Concentration fluctuations atoms

Surface diffusion can be studied with a wide variety of methods using both macroscopic and microscopic techniques of great diversity.98 Basically three methods can be used. One measures the time dependence of the concentration profile of diffusing atoms, one the time correlation of the concentration fluctuations, or the fluctuations of the number of diffusion atoms within a specified area, and one the mean square displacement, or the second moment, of a diffusing atom. When macroscopic techniques are used to study surface diffusion, diffusion parameters are usually derived from the rate of change of the shape of a sharply structured microscopic object, or from the rate of advancement of a sharply defined boundary of an adsorption layer, produced either by using a shadowed deposition method or by fast pulsed-laser thermal desorption of an area covered with an adsorbed species. The derived diffusion parameters really describe the overall effect of many different atomic steps, such as the formation of adatoms from kink sites, ledge sites... 

Fluctuations of Pi,(r) can be caused by density or concentration fluctuations. Real density fluctuations in simple (one-component) fluids occur near the critical point, where the isothermal compressibility of the liquid diverges. In multicomponent fluids, the concentration fluctuations can be observed when (at least one of) the components want to demix. Naturally, in more complex samples, like colloids in some solvents, fluctuations are present because, on the atomic length scale point of view, there are several (at least two) different phases present. SANS (and also, its X-ray counterpart SAXS) can be applied with success in these latter instances. 

To study the depletion effect, let us consider the simple model of an isolated binary nanoparticle. Let us also assume that there is no constraint on lattice rearrangement Then, the process of nucleation of the new phase in the initially homogeneous system is related to the concentration fluctuations. Let Co be the molar fraction of species B in the particle before nucleation, C is the molar fraction of species B in the new phase (the new phase nucleus will have a concentration different from the parent phase, and C Co, No, and N are the number of atoms in the parent and new phases, respectively. 

The key physics of our model (see Eqs. (9) and (10)) is contained in the nonlocal diffusion kernels which occur after integrating over the atomic processes which produce step fluctuations. We have calculated these kernels for a variety of physically interesting cases (see Appendix C) and have related the parameters in those kernels to atomic energy barriers (see Appendix B). The model used here is close in spirit to the work of Pimpinelli et al. [13], who developed a scaling analysis based on diffusion ideas. The theory of Einstein and co-workers and Bales and Zangwill is based on an equihbrated gas of atoms on each terrace. The concentration of this gas of atoms obeys Laplace s equation just as our probability P does. To make complete contact between the two methods however, we would need to treat the effect of a gas of atoms on the diffusion probabilities we have studied. Actually there are two effects that could be included. (1) The effect of step roughness on P(J) - we checked this numerically and foimd it to be quite small and (2) The effect of atom interactions on the terrace - This leads to the tracer diffusion problem. It is known that in the presence of interactions, Laplace s equation still holds for the calculation of P(t), but there is a concentration... 

However, the intra-atomic Coulomb interaction Uf.f affects the dynamics of f spin and f charge in different ways while the spin fluctuation propagator x(q, co) is enhanced by a factor (1 - U fX°(q, co)) which may exhibit a phase transition as Uy is increased, the charge fluctuation propagator C(q, co) is depressed by a factor (1 -H UffC°(q, co)) In the case of light actinide materials no evidence of charge fluctuation has been found. Most of the theoretical effort for the concentrated case (by opposition to the dilute one-impurity limit) has been done within the Fermi hquid theory Main practical results are a T term in electrical resistivity, scaled to order T/T f where T f is the characteristic spin fluctuation temperature (which is of the order - Tp/S where S is the Stoner enhancement factor (S = 1/1 — IN((iF)) and Tp A/ks is the Fermi temperature of the narrow band). 

As discussed in Section IV, we may derive from (5.9) a kinetic equation for the distribution function of the electrons, which takes into account reactions and large-scale fluctuations. From such an equation we get a Langevin-type rate equation for the concentrations of electrons, ions, and atoms, which reads in the case of local equilibrium35... 

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