Copyright 2004 by Marcel Dekker, Inc. All Rights Reserved.

where e F = e F ieV and e F F = e x F F ie y F . Substituting Eq. (31) into Eq. (24), weobtain the relative values of the optical transition probability to/from theexciton state having the total angular momentum projection F coursed by theabsorption/emission of the r F polarized light.For the exciton state with F = 0, we obtainP U;L0ðr F Þ ¼ jh0je F pˆb jC ˜ U;L0ij 2 ¼ je F z h0jpˆzj˜CU;L0ij 2¼ ðe 2 z þ eV2 z ÞNU;L 0¼ N U;L0sin 2 ðhÞ ð32Þwhere h is the angle between the crystal hexagonal axis and the light propagationdirection. In deriving Eq. (32), we used the identity for three orthogonalvectors (e, eV, and c): cos 2 (h e ) + cos 2 (h e V) + cos 2 (h) = 1, where h e and h e V are theangles between the crystal hexagonal axis and the vectors e and eV, respectively.One can see from Eq. (32) that the excitation probability of the upper (+) F = 0state does not depend on the NC size, and that for the lower state ( ), it isidentically equal to zero. The lower F = 0 exciton state is always opticallypassive.For the exciton states with F = +1, we obtainP U;LF¼1 ðrF Þ ¼ jh0je b pˆF jC ˜ U;L1i 2 ¼ 1 4 jeb h0jpˆþ jC ˜ U;L1ij 2¼ 1 4 je bieV j 2 N U;L1¼ N U;L1ð1Fcos hÞ 2 ð33ÞSimilar calculations yield the following expression for the excitation probabilityof the F = 1 state:P U;LF¼1 ðrF Þ ¼ N U;L1ð1bcos hÞ 2 ð34ÞDeriving Eqs. (33) and (34), we used the orthogonality condition (ec) = 0. In azero magnetic field, the exciton states F = 1 and F = 1 are degenerate andwe cannot distinguish them in a system of randomly oriented crystals.To find the probability of exciton excitation for a system of randomlyoriented NCs, we average Eqs. (26) and (28) over all possible solid angles. Therespective excitation probabilities are proportional toP L 0 ¼ 0; PU 0 ¼ NU 03 ;P L 1 ¼ PL 1 ¼ 2NL 13 ; PU 1 ¼ PU 1 ¼ 2NU 13ð35ÞThere are three optically active states with relative oscillator strengths P0 U,2P1 U, and 2PL 1 . The size dependence of these strengths for different NC shapesis shown in Fig. 4 for hexagonal CdSe nanoparticles. It is seen that the NCCopyright 2004 by Marcel Dekker, Inc. All Rights Reserved.