Three-dimensional hemispherical nucleus

Analysis of deposition transients shows that deposition of copper on TiN from 50 mM copper (II) pyrophosphate solution proceeds through instantaneous nucleation of three dimensional hemispherical clusters and diffusion limited growth. Determination of the diffusion coefficient from the current maximum and analysis of the current decay using the Cotrell equation yielded values of 1 x 10 6 to 2 x 10 6 cm2 s, slightly lower than the value for Cu2t ions due to the presence of the pyrophosphate ligand. The potential dependence of inux and t,n suggest that the nucleus density is the only potential dependent parameter. 

When the charge-transfer step in an electrodeposition reaction is fast, the rate of growth of nuclei (crystallites) is determined by either of two steps (I) the lattice incorporation step or (2) the diffusion of electrodepositing ions into the nucleus (diffusion in the solution). We start with the first case. Four simple models of nuclei are usually considered (a) a two-dimensional (2D) cylinder, (b) a three-dimensional (3D) hemisphere, (c) a right-circular cone, and (d) a truncated four-sided pyramid (Fig. 7.2). 

Models allowing to determine A and No obtain the current transient from the material flux to free , noninteracting, growth centers, considering circular diffusion zones around them, with time-dependent radii r. As shown in Fig. 4, these are two-dimensional projections of three-dimensional fields that define, for a hemispherical nuclei of radius ro, an equivalent area toward which the same amount of matter that diffuses spherically to a three-dimensional nucleus diffuses by planar diffusion. 

Fig. 3. A two-dimensional diagram for the Monte Carlo method showing the development of a cascade. The entering nucleon has 400 Mev of kinetic energy the numbers indicate the energy of the nucleons involved. The path of each nucleon is shown until it leaves the nucleus or its energy becomes less than 35 Mev, indicated by a broken line and arrow. An open circle indicates the position at which a collision would have occurred but was forbidden by the Pauli exclusion principle. In this collision three cascade particles emerge in the forward hemisphere and the nucleus is left with a thermal excitation of 66 Mev. (Figure is redrawn from the work of Booth et al.)...

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