## 5 FRET in Ensembles of Nanocrystals
The broadening of the exciton transfer resonance, \(\Gamma\) can be related to the spectral overlap of the emission and absorption spectra of the donor and acceptor, respectively [50], and the transfer rate can be expressed as [187]  
\[\frac{k_{F}}{\gamma_{0}^{(D)}}=\frac{3\kappa^{2}c^{4}}{4\pi\epsilon_{1}^{2}b^{6 }}Q_{A}\int_{-\infty}^{+\infty}\frac{d\omega}{\omega^{4}}I_{A}(\omega)L_{D}( \omega)\,, \tag{13}\]  
where \(\gamma_{0}^{(D)}\) is the spontaneous emission rate of the donor, \(I_{A}(\omega)\) is the absorption lineshape function of the acceptor, \(L_{D}(\omega)\) is the emission lineshape function of the donor (both are normalized to unity) and \(Q_{A}\) is the frequency-integrated absorption cross section of the acceptor QD,  
\[Q_{A}=\bigg{(}\frac{\pi\hbar c}{E_{g}(R_{2})}\bigg{)}^{2}\gamma_{0}^{(A)} \tag{14}\]with \(\gamma_{0}^{(A)}\) being the spontaneous emission rate of the acceptor. Notice that the depolarization factor \(\kappa^{2}\), Equation (3), which distinguishes QD donors and acceptors from, e.g., molecules, has been included explicitly in Equation (13). The FRET rate decreases rather quickly with the donor-acceptor distance, \(k_{F}=\gamma_{0}^{(D)}(\frac{b_{0}}{b})^{6}\), where \(b_{0}\) is a parameter of the dimension of length named \"Forster radius\" whose definition is clear from Equation (13). In ensembles formed of direct-band-gap II-VI or III-V semiconductor crystallites (QDs), the exciton transfer has the rates \(\sim\)10\({}^{8}\)-10\({}^{9}\)-s\({}^{-1}\), as measurements [52; 53; 55; 56] and calculations [184; 186; 188] show. Notice that \(k_{F}\) depends on \(b\) and the above values probably correspond to somewhat different donor-acceptor distances. Experimental observations of the exciton transfer were carried out with ensembles of closely packed monodisperse CdSe nanocrystals as well as of two-size three-dimensional mixtures [55] and bilayered CdSe nanocrystal systems [56]. It was shown that the inter-layer transfer in bilayered ensembles with controlled donor-acceptor separation turns out to be more efficient than in the 3D ensembles of the monodisperse and the two-size-mixed crystallites [56]. Note that the measured and computed FRET rates are of the same order of magnitude as the radiative recombination rates in high-density ensembles of colloidal II-VI crystallites. Moreover, a few measurements of the Forster radius have been performed by controlling the distance between two different groups of QDs of different size, acting as donors and acceptors [189], or by preparing a homogeneously blended solid-state films [190] composed of two groups of dots. In both works, CdSe/ZnS core-shell QDs were used and the reported results for the Forster radius are 14-22 nm and 8-9 nm, respectively; notice that the former work [189] used larger nanocrystals. Despite the uncertainty in experimental conditions and difficulty to evaluate the spectral overlap in Equation (13) for _individual_ QDs, there is a consensus that \(R_{F}\) typically is of the order of 10 nm for highly luminescent colloidal nanocrystals. In ensembles of silicon crystallites the exciton transfer turns out to be much slower than the radiative recombination, its rates are two to three orders of magnitude lower [184; 191; 192] (smaller than \(\sim\)10\({}^{3}\cdot\)s\({}^{-1}\)). Doping of silicon nanocrystals with phosphorus allows to increase the rates up to the values comparable with those of radiative recombination [193] (\(\lesssim 10^{7}\cdot\)s\({}^{-1}\) for nanocrystal radius \(R\gtrsim 1\) nm). Nevertheless, these values remain still smaller than the FRET rates for colloidal QDs made of direct-gap semiconductors. For the sake of completeness, Forster-type resonant energy transfer from a QD exciton can occur also when the final elementary excitations have a different nature, for instance, they are phonons or molecular vibrations. The latter can involve ligand molecules present in the vicinity of a colloidal QD or solvent molecules and is particularly important for quantum dots emitting in the infrared [194; 195].