CHAPTER 1. FLUORESCENT SILICON CLUSTERS 6 because these two processes would have to happen simultaneously, this scenario is clearly a rare occurrence. As a consequence, the fluorescence lifetime from bulk crystalline silicon is very long and fluorescence intensity is very weak. When bulk crystalline silicon is reduced in size to a scale approaching that of the nanoscale regime, translational symmetry is gradually lost. A consequence of small size of nanocrystals is, therefore, that materials that are indirect band gap semiconductors in the bulk phase become quasi-direct semiconductors at the nanoscale. For silicon, this means that high fluorescence intensities are possible if nanocrystallites are only small enough to display a quasi-direct band gap. Another feature of such reductions in crystal size is quantum confinement. Quantum confinement relates to the shift of energy levels with size, and neatly explains the energy spectrum of, for example, colour-centre defects in crystals (Hayes and Stoneham, 2004; Fox, 2010), and electrons confined in nanoscale bubbles in liquid helium (Fowler and Dexter, 1968; Grimes and Adams, 1990, 1992). These energy shifts can be understood using the popular particle-in-a-box model that is almost perennially reviewed in quantum mechanical text books. The energy spectrum, E ( n ), of an electron confined in a one-dimensional box of length ℓ with infinitely high box walls can be straightforwardly derived to give the following equation: where n is the principal quantum number, ¯ h the reduced Planck (or Dirac) constant and m$_{e}$ the electron mass. The quantum number, n , is indexed from n = 1, and the energy difference E( n = 2) and E( n = 1) would be equivalent to the fluorescence energy from the first excited state to the ground state. The analogy of the one-dimensional model can be straightforwardly extended E ( n ) = n $^{2}$π$^{2}$¯ h 2 2 m$_{e}$ℓ 2 (1.1)