## 3 Formation of Cobalt Nanoparticles by Ion Implantation
### Results
\begin{table}
\begin{tabular}{l c c c c c c} \hline \hline
**Sample** & **Annealing** & \(\bigtriangleup\)**\(BL\)** & \(\mathbf{\sigma^{2}}\)** & \(CN\)** & \(\mathbf{C_{3}}\)** & **size** \\  & **temp( \({}^{\circ}\)C)** & **(Å)** & **(10\({}^{-3}\)Å)** & **(atoms)** & **(10\({}^{-5}\)Å)** & **(nm)** \\ \hline \(0.3\) at.\% & \(1,100\) & \(2.485(0.004)\) & \(3.04(0.12)\) & \(11.4(0.2)\) & \(-3.4(1.6)\) & \(11.4(2.5)\) \\  & \(800\) & \(2.453(0.01)\) & \(5.78(0.36)\) & \(7.6(0.4)\) & \(-18.3(5.0)\) & \(2.1(0.5)\) \\ \hline \hline \end{tabular}
\end{table} 
Table 2: _Cont._The reconstructed interatomic distance distribution for selected samples is shown in Figure 4 following Reference [42] and using a photoelectron mean free path of 8 A, the latter following from [43]. This figure better demonstrates finite-size effects. The observed structural changes with decreasing NP size relative to the bulk standard are as follows: (a) the decrease in peak height (or lower coordination) stems from the under-coordinated surface atoms; (b) the increasing FWHM (or increasing disorder) results from the relaxation and reconstruction of the under-coordinated surface atoms; (c) surface-curvature-induced capillary pressure yields a \(BL\) contraction; and finally (d) the asymmetric bondlength distribution (or negative \(C_{3}\)) results from the superposition of the different \(BL\)s from the core and surface/interface (where the latter are shorter). For semiconductor NPs embedded in SiO\({}_{2}\), simulations [44] and experiments [45] have demonstrated that the interfacial atom environment is highly distorted to accommodate material and bonding differences across the interface. Similar trends have been reported for other embedded elemental metal NPs [46, 47, 48, 49]. Figure 4: Reconstructed interatomic distance distributions for bulk HCP Co, 13 nm, 5.2 nm and 3.3 nm diameter Co NPs. Figure 3: _Cont_. Figure 5: \(FT\) EXAFS spectra for (**a**) bulk HCP Co, (**b**) 5.2 nm and (**c**) 2.8 nm diameter HCP Co NPs, as a function of measurement temperature. Temperature-dependent, phase-corrected FT EXAFS spectra are presented in Figure 5 for bulk HCP Co and both 5.2 nm and 2.8 nm diameter HCP Co NPs. The amplitude of the signal decreases with an increase in measurement temperature due to the increase in thermal disorder, while, as mentioned above, the amplitude for bulk material exceeds that for NPs at any given temperature due to the reduced \(CN\) and enhanced structural disorder associated with NPs. For any given sample, the general increase in \(\sigma^{2}\) and \(C_{3}\) as a function of measurement temperature is due to an increase in thermal disorder. Included in Figure 6(a,b) are fits to the correlated Einstein model [50] given for \(\sigma^{2}\) and \(C_{3}\) respectively as:  
\[\sigma^{2} =\sigma_{S}^{2}+\frac{\hbar\omega_{E}}{2k_{eff}}\frac{1+z}{1-z} \tag{1}\] \[C_{3} =C_{3S}+\frac{\beta(\hbar\omega_{E})^{2}}{2k_{eff}^{3}}\frac{1+10 z+z^{2}}{(1-z)^{2}} \tag{2}\]  
Here, \(\hbar\) is Planck's constant divided by 2\(\pi\), \(\omega_{E}\) is the Einstein frequency (\(\omega_{E}=\sqrt{\frac{k_{eff}}{\mu}}\)), \(\mu\) is the reduced mass of the atomic pair under investigation, \(k_{eff}\) is the effective harmonic spring constant, \(\beta\) is the cubic anharmonicity constant, \(z=exp(-\Theta_{E}/T)\) and \(T\) is the measurement temperature. The Einstein temperature is given by \(\Theta_{E}=\hbar\omega_{E}/k_{B}\) where \(k_{B}\) is Boltzmann's constant. \(\sigma_{S}^{2}\) and \(C_{3S}\) are the temperature independent (or static) contributions to the total disorder and asymmetry, respectively. Further discussion of the correlated Einstein model and application to various materials can be found in [42, 51, 52, 53, 54]. Table 3 lists values for \(\Theta_{E}\), \(\sigma_{S}^{2}\) and \(k_{eff}\). Figure 6: Thermal evolution of the first NN; (**a**) \(\sigma^{2}\); (**b**) \(C_{3}\); and \(BL\) for the Co standards and NP samples. Solid lines are fits with a correlated anharmonic Einstein model (Equations (1) and (2)).