Abstract
One hundred years after the original formulation by Petrus J.W. Debije (aka Peter Debye), the Debye Scattering Equation (DSE) is still the most accurate expression to model the diffraction pattern from nanoparticle systems. A major limitation in the original form of the DSE is that it refers to a static domain, so that including thermal disorder usually requires rescaling the equation by a Debye-Waller thermal factor. The last is taken from the traditional diffraction theory developed in Reciprocal Space (RS), which is opposed to the atomistic paradigm of the DSE, usually referred to as Direct Space (DS) approach. Besides being a hybrid of DS and RS expressions, rescaling the DSE by the Debye-Waller factor is an approximation which completely misses the contribution of Temperature Diffuse Scattering (TDS). The present work proposes a solution to include thermal effects coherently with the atomistic approach of the DSE. A deeper insight into the vibrational dynamics of nanostructured materials can be obtained with few changes with respect to the standard formulation of the DSE, providing information on the correlated displacement of vibrating atoms.
Highlights
Based on the orientational average of the intensity distribution, in 1915 Debye derived his equation for scattering[1], I (Q)
As the summation runs over all N atoms, equation 1 requires calculation of N2 terms, which is computationally demanding if particles exceed a few tens of nanometers; the sample size and shape distributions must be properly represented, implying a further increment in the number of terms to be computed
As assumed in the following for simplicity and coherently with the experimental case study, the DW factor is written as exp(−2M) = exp(−2B sin[2] Θ/λ2) = exp(−Q2 uQ2 ), (MSD) projected along Q, whereas the B-factor is defined where as B =
Summary
Scattering Equation received: 15 November 2015 accepted: 09 February 2016 Published: 26 February 2016. A major limitation in the original form of the DSE is that it refers to a static domain, so that including thermal disorder usually requires rescaling the equation by a Debye-Waller thermal factor. The effect of atomic vibrations has usually been included in a simplified way, multiplying equation 1 by the Debye-Waller (DW) factor[8]. The instantaneous scattering amplitude from a small crystal is a sum of phase terms, weighted on the atomic scattering factors[8], N. where the instantaneous atomic position, ri + δi(t), includes a static component, ri, referred to an average position, plus a dynamic term, the time-dependent (thermal) displacement δi(t).
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