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The Debye equation: Powder diffraction patterns directly from atom clusters. What we can really do and when it is convenient.

Antonio Cervellino 1Antonella Guagliardi 2Cinzia Giannini 2

1. Swiss Light Source, Paul Scherrer Institute, Villigen PSI 5232, Switzerland
2. CNR-Istituto di Cristallografia (IC), via Amendola 122/O, Bari 70126, Italy


The classical way of computing powder diffraction patterns is the Bragg approach, where the atomic structure of the material is intrinsically supposed to be periodic and extended to the whole space. These conditions (that can be relaxed somehow) allow to work directly in reciprocal space. The alternative is to use the Debye equation, that makes no hypothesis on the atomic structure periodicity and on its spatial extension. One sees clearly that the latter is particularly suited for nanomaterials, where one (and often both) conditions fail. The computation of a powder diffraction pattern via Debye equation involves all interatomic vector lengths; these are of order N2, where N is the number of atoms. This is clearly impossible except in few cases. However, a careful analysis shows that it is possible to proceed in two stages - first encoding the  interatomic distance vectors in a sampled linear density (PDF) and then evaluating the diffraction pattern from the latter. The second stage is especially fast (order N1/3 + special FFT[1]) allowing to evaluate a full pattern in μs times. The first stage i) can be easily prototyped and it is therefore seldom required (once-over for a structure type), ii) it can be carefully tailored on the specific regularities of the system, and doing so we can lower its complexity to the order of N to N4/3. Using special methods we can also encode continuous (strain fields, dislocations) and discontinuous (stacking faults, twins) disorder types, described it by suitable statistical parameters. Such disorder forms can be treated in a variational fashion, adding little to the computational load. Therefore the Debye approach becomes suitable to describe real-world nanomaterials without restriction to crystallinity and, exploiting the optimized two-stage method, it becomes computationally effective up to the 100 nm - 108 atom limit. The computing time (first stage) extends to at most hours for the extreme cases, the second stage remains as fast. This covers completely the gap to the region of full validity of the Bragg method. Examples, simulations and applications to real systems [2-6] are shown.

  1. J. Comput. Chem. 27 (2006) 995-1008
  2. Phys. Rev. B 72 (2005) 035412
  3. J. Appl. Crystallogr. 36 (2003) 1148-1158
  4. Eur. Phys. J. B 41 (2004) 485-493
  5. Nano Letters 6 (2006) 1966-1972
  6. Phys. Rev. Lett. 100 (2008) 045502


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Related papers

Presentation: Oral at 11th European Powder Diffraction Conference, Microsymposium 13, by Antonio Cervellino
See On-line Journal of 11th European Powder Diffraction Conference

Submitted: 2008-04-30 11:16
Revised:   2009-06-07 00:48