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

International Tables for Crystallography (2018). Vol. H, ch. 3.4, pp. 277-278

Section 3.4.4.3.2. GAIN: indexing via a genetic-algorithm search method

A. Altomare,a* C. Cuocci,a A. Moliternia and R. Rizzia

aInstitute of Crystallography – CNR, Via Amendola 122/o, Bari, I-70126, Italy
Correspondence e-mail:  angela.altomare@ic.cnr.it

3.4.4.3.2. GAIN: indexing via a genetic-algorithm search method

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The use of genetic algorithms (GAs) for indexing powder diffraction data by exploiting the diffraction geometry (as in the traditional indexing methods) was firstly proposed by Tam & Compton (1995[link]) and Paszkowicz (1996[link]). Subsequently, Kariuki et al. (1999[link]) applied GA techniques by using whole profile fitting with the aim of exploring the parameter space {a, b, c, α, β, γ} and finding the global minimum of the R-factor {a, b, c, α, β, γ} hypersurface, yielding the parameter set able to generate the best agreement between the observed and calculated powder diffraction patterns.

This new strategy has been implemented in the program GAIN (Harris et al., 2000[link]), whose main features are:

  • (1) Starting from a population of Np sets of lattice parameters and using the evolutionary operations of mating, mutation and natural selection, the population is allowed to evolve through several generations, with the aim of generating sets of possible trial cell parameters.

  • (2) The search procedure, using a GA, is performed in restricted, sensible cell-volume ranges consistent with the knowledge of the system under study.

  • (3) For each set of trial parameters a calculated powder diffraction pattern is constructed. The peak positions and parameters describing the shape and width of each peak are used in the Le Bail profile-fitting procedure (Chapter 3.5[link] ).

  • (4) The pattern is split into different regions (defined by the user), and the weighted profile R factor is calculated for each region; all the values are summed to obtain the overall [R_{\rm wp}^{\prime}]:[R_{\rm wp}^{\prime} = \sum\limits_{\rm regions} \left[{\textstyle\sum_i w_i(y_i - y_{ci})^2 \over \textstyle\sum_i w_iy_i^2} \right] ^{1 / 2}, ]where the summation is over the regions, i runs over the experimental points belonging to each region and [{y_i}] and [{y_{ci}}] are the observed and calculated profile at the ith experimental step, respectively. Via the [R_{\rm wp}^{\prime}] formula the residual for each region is scaled according to the total intensity in the region, so a region with only low-intensity peaks can make an important contribution to [R_{\rm wp}^{\prime}].

This approach is robust at handling the problems that may affect the experimental powder pattern: peak overlap, (hkl)-dependent effects and zero-point errors. It is time consuming (particularly in the case of low symmetry) but not very sensitive to the presence of minority impurity phases.

References

Harris, K. D. M., Johnston, R. L., Chao, M. H., Kariuki, B. M., Tedesco, E. & Turner, G. W. (2000). Genetic algorithm for indexing powder diffraction data. University of Birmingham, UK.Google Scholar
Kariuki, B. M., Belmonte, S. A., McMahon, M. I., Johnston, R. L., Harris, K. D. M. & Nelmes, R. J. (1999). A new approach for indexing powder diffraction data based on whole-profile fitting and global optimization using a genetic algorithm. J. Synchrotron Rad. 6, 87–92.Google Scholar
Paszkowicz, W. (1996). Application of the smooth genetic algorithm for indexing powder patterns – tests for the orthorhombic system. Mater. Sci. Forum, 228–231, 19–24.Google Scholar
Tam, K. Y. & Compton, R. G. (1995). GAMATCH – a genetic algorithm-based program for indexing crystal faces. J. Appl. Cryst. 28, 640–645.Google Scholar








































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