Spheroids extinction

The sum rule (4.81) for extinction was first obtained by Purcell (1969) in a paper which we belive has not received the attention it deserves. Our path to this sum rule is different from that of Purcell s but we obtain essentially the same results. Purcell did not restrict himself to spherical particles but considered the more general case of spheroids. Regardless of the shape of the particle, however, it is plausible on physical grounds that integrated extinction should be proportional to the volume of an arbitrary particle, where the proportionality factor depends on its shape and static dielectric function. 

A few particles, such as spores, seem to be rather well approximated by spheroids, and there are many examples of elongated particles which may fairly well be described as infinite cylinders. Our next step toward understanding extinction by nonspherical particles is to consider calculations for these two shapes. To a limited extent this has already been done spheroids small compared with the wavelength in Chapter 5 and normally illuminated cylinders in Chapter 8. We remove these restrictions in this section measurements are presented in the following section. Because calculations for these shapes are more difficult than for spheres, we shall rely heavily on those of others. 

Figure 11.15 shows Asano s calculations of extinction by nonabsorbing spheroids for an incident beam parallel to the symmetry axis, which is the major axis for prolate and the minor axis for oblate spheroids. Because of axial symmetry extinction in this instance is independent of polarization. Calculations of the scattering efficiency Qsca, defined as the scattering cross section divided by the particle s cross-sectional area projected onto a plane normal to the incident beam, are shown for various degrees of elongation specified by the ratio of the major to minor axes (a/b) the size parameter x = 2ira/ is determined by the semimajor axis a. 

A glance at the curves in Fig. 11.15 reveals extinction characteristics similar to those for spheres at small size parameters there is a Rayleigh-like increase of Q a with x followed by an approximately linear region broad-scale interference structure is evident as is finer ripple structure, particularly in the curves for the oblate spheroids. The interference structure can be explained... 

Figure 11.15 Calculated extinction by spheroids the incident light is parallel to the symmetry axis. From Asano (1979).

Perhaps the greatest difference between the extinction calculations for prolate and oblate spheroids is in the ripple structure, which is much more obvious for the latter and even persists to the largest a/b ratios shown, although with reduced amplitude. 

I or the prolate spheroid a similar, but opposite, effect is evident the first extinction maximum decreases and shifts to larger values of x with increasing... 

Microwave ( = 3 cm) extinction measurements for beams incident parallel ( = 0°) and perpendicular (f = 90°) to the symmetry axis of prolate spheroids... 

Figure 11.23 Measured extinction of microwave radiation by prolate spheroids. From Greenberg et al. (1961).

Wang and Kerker" and Chew and Wang" " have presented a theoretical treatment, based on the electrodynamic approaches used for the SERS problem, and provide a full description of the extinction of the dye-coated spheroids. They also calculated the luminescence enhancement, and find it to be up to 10" on silver for optimal wavelengths and particle shapes. 

Fig. 4.3. Extinction spectra of oblate spheroids for minonmajor axis ratios ranging from 1 1 (a sphere) to 1 10. Each spheroid has the same volume, taken to be that for a sphere whose radius is 80 nm.

For a cloud of randomly oriented small spheroids, the spectral absorption ( extinction) coefficient is given as (Lee and Tien [189]) ... 

Figure 7 shows extinction spectra calculated for randomly oriented linear chains of 13-nm gold spheres with interparticle spacing 1.1 mn. It is the same model that has been considered in Ref [70]. We note two spectral resonances related to the transversal and longitudinal (red-shifted) plasmon excitations. Such properties are analogous to the randomly oriented metal spheroids [45, 72]. By contrast to Fig. 8 in Ref [70], we conclude that random chain orientations do not eliminate the red-shifted longitudinal resonance. Perhaps, the spectra of Fig. 8 from [70] were calculated with an insufficiently large multipole expansion order. 

P.L. Dutton, K.M. Petty, H.S. Bonner, and S.D. Morse, Cytochrome C2 and Reaction Center of Rhodopseudomonas spheroides Ga. Membranes. Extinction Coefficients, Content, Half-Reduction Potentials, Kinetics and Electric Field Alterations, Biochim. Biophys. Acta 387 536 (1975). 

The inclusion of fullerenes into these cyclic dimers and jaws implies a close proximity of the MP with the spheroid Ji-system. The effects of this interaction are detectable in the red shifts of the absorption bands (Soret and Q-bands) that usually come together with low extinction coefficients. This is a typical sign of a mutual perturbation of a jt-systan. Moreover, the emission of the chromo-phores is also progressively quenched after the addition of fuUerene to the solution. 

The electronic plasmon absorption of a 6 nm Ag island film is shown in Figure 3. In the same figure, the extinction cross section calculated for a silver sphere and a silver prolate with a 3 1 aspect ratio, within the long wavelength limit of Mie theory, are included for comparison. The computation clearly illustrates the considerable shift to the red of the main plasmon absorption of the prolate spheroids in reference to the silver sphere. The broad plasmon absorption indicates a large distribution of sizes and shapes of Ag nanoparticles, and has a maximum at 494 nm. Notably, the SERS excitation profile follows closely the measured plasmon absorption, confirming the EM nature of the observed enhanced intensities. ... 

In Fig. 3.3 the extinction efficiency is reported for a prolate spheroid excited with an electric field E polarized at tt/4 respect to the y-axis. As we can observe, the dipolar resonance mode splits into one mode at small wavelengths (the transverse mode) and one mode at longer wavelengths (the longitudinal mode) this in agreement with Eq. (3.7) which for axially symmetric ellipsoidal particles provides two resonances. 

Figure 3.3 Extinction efficiencies of an Ag proiate spheroid with a = 45 nm and b = 15 nm. The three curves represent the responses of the nanoparticle for an impacting plane-wave with different polarization directions along the minor axis E° = (0, 0,1), along the major axis 6 = (0,1, 0) and with both the components E° = (0, V2/2, V2/2).

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