Chemical composition dispersion in bi-metallic nanoparticles: semi-automated analysis using HAADF-STEM

The FePd-L1₀ structure is a tetragonal distortion of the fcc phase, with lattice parameter a = b varying from 0.385 to 0.388 nm in the [0.37, 0.58] x-range, and c almost constant equal to 0.3718 nm (Shima et al. 2004). In terms of cell volume and atomic densities, we can thus treat this tetragonal structure as a cubic structure with a equivalent cubic lattice parameter given by:In this general description, the atomic density variation can easily be taken into account. We thus consider a cubic structure with a lattice parameter a, filled with a constant number of atoms N over the whole homogeneity range of the alloy A _x B _1−x . The cell parameter is then given by a Vegard law as written in relation (9):The unit cell contains N _A(x) atoms of A and N _B(x) atoms of B, which are the product of their atomic densities ρ_A(x) and ρ_B(x) (expressed in number of atoms per unit volume) by the unit cell volume a(x)³. Thus, relation (1) becomes:Since the atomic densities vary linearly as a function of x, elementary algebraic manipulations lead to:Introducing the intensities corresponding to the monoatomic phases I _0A = I ₀(x = 1) and I _0B = I ₀(x = 0) finally gives for any composition:It can be seen that this expression differs from the simple relation given in (8), which becomes valid only if the relative parameter variation \(\Updelta/\hbox{a}\) can be neglected. A close inspection of these relations shows that the maximum of error while using relation (8) instead of (13) will be reached for the stoichiometric composition A_0.5B_0.5. Describing the FePd-L1₀ tetragonal phase as a cubic structure with a equivalent cubic lattice parameter given by relation (9), we can then calculate that the maximum relative error in the HAADF intensity remains lower than 0.64 % if relation (8) is used instead of (13). This very negligible value shows that the influence of the variation of atomic densities on the chemical dependence of the HAADF intensity can safely be ignored in practice for a narrow compositional range, such as in the case of the FePd system.

The error in the HAADF intensity I ₁₀ comes from different sources: the uncertainty in the value of D _proj and of the thickness t and detection errors. The first error source has been evaluated to 3 % assuming an error of two pixels on the measurement of the radius of projected particles in images such as shown in Fig. 2. In principle, one should also add the effect of the probe size [full width at half-maximum (FWHM) ϕ] which leads to an apparent particle diameter given by (Treacy et al. 1989): D _measured = (D _particle ²  + ϕ²)^1/2. In practice, this convolution effect is rapidly negligible: with a conventional FEG microscope, for example, a typical probe size of 0.5 nm leads to a broadening less than 0.1 nm for particles >1.5 nm (about 5 % error). In the present case of particles with a mean size of 10 nm, the error is safely negligible (0.5 % even with a probe FWHM of 1 nm). The second error source is related to the quality of the calibration of the flattening effect. According to the holographic measurements (Sato et al. 2005), the relative error in the thickness determination is about 7 % for the mean projected size, that is 10 nm. The other detection errors, related to beam intensity variations during the analysis, or to diffraction effects discussed in section IV are estimated to be around 5 % at maximum. Finally, the total error in intensity is assumed to be smaller than 15 %.