Debye equation versus Whole Powder Pattern Modelling: Real versus reciprocal space modelling of nanomaterials

Abstract

Real space methods like the Debye equation are increasingly being employed as an alternative to traditional Line Profile Analysis (LPA) techniques for the study of size and strain effects in nanomaterials. Until recently, the use of this technique in modelling was hindered by the time necessary to complete a calculation. This limitation encouraged development of the alternative, reciprocal-space Whole Powder Pattern Modelling, which on the other hand lacks physical validation when applied to the study of very small atomic clusters ( V) of 2.7 nm and 4.0 nm, respectively; while the standard deviations of the natural logarithm (σ) used were 0.15 and 0.40, resulting in a narrow and wide distribution for each set. (See ref. 6, and references therein, for a precise definition of V and σ.) In order to perform the Debye function simulation [3,7], a series of increasingly large spherical Au clusters with fcc structure were created. The diameter that related to less than .1 percent of the assumed volume weighted size distribution defined the maximum sphere size used in the simulations. Since in these constructions the diameter was not a continuous variable, two rules to govern the step size have been assumed: i) that the radius difference Δr between consecutive clusters is constant, ii) that the volume increment between consecutive clusters is an integer multiple of the Wigner-Seitz unit cell volume. Using these two assumptions it is simple to see that Δr = a(3/2π)/2=0.39a. This discrete spacing is arbitrary but also sensible, as confirmed by our results. Clusters have been processed in order to obtain sets of interatomic pair distances and multiplicities. If we imagine that each unique distance is a Dirac delta weighted by its multiplicity, we can see that the pair distance density distribution is a Dirac comb. Immediately, though, the generated set of unique distances is fed into the sampling routine, which convolutes the distance Dirac comb with a suitable Gaussian profile [7], sampling the resulting continuous distance density. As the sampling step is inversely related to the maximum diffraction vector length Qmax, the sampling is performed simultaneously with different steps ranging from 0.03 to 0.96 A, covering quite all possible experimental conditions. The sampled pseudodistance sets were then used to calculate diffraction patterns via a suitably adapted Debye formula, which is amenable to a fast transZ. Kristallogr. Suppl. 30 (2009) 87 form. More details on the fast Debye simulation algorithms can be found elsewhere [3,7]. Three levels of Poisson noise were then added to the simulated intensity to obtain patterns with signal-to-noise ratios (SNR=√Imax) of 316.2, 100, and 31.6 (max noise added). Then simulated patterns were modelled with the WPPM approach [4,5,8], refining the parameters of the fcc lattice, size distribution, small angle scattering contribution and a Chebyshev polynomial background. A range of trial size distribution forms was assumed including: continuous and discrete lognormal distributions, and a continuous gamma distribution. The results of these analyses for each distribution are given in the following sections. Continuous lognormal distribution The patterns calculated by the Debye approach were first modelled assuming a continuous lognormal distribution of spheres in the WPPM framework. The obtained size distributions matched exactly the expected Debye distribution for all studied patterns. Even at the small particle size range of 1-10 nm, the WPPM method was able to accurately distinguish the different lognormal parameters of two size distributions with the same Scherrer size (integral breadth). The exact match between the discrete distribution used in the Debye simulations and the continuous curve employed in WPPM was beyond expectations (cf. figure 2). The lower residual and weighted sum of squares (wss ≡∑[(IDebye-IWPPM)/IDebye]) for the distributions with a larger Scherrer size of 4.0nm was an expected result (cf. figure 1). As the size increases the differences between the discrete Debye crystal, and the spatially averaged reciprocal space method, become less influential. Furthermore, at larger sizes the shape of the particle created in the Debye approach is increasingly well represented by a sphere. 20 40 60 80 100 120 140 0.0 5.0x10 1.0x10 20 40 60 80 100 120 140 0.0 5.0x10 1.0x10 20 40 60 80 100 120 140 0.0 5.0x10 1.0x10 -1500