Tables for
Volume H
Powder diffraction
Edited by C. J. Gilmore, J. A. Kaduk and H. Schenk

International Tables for Crystallography (2018). Vol. H, ch. 3.9, p. 357

Section Generation of calibrated PONKCS models

I. C. Madsen,a* N. V. Y. Scarlett,a R. Kleebergb and K. Knorrc

aCSIRO Mineral Resources, Private Bag 10, Clayton South 3169, Victoria, Australia,bTU Bergakademie Freiberg, Institut für Mineralogie, Brennhausgasse 14, Freiberg, D-09596, Germany, and cBruker AXS GmbH, Oestliche Rheinbrückenstr. 49, 76187 Karlsruhe, Germany
Correspondence e-mail: Generation of calibrated PONKCS models

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The generation of a suitable PONKCS model requires that:

  • (1) The unknown phase is available as either a pure specimen or as a component of a mixture where its abundance is known (in some instances, this may be achieved by other means, such as the measurement of bulk and/or microchemical composition.)

  • (2) The unknown phase does not vary considerably from the material used to derive the relative intensities of the model. Preferred orientation and other sample-related effects may be compensated for based upon an indexed diffraction pattern.

The initial step in the generation of a PONKCS model is to describe the contribution to the diffraction pattern of the phase with a series of peaks. If the phase of interest has been indexed, the Le Bail or Pawley methods (see Chapter 3.5[link] ) can be used to constrain peak positions to the space group and unit-cell parameters while the individual reflection intensities are allowed to vary to best match the observed peaks (i.e. an hkl_phase). If the phase has not been indexed, a series of unrelated peaks can be refined using a standard material and scaled as a group during analysis (i.e. a peaks_phase). While this approach is effective in most cases, it restricts the refinable parameters that may be used in the treatment of systematic errors such as preferred orientation.

The next step is to calibrate the hkl_phase or peaks_phase and derive a `phase constant' that is equivalent to the ZMV value in crystal-structure-based quantification. This is achieved by the preparation of a mixture in which there are known amounts Wα and Ws of the unknown and standard, respectively. Recalling equation (3.9.25)[link], the ratio of the weight fractions is then given by[{{{W_\alpha }} \over {{W_s}}} = {{{S_\alpha }{{(ZMV)}_\alpha }} \over {{S_s}{{(ZMV)}_s}}} ,\eqno(3.9.42)]where Sα and Ss are the refined scale factors for the unknown and standard, respectively.

Rearrangement of equation (3.9.42)[link] then provides the means for determining an empirical value of (ZMV)α, which is required for the calibration of a peaks_phase:[{(ZMV)}_{\alpha }={{{W}_{\alpha }}\over{{W}_{s}}}{{{S}_{s}}\over{{S}_{\alpha }}} {(ZMV)}_{s}. \eqno(3.9.43)]For an hkl_phase the value of V can be determined from the refined unit-cell parameters and hence can be removed from the phase constant resulting in[{(ZM)}_{\alpha }={{{W}_{\alpha }}\over{{W}_{s}}}{{{S}_{s}}\over{{S}_{\alpha }}} {{{(ZMV)}_{s}}\over{{V}_{\alpha }}}. \eqno(3.9.44)]Unlike the ZMV value derived from the unit-cell contents of a crystal structure, the phase constants derived using equations (3.9.43)[link] and (3.9.44)[link] have no physical meaning, since they have been derived by empirical measurement. For an hkl_phase, a more physically meaningful value of ZM can be obtained by deriving the true unit-cell mass from the measured phase density according to[{(ZM)_{\alpha ({\rm true})}} = {{{\rho _\alpha }{V_\alpha }} \over {1.6604}}. \eqno(3.9.45)]The empirical `structure factor' values in the hkl_phase could then be scaled according to the relation ZMα(true)/ZMα, making them approximate `real' structure factors for the material. Note that this final step is not necessary for quantification, but may make the method more generally applicable.

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