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

Figure 1a shows the surface plasmon absorption of 9-,22-,48-, and99-nm gold nanoparticles [15]. The particles were prepared in aqueous solutionsby reduction with sodium citrate. From Fig. 1a, one can see that thesurface plasmon resonance red shifts with increasing particle size, which isaccompanied by an increase in the bandwidth. Both effects will be discussedin more detail later. Figure 1b shows the molar extinction coefficients for thesame four samples. These coefficients are on the order of 1 10 8 to 1 10 11M 1 cm 1 ; they increase linearly with the increasing volume of the particles.Note that these extinction coefficients are several orders of magnitude greaterthan those for strongly absorbing organic dye molecules. The large extinctioncoefficients are due to a large number of free electrons contributing to thepolarizability and, hence, to the extinction of the particles.Theoretically, the surface plasmon absorption can be described usingthe Maxwell equations. This was first done by Gustav Mie [11] in 1908. Hesolved the Maxwell equations for the electromagnetic wave interacting witha small sphere, assuming the same macroscopic, frequency-dependent material’sdielectric constant as for a bulk metal. The solution of these equationsfor a spherical object with appropriate boundary conditions leads to a seriesof multipole oscillations for the extinction cross section. Mie obtained thefollowing expressions for the extinction and scattering cross sections (r ext andr sca , respectively) [8–11] for a nanoparticle with radius r:r ext ¼2p X lDjkj 2L ¼1r sca ¼ 2p X ljkj 2 L ¼1wherea L ¼ mw LðmxÞw V L ðxÞmw L ðmxÞgL V ðxÞð2L þ 1Þ Refa L þ b L g ð1Þð2L þ 1Þ ja Lj 2 þjb L j 2w VL ðmxÞwLðxÞw VL ðmxÞgLðxÞð2Þb L ¼ w LðmxÞw V L ðxÞw L ðmxÞgL V ðxÞmw V L ðmxÞwLðxÞmw VL ðmxÞgLðxÞn is the complex index of refraction of the metal particle, n m is the real indexof refraction of the surrounding medium, m = n/n m , k is the wave vector, x =jkjr, w L and g L are the Ricatti–Bessel cylindrical functions, L is the summationindex of the partial waves (L = 1 corresponds to the dipole oscillation, L = 2is associated with the quadrupole oscillation, and so on), and the primeindicates differentiation with respect to the argument in the parentheses. Theabsorption cross section can be calculated as r abs = r ext r sca .Copyright 2004 by Marcel Dekker, Inc. All Rights Reserved.