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Photoselection does not directly measure the absorption and emission dipoles. In many molecular fluorophores, it measures the angle, β, between transition dipoles according to149 2 5 3cos 1 2 (3.10)

However, as discussed above, equation 3.10 applies to fluorophores with absorption and emission that are simple linear dipoles. Because the spectroscopic properties of CdSe nanocrystals are determined by both linear and planar transitions, the relative distribution of intensity along particular axes of the nanorod determines the measurable anisotropy.175

We sought to understand the microscopic basis of ensemble measurements by using single-particle fluorescence and excitation polarization measurements. In the existing literature, absorption transition intensity is described by , , , with

photoluminescence intensity , , , in which is the projection of transition

intensity along the i-axis of the fluorophore.169 These values are normalized such that

1 and 1 . By symmetry, the component of the intensity

along the minor axes is assumed to be equivalent ( ). Following these

simplifications, a cylindrically-symmetrical cone notated in short by , , describes the average spatial distribution of the transition intensities for a nanorod ensemble with respect to the nanorod long axis.

This approximation allows a reduction of these intensity ratios to the absorption anisotropy (r) and emission anisotropy (q), defined by analogy with equation 3.4:

2 (3.11)

Values of r or q can range from 1, polarized along the unique axis, to -0.5, equivalently intense along the minor axes. One key difference between emission anisotropy q and absorption anisotropy, r, is that q is fixed by the emission pattern of the nanorod whereas r is a function of the incident energy because the excitation changes with energy but not the photoluminescence. Because r and q indicate specific orientations on the nanorod axes, they allow a prediction of the anisotropy measureable using photoselection, R, as in equation 3.3 and shown in Figure 3.1. Physically-reasonable assumptions and/or direct measurements of any two of R, r, or q allow determination of the third by equation 3.3. One approach, taken by Sitt et al. is to assume that band-edge absorption is completely polarized (i.e. 1).169 Using this methodology, the fitted anisotropy of the first absorption (R1) from Figure 3.5c (0.284) yields an emission anisotropy 0.71, slightly lower than reported emission anisotropy from single-particle measurements for CdSe nanorods.160 To obtain an experimental value of emission anisotropy for the CdSe/Cd1- xZnxS nanorods analyzed in Figure 3.6, single-particle fluorescence anisotropy was measured for the same sample.

Figure 3.7 (a) Configuration for single nanorod emission anisotropy measurements.

emission is directed through a polarized beam splitter on to a CCD detector. (b) Fluorescence data for a single CdSe/Cd1-xZnxS nanorod sample in black open circles,

fitted to Acos curve in red.

The configuration of the fluorescence polarization measurement is shown in Figure 3.7a: a circularly polarized 532nm laser excites the sample of nanorods and the emission is directed through a polarized beam splitter on to a CCD detector. Figure 3.7b shows a typical single nanorod emission polarization result in open circles measured as a function of polarizer angle ( ) and fitted to a solid curve of cos , where is the angle between the (laboratory frame) polarizer zero and the maximum of the polarized fluorescence curve. The fluorescence anisotropy q averaged from 191 nanorods was 0.74 (P=0.81). This value is slightly higher than the value of q calculated by equation 3.3 if one assumes a completely anisotropic absorption of the band edge state. Direct measurement of q provides an estimate of band-edge absorption anisotropy of 0.96

by Eq. 10 using 0.284. Thus, assuming 1 necessarily overstates r and

underestimates q but the difference the model assumption and empirical measurement is small (<5%).

Table 3.1 Transition Anisotropy Assignments from Deconvolution Procedure

Transition Lit. Transition182 Ri[fitted] r (q=0.74) [eq. 3.3]

1 , 1 , 0.284 0.96

1 , 1 , 0.061 0.21

2 , 2 , 0.141 0.48

N/A 0.090 0.31

N/A 0.096 0.32

Using results from the fitting procedure in Figure 3.6, we can extend the analysis of absorption anisotropy from the band-edge to higher-energy excitations. Table 3.1 catalogues the values of of each state fitted in Figure 3.6 using the measured values of q and . The results offer a quantitative measure of absorption anisotropies in qualitative agreement with pseudopotential calculations for CdSe nanorods.182 Figure 3.8 is a graphical representation of the transition intensity cones of the ensemble absorptions on

the nanorod axis. This figure represents the model of cylindrically-symmetrical nanorod electronic structure underpinning the existing literature.161,169,170 This figure reflects the prevailing understanding of CdSe nanorod absorption, based on exciton fine structure and crystal geometry, that electronic transitions are linearly-polarized along the nanorod long axis or plane-polarized transverse to the long axis. In the next section, the efficacy of this explanation for explaining the observed properties of single nanorod absorption and emission polarization properties is tested.

Figure 3.8 Emission and absorption intensity cones for an ensemble of core/shell

CdSe/Cd1-xZnxS nanorods are plotted on the nanorod axis at their approximate location along the absorption (solid) and photoluminescence (dashed) curves. Each cone represents the intensity distribution consistent with the measured values of q and fitted values of R_i for the sample of nanorods, assuming cylindrical symmetry of the excitation and emission.

3.6 Emission and Excitation Anisotropy of Single Nanorods