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

International Tables for Crystallography (2006). Vol. C, ch. 5.2, pp. 497-498

Section 5.2.8. Camera methods

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

5.2.8. Camera methods

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The types of powder camera frequently used in the determination of lattice parameters are described in Section 2.3.4[link] . The main geometrical aberrations affecting measurements made with them are summarized in Table 5.2.8.1[link]. At high angles, most of them vary approximately as (π − 2θ)2, and one would thus expect to obtain an approximately straight-line extrapolation if the apparent values of the lattice parameter were plotted against a function something like (π − 2θ)2. A function that has been found very satisfactory in practice was suggested by Nelson & Riley (1945a[link], b[link]) [see also Taylor & Sinclair (1945a[link], b[link])]: [\cos^2\theta({\rm cosec}\,\theta+\theta^{-1})/2. \eqno (5.2.8.1)]This function gives linear plots down to quite small values of θ.

Table 5.2.8.1| top | pdf |
Some geometrical aberrations in the Debye–Scherrer method [increase in θ = +, decrease = −]

Source of aberrationEffect on θAngle variation of ΔdRemarks
Specimen displacement      
 towards exit [\cos^2\theta] Minimized by accurate construction and centring
 towards entrance + [\cos^2\theta] Extrapolates to zero
 sideways [\sim0] [\sim\theta]  
Beam divergence      
 perpendicular to axis + [\cos \theta\cot\theta] or [\cos^2\theta/2\theta] Minimized by reducing collimator dimensions
 parallel to axis + or − Complex See Langford, Pike & Beu (1964[link])
Film shrinkage + [(\pi-2\theta)\cot\theta] Affects only van Arkel arrangement
Knife-edge calibration + or − [\theta \cot \theta] Affects only Bradley–Jay arrangement. Partly eliminated by usual extrapolation
Specimen absorption + [\cos \theta\cot\theta] or [\cos^2\theta/2\theta] Minimized by reducing specimen diameter or dilution. Extrapolates to zero
For van Arkel and Bradley–Jay arrangements. For Straumanis–Ievins', + or − and [(\pi -2\theta)\cot\theta], respectively.

References

Langford, J. I., Pike, E. R. & Beu, K. E. (1964). Precise and accurate lattice parameters by film powder methods. IV. Theoretical calculation of axial (vertical) divergence profiles, centroid shifts, and variances for cylindrical powder diffraction cameras. Acta Cryst. 17, 645–651.
Nelson, J. B. & Riley, D. P. (1945a). An experimental investigation of extrapolation methods in the derivation of accurate unit-cell dimensions of crystals. Proc. Phys. Soc. London, 57, 160–177.
Nelson, J. B. & Riley, D. P. (1945b). An experimental investigation of extrapolation methods in the derivation of accurate unit-cell dimensions of crystals. Proc. Phys. Soc. London, 57, 477–495.
Taylor, A. & Sinclair, H. B. (1945a). The influence of absorption on the shapes and positions of lines in Debye–Scherrer powder photographs. Proc. Phys. Soc. London, 57, 108–125.
Taylor, A. & Sinclair, H. B. (1945b). On the determination of lattice parameters by the Debye–Scherrer method. Proc. Phys. Soc. London, 57, 126–135.








































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