CHAPTER 1. FLUORESCENT SILICON CLUSTERS 7 to three dimensions. A more realistic model, using the work function rather than infinitely high potential walls, requires solving transcendental equations. This latter, more rigorous, treatment lowers the energy values by not more than 10 %, for which decreases in the energy difference and effective magnitude of the box diameter for a given transition energy are implicit. Furthermore, a more realistic potential surface than a square well would also lower the energy levels as the constraints imposed by the former must, of their very nature, be more relaxed. Lockwood and co-workers observed that the fluorescence of a Si/SiO$_{2}$ su- perlattice depended on the silicon film thickness (Lockwood et al., 1996). They attributed this behaviour to one-dimensional confinement of the excited elec- tron within the silicon film. To explain the shift in the fluorescence energy, they adopted the particle-in-a-box model and showed that the peak energy of the observed red fluorescence band followed equation 1.2. Here, d is the silicon layer thickness, m ∗ e and m ∗ h are the ’effective masses’ of electron and hole, although the authors acknowledge that, strictly speaking, the concept of effective electron and hole masses has no physical meaning in nanoscale systems that do not exhibit the translational symmetry of crystals. A good fit was reported for E ( eV ) = 1 . 60 + 0 . 72 d − 2 eV, with d given in nm, which implies effective masses m ∗ e ≈ m ∗ h ≈ 1 m$_{e}$ in good agreement with bulk crystalline silicon were m ∗ $_{e}$( Si$_{bulk}$ ) = 1.18 m$_{e}$ and m ∗ $_{h}$( Si$_{bulk}$ ) = 0.81 m$_{e}$. The fit shows that the band gap energy, at E$_{g}$ = 1.60 eV, is considerably larger than that of bulk crystalline silicon ( E$_{g}$ (c-Si) = 1.12 eV at 295 K), and is, in fact, more similar to that of amorphous silicon ( E$_{g}$ (a-Si) = 1.5 - 1.6 eV at 295 K). The good fit with experimental data and the similarity between the E ( n ) = E$_{g}$ + π $^{2}$¯ h 2 2 d 2 ( 1 m ∗ e + 1 m ∗ h ) (1.2)