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Metrics, measures, and parametrizations for grain boundaries: a dialog

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Abstract

Analysis of experimental data on grain boundaries (GBs) can involve putting data into angle “bins.” An example is Brandon’s classification: if rotation angle and axis are each within \(15^\circ/\sqrt{\Sigma}\) of a perfect coincidence site lattice (CSL) with density 1/Σ, the GBs can also be considered to be in that CSL relationship, and not if otherwise. Other examples of binning are studies of GB distributions in the full 5D angle space. To determine the size of a bin (necessary for densities, gradients, etc.) one must find a useful way, respecting symmetry, of determining metrics and measures on the full 5-dimensional space of both misorientation and interface normal. For a pair of low-angle GBs, the issue of metric is complicated by the fact that both their rotation axes and their GB normals can stay far apart as their rotation angles approach zero. We address all these issues as a dialog, and provide a framework for choosing metrics.

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Notes

  1. Write e = (e 0, v) and f = (f 0, w); then ef = (e 0 f 0v\cdot w, e 0 wf 0 vv\times w).

  2. If A and B are the matrices of the rotations, this is also \(\sqrt{3 - \hbox{trace}(A^{-1}B)}= \sqrt{{\frac{1}{2}}{\hbox{trace}}(A-B)(A-B)^T}.\)

References

  1. Brandon DG (1966) Acta Metall 14:1479

    Article  CAS  Google Scholar 

  2. Maksimova EL, Shvindlerman LS, Straumal BB (1989) Acta Metall 37:2855

    Article  CAS  Google Scholar 

  3. Adams BL, Zhao J, Grimmer H (1990) Acta Cryst A 46:620

    Article  Google Scholar 

  4. Warrington DH, Boon M (1975) Acta Metall 23:599

    Article  Google Scholar 

  5. Morawiec A (2000) Acta Mater 48:3525

    Article  CAS  Google Scholar 

  6. Saylor DM, Morawiec A, Rohrer GS (2003) Acta Mater 51:3663

    Article  CAS  Google Scholar 

  7. Saylor DM, El-Dasher BS, Adams BL, Rohrer GS (2004) Metall Mater Trans A, Metall Mater Sci 35:1981

    Article  Google Scholar 

  8. Saylor DM, EL-Dasher BS, Rollett AD, Rohrer GS (2004) Acta Mater 52:3649

    Article  CAS  Google Scholar 

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Correspondence to Jean E. Taylor.

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Cahn, J.W., Taylor, J.E. Metrics, measures, and parametrizations for grain boundaries: a dialog. J Mater Sci 41, 7669–7674 (2006). https://doi.org/10.1007/s10853-006-0592-8

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  • DOI: https://doi.org/10.1007/s10853-006-0592-8

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