Tables for
Volume C
Mathematical, physical and chemical tables
Edited by E. Prince

International Tables for Crystallography (2006). Vol. C, ch. 5.2, pp. 496-497

Section 5.2.7. Energy-dispersive techniques

W. Parrish,a A. J. C. Wilsonb and J. I. Langfordc

aIBM Almaden Research Center, San Jose, CA, USA,bSt John's College, Cambridge CB2 1TP, England, and cSchool of Physics & Astronomy, University of Birmingham, Birmingham B15 2TT, England

5.2.7. Energy-dispersive techniques

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There are now two basic energy-dispersive techniques available. In both, the specimen and detector are fixed in a selectable θ–2θ setting. The method (Giessen & Gordon, 1968[link]) first described and most widely used requires a solid-state detector and a multichannel pulse-height analyser (Section 2.3.2[link] and Chapter 2.5[link] ). The resolution of the pattern is determined by the energy resolution of the detector and is considerably poorer than that of conventional angle-dispersive techniques, thereby greatly limiting its applications. The second method uses an incident-beam monochromator, a conventional scintillation counter, and a single-channel pulse-height analyser. The monochromator is step-scanned to select a gradually increasing (or decreasing) single wavelength (Parrish & Hart, 1987[link]). This method permits much higher count rates, thereby reducing the time required for the experiment. Since the resolution is determined by the X-ray optics, the resolution is the same as in angle-dispersive diffractometry (Subsection[link] ). The method has, however, the disadvantage that the widths of the profiles vary with energy, and unless care is taken with the step size there may be too few points per reflection to define the profile adequately. The method is particularly applicable to synchrotron radiation, but there have been no publications to date on its use for lattice-parameter determination.

Energy-dispersive techniques (Section 2.2.3[link] and Chapter 2.5[link] ) are not ordinarily the method of choice for lattice-parameter determination. Relative to angle-dispersive techniques, they suffer from the following disadvantages:

  • (1) lower resolution;

  • (2) need for absolute energy calibration of the multichannel pulse-height analyser;

  • (3) need to know the energy distribution in the incident beam;

  • (4) specimen transparency varies with energy; even tungsten becomes transparent for 35 keV radiation.

Nevertheless, the advantage of stationary specimen and detector may outweigh these disadvantages for special applications.

A diffractometer can be converted from angle-dispersive to energy-dispersive by (i) replacing the usual counter by a solid-state detector, (ii) replacing the usual electronic circuits by a multichannel pulse-height analyser, and (iii) keeping the specimen and detector stationary while the counts are accumulated. When so used, the geometrical aberrations are essentially the same as those of an angle-dispersive diffractometer, though the greater penetrating power of the higher-energy X-rays means that greater attention must be paid to the irradiated volume and the specimen transparency (Langford & Wilson, 1962[link]; Mantler & Parrish, 1977[link]). As Sparks & Gedcke (1972[link])1 emphasize, spacing measurements made with such an arrangement are subject to large specimen-surface displacement and transparency aberrations, and the corrections required to allow for them are difficult to make. Fukamachi, Hosoya & Terasaki (1973[link]) and Nakajima, Fukamachi, Terasaki & Hosoya (1976[link]) showed that this difficulty can be avoided if the Soller slits are rotated about the beam directions by 90°, so that they limit the equatorial divergence instead of the axial; this was, of course, the orientation used by Soller (1924[link]) himself. Any effect of specimen-surface displacement and transparency is then negligible if ordinary care in adjustment is used, and the specimen may be placed in the reflection, or the symmetrical transmission, or the unsymmetrical transmission position (Wilson, 1973[link]). The geometrical aberrations are collected in Table[link], and apply to the original orientation of the Soller slits; in the Sparks & Gedcke (1972[link]) orientation, the usual ones apply. In general, the physical aberrations are the same for both orientations. The most difficult correction is that for the energy distribution in the incident X-ray beam; aspects of this have been discussed by Bourdillon, Glazer, Hidaka & Bordas (1978[link]), Glazer, Hidaka & Bordas (1978[link]), Buras, Olsen, Gerward, Will & Hinze (1977[link]), Fukamachi, Hosoya & Terasaki (1973[link]), Laguitton & Parrish (1977[link]) and Wilson (1973[link]). Only the last of these is directly relevant to the lattice-spacing problem. The best results reported so far seem to be those of Fukamachi, Hosoya & Terasaki (1973[link]) (0.01% in the lattice parameter).

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Centroid displacement [\langle \Delta E/E\rangle] and variance W of certain aberrations of an energy-dispersive diffractometer [mainly from Wilson (1973[link]), where more detailed results are given for the aberrations marked with an asterisk]

The Soller slits are taken to be in the original orientation (Soller, 1924[link]). For the notation, see the footnote.

Aberration [\langle \Delta E/E\rangle] W
Specimen displacement [\sim0] Included in equatorial divergence
Specimen transparency* [\sim0] ?
Equatorial divergence* [\sim 0] [\cot^2\theta(A^2+B^2)/24] for narrow Soller slits
Axial divergence [-R^{-2}\,{\rm cosec}^2\,\theta[X^2\cos2\theta+4Y^2\cos^2\theta+Z^2\cos2\theta]/24] [\eqalign{R^{-4}\,&{\rm cosec}^4\,\theta[X^4\cos^2 2\theta + 4Y^4(1+\cos2\theta)^2 \cr & +Z^4\cos^22\theta+5X^2Z^2+5Y^2(X^2+Z^2) \cr &\times(1+\cos2\theta)^2]/720}]
Refraction* Probably negligible at the present stage of technique
Response variations
 Centroid [[Vf'+f''(\mu_3/2-V^2f'/f)]/Ef] ?
 Peak [-f'I/E\,f I''] ?
Interaction of Lorentz etc. factors and geometrical aberrations [\eqalign{\langle(\Delta&\theta)^2\rangle/2-\cot\theta[\langle\Delta\theta\rangle+(g'/g)\langle(\Delta\theta)^2\rangle] \cr &+\cot^2\theta(EI'/I)\langle(\Delta\theta)^2\rangle}] [\eqalign{-&\cot\theta[\langle(\Delta\theta)^3\rangle-\langle\Delta\theta\rangle\langle(\Delta\theta)^2\rangle] \cr& +\cot^2\theta\{\langle(\Delta\theta)^2\rangle-\langle\Delta \theta\rangle^2 \cr &+(2g'/g)[\langle(\Delta\theta)^3\rangle-\langle\Delta\theta\rangle\langle(\Delta\theta)^2\rangle]\}}]

Notation: A and B are the angular apertures (possibly equal) of the two sets of Soller slits; E is the energy of the detected photon; f (E) is the variation of a response (energy of the continuous radiation, absorption in the specimen etc.) with E; g(θ) is an angle-dependent response (Lorentz factor etc.); I(EE1) dE is the counting rate recorded at E when the energy of the incident photons is actually E1; R is the diffractometer radius; V is the variance and μ3 is the third central moment of the energy-resolution function I; 2X, 2Y, 2Z are the effective dimensions (possibly equal) of the source, specimen, and detector; the primes indicate differentiation; the averages <(Δθ)2> etc. are over the range of Bragg angles permitted by the slits etc.

Okazaki & Kawaminami (1973[link]) have suggested the use of a stationary specimen followed by analysis of the diffracted X-rays with a single-crystal spectrometer. This would give some of the advantages of energy-dispersive diffractometry (easy control of temperature etc., because only small windows would be needed), but there would be no reduction in the time required for recording a pattern.


Bourdillon, A. J., Glazer, A. M., Hidaka, M. & Bordas, J. (1978). High-resolution energy-dispersive diffraction using synchrotron radiation. J. Appl. Cryst. 11, 684–687.
Buras, B., Olsen, J. S., Gerward, L., Will, G. & Hinze, E. (1977). X-ray energy-dispersive diffractometry using synchrotron radiation. J. Appl. Cryst. 10, 431–438.
Fukamachi, T., Hosoya, S. & Terasaki, D. (1973). The precision of interplanar distances measured by an energy-dispersive X-ray diffractometer. J. Appl. Cryst. 6, 117–122.
Giessen, B. C. & Gordon, G. E. (1968). X-ray diffraction: new high-speed technique based on X-ray spectroscopy. Science, 159, 973–975.
Glazer, A. M., Hidaka, M. & Bordas, J. (1978). Energy-dispersive powder profile refinement using synchrotron radiation. J. Appl. Cryst. 11, 165–172.
Laguitton, D. & Parrish, W. (1977). Experimental spectral distribution versus Kramers' law for quantitative X-ray fluorescence by the fundamental parameters method. X-ray Spectrom. 6, 201–203.
Langford, J. I. & Wilson, A. J. C. (1962). Counter diffractometer: the effect of specimen transparency on the intensity, position and breadth of X-ray powder diffraction lines. J. Sci. Instrum. 39, 581–585.
Mantler, M. & Parrish, W. (1977). Energy dispersive X-ray diffractometry. Adv. X-ray Anal. 20, 171–186.
Nakajima, T., Fukamachi, T., Terasaki, O. & Hosoya, S. (1976). The detection of small differences in lattice constant at low temperature by an energy-dispersive X-ray diffractometer. J. Appl. Cryst. 9, 286–290.
Okazaki, A. & Kawaminami, M. (1973). Accurate measurement of lattice constant in a wide range of temperature: use of white X-rays and double-crystal diffractometry. Jpn. J. Appl. Phys. 12, 783–789.
Parrish, W. & Hart, M. (1987). Advantages of synchrotron radiation for polycrystalline diffractometry. Z. Kristallogr. 179, 161–173.
Soller, W. (1924). A new precision X-ray spectrometer. Phys. Rev. 24, 158–167.
Sparks, C. J. & Gedcke, D. A. (1972). Rapid recording of powder diffraction patterns with Si(Li) X-ray energy analysis system: W and Cu targets and error analysis. Adv. X-ray Anal. 15, 240–253.
Wilson, A. J. C. (1973). Note on the aberrations of a fixed angle energy-dispersive diffractometer. J. Appl. Cryst. 6, 230–237.

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