Lattice parameter of small particles

Substrate effects may have an important influence on the lattice parameter of small particles. There are several reports of lattice contraction,142-144 but... 

The EXAFS results indicate that the lattice parameter of small particles is contracted as compared with that of the bulk metal (23). This fact is confirmed by radial electron distribution data obtained from X-ray experiments (257a,b) and also by electron diffraction measurements (257c). Both this contraction and the anomalous pentagonal symmetry can be eliminated in the case of Pt particles onto which hydrogen is adsorbed. These relaxation effects, which are found to be reversible, will not be... 

J. Woltersdorf, A.S. Nepijko, E. Pippel, Dependence of lattice-parameters of small particles on the size of the nuclei. Surf. Sci. 106(1-3), 64-69 (1981)... 

The stress at the surfaces of a solid body leads to a compression (or expansion) of the material. While the deformation is small it becomes measurable for small particles. The measurement of the lattice parameter of small particles may therefore be used to determine the surface stress. For simplicity we consider a sphere of a radius R of an elastically isotropic solid with a compressibility K subject to an isotropic surface stress The incremental work against the surface stress and the bulk elasticity by expanding the body is... 

Gamamik MY. 1993. The physical nature of changes of lattice parameters in small particles. 

R. Lamber, S. Wetjen, N.I. Jaeger, Size dependence of the lattice-parameter of small palladium particles. Phys. Rev. B 51(16), 10968-10971 (1995)... 

Most EXAFS data show that the lattice parameter of a small particle is contracted as compared with the bulk metal. For example, Renouprez and co-workers125,135 observe for Pt encased in Y-zeolite a Pt-Pt distance contracted from 0.277 to 0.270 nm. Similarly, Hamilton et al.123 observe Cu-Cu distances varying from 0.233 for very small clusters, to 0.250 for... 

When the particles are in epitaxy with a crystalline support, the lattice of the small particles can be accommodated to the substrate lattice. If the lattice parameter of the support is larger than those of the particle, its lattice may expand. This has been observed by HRTEM for 2-nm Pd particles supported on MgO(lOO), where the expansion is 8% [34]. However, when the size of the particles increases, they progressively recover their bulk lattice parameter by the introduction of dislocations as shown experimentally [35] and simulated by molecular dynamics [36]. 

A supersonic free jet mixture of metal particles in argon has been studied by electron diffraction. Particles of Bi, Pb, and In of 40-95 A were measured. Changes in crystal structure from that of bulk metal were observed for clusters in the 50-60 A diameter range (2000-4000 atoms/particle). Indium growing particles changed from tetragonal to face-centered cubic as size increased. Lattice parameters of the microcrystals were found to decrease as the cluster size increased. Apparently a high proportion of surface atoms in small clusters favors crystal defects... 

Another informative method of nanocomposites characterization is the wide angle X-ray structural analysis (WAXS) and especially small-angle X-ray analysis (SAXS). X-ray analysis allows one to determine the lattice parameters of nanoparticles and, as it was found [2, 12] for the majority of suliides and oxides, the small particles exhibit the same lattice parameters and crystallographic structure as bulk substances up to the size of 20 A. At this size, the deformation of the lattice takes place which accompanies the transition to cluster structure with minimized full energy, induding free energy of the surface [11]. 

The lattice gas has been used as a model for a variety of physical and chemical systems. Its application to simple mixtures is routinely treated in textbooks on statistical mechanics, so it is natural to use it as a starting point for the modeling of liquid-liquid interfaces. In the simplest case the system contains two kinds of solvent particles that occupy positions on a lattice, and with an appropriate choice of the interaction parameters it separates into two phases. This simple version is mainly of didactical value [1], since molecular dynamics allows the study of much more realistic models of the interface between two pure liquids [2,3]. However, even with the fastest computers available today, molecular dynamics is limited to comparatively small ensembles, too small to contain more than a few ions, so that the space-charge regions cannot be included. In contrast, Monte Carlo simulations for the lattice gas can be performed with 10 to 10 particles, so that modeling of the space charge poses no problem. In addition, analytical methods such as the quasichemical approximation allow the treatment of infinite ensembles. 

While it may be reasonable (at least to a first approximation) to distinguish between resonant atoms in a surface shell and those in the bulk, the application of this model to particle size determination is also complicated by the possible presence of lattice modifications (e.g., expansions or contractions) for small particles. This point has been emphasized by Schroeer (752) and interpreted in terms of an internal pressure. For example, Ktindig et al. (143) found a nearly linear dependence with dispersion of the quadrupole splitting e for a-Fe203 microcrystals. As noted above, this was explained in terms of the shell model. However, if the lattice parameter a of a 5-nm particle is increased over the bulk value by 2% (755), then the increase in the quadrupole splitting Ae with decrease in particle size may be related to a corresponding increase in the lattice parameter A a by (Ae/e)/(Aa/a) 65... 

---