thus, many Bragg reflections can be measured at the same time. For an ideally imperfect crystal, with volume V_{s} and unitcell volume v_{c}, the number of neutrons of wavelength λ reflected at Bragg angle by the planes with structure factor F is given by where is the number of incident neutrons per unit wavelength interval. In practice, the intensity in equation (2.5.2.3) must be corrected for wavelengthdependent factors, such as detector efficiency, sample absorption and extinction, and the contribution of thermal diffuse scattering. Jauch, Schultz & Schneider (1988) have shown that accurate structural data can be obtained using the singlecrystal timeofflight method despite the complexity of these wavelengthdependent corrections.

Construction in reciprocal space to illustrate the use of multiwavelength radiation in singlecrystal diffraction. The circles with radii k_{max} = 2π/λ_{min} and k_{min} = 2π/λ_{max} are drawn through the origin. All reciprocallattice points within the shaded area may be sampled by a linear positionsensitive detector spanning the scattering angles from 2θ_{min} to 2θ_{max}. With a positionsensitive area detector, a threedimensional portion of reciprocal space may be examined (after Schultz, Srinivasan, Teller, Williams & Lukehart, 1984). 
This technique, first developed by Buras & Leciejewicz (1964), has made a unique impact in the study of powders in confined environments such as highpressure cells (Jorgensen & Worlton, 1985). As in singlecrystal Laue diffraction, the time of flight is measured as the elapsed time from the emergence of the neutron pulse at the moderator through to its scattering by the sample and to its subsequent detection. This time is given by equation (2.5.2.2). Many Bragg peaks, each separated by time of flight, can be observed at a single fixed scattering angle, since there is a wide range of wavelengths available in the incident beam.
A good approximation to the resolution function of a timeofflight powder diffractometer is given by the secondmoment relationship where , and are, respectively, the uncertainties in the d spacing, time of flight, and Bragg angle associated with a given reflection, and is the uncertainty in the total path length (Jorgensen & Rotella, 1982). Thus, the highest resolution is obtained in back scattering (large ) where cot is small. Timeofflight instruments using this concept have been described by Steichele & Arnold (1975) and by Johnson & David (1985). With pulsed neutron sources a large source aperture can be viewed, as no chopper is required of the type used on reactor sources. Hence, long flight paths can be employed and this too [see equation (2.5.2.4)] leads to high resolution. For a well designed moderator the pulse width is approximately proportional to wavelength, so that the resolution is roughly constant across the whole of the diffraction pattern. For an ideal powder sample the number of neutrons diffracted into a complete Debye–Scherrer cone is proportional to (Buras & Gerward, 1975). j is the multiplicity of the reflection and the remaining symbols in equation (2.5.2.5) are the same as those in equation (2.5.2.3).
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