# Optimization of Nonspherical Gold Nanoparticles for Photothermal Therapy
## 2 Light Absorption by Nonspherical Gold Nanoparticles  
The geometrical models of a gold nanospheroid, nanocylinder, and nanorod are shown in Figure 1. The gold nanorod is considered to have the geometry of a cylinder capped with two hemispheres. Two size parameters, length \(L\) and diameter \(D\), are used to describe all three types of shapes. The length \(L\) and the aspect ratio \(R=L/D\) are used for the optimization of these nanoparticles. Notably, a prolate spheroid and cylinder, with their length larger than their diameter (\(L>D\)), are considered in this study. All three types of nanoparticles belong to the class of rotationally symmetric particles. Moreover, the nanoparticles are randomly oriented in therapeutic applications. Fortunately, the light absorption properties of randomly oriented, rotationally symmetric metal nanoparticles can be easily simulated by the T-matrix method [15]. Furthermore, the absorption cross-section of homogeneous, randomly oriented, and rotationally symmetric particles is given by the Formula (1) [16] as follows:  
\[C_{abs}=-\frac{2\pi}{k^{2}}\text{Re}{\sum_{n=1}^{\infty}\sum_{m=-n}^{n}\left[T _{mmm}^{11}+T_{mmm}^{12}\right]}-\frac{2\pi}{k^{2}}{\sum_{n=1}^{\infty}\sum_{n ^{\prime}=1}^{\infty}\sum_{m=-n}^{n}\sum_{m^{\prime}=-n^{\prime}}^{n^{\prime} }\sum_{i=1}^{2}\sum_{j=1}^{2}\left|T_{mmm^{\prime}n^{\prime}}^{jj}\right|^{2}}, \tag{1}\]  
where \(k\) is the wavenumber in the surrounding medium and \(T\) is a matrix. The details of the T-matrix method can be found in a book written by [PERSON] et al. [16]. The absolute magnitude of the  
Figure 1: Geometrical models of gold (**a**) nanospheroid, (**b**) nanocylinder, and (**c**) nanorod. The \(L\) represents length and the \(D\) represents diameter. absorption cross-section does not provide a reliable measure for the light absorption of an ensemble of nanoparticles employed in real-life applications, because smaller particles can be loaded in a given volume in greater numbers compared to particles with a larger size. Therefore, the more meaningful parameter for comparison across a range of sizes includes the absorption cross-section per unit particle volume called volume absorption coefficient \(\alpha_{abs}\). Volume absorption coefficient \(\alpha_{abs}\) is calculated herein to compare the light absorption ability of the three types of nanoparticles. The \(\alpha_{abs}\) is given by Formula (2) as follows:  
\[\alpha_{abs}=\frac{C_{abs}}{V}, \tag{2}\]  
where \(V\) is the particle volume. The refractive index of particles is an important parameter for the calculation. When the light interacts with the metal nanoparticles, especially for sizes smaller than the mean free path of the free electrons, the complex refractive index of the nanoparticle deviates from the bulk value because the collision of the free electrons with the particle surface becomes important and cannot be neglected [17]. The complex refractive index becomes size-dependent, and it can be expressed as [17]:  
\[n_{p}=\sqrt{n_{bulk}^{2}(\omega)+\frac{\omega_{p}^{2}}{\omega^{2}+i\omega v_{f} /l_{\infty}}-\frac{\omega_{p}^{2}}{\omega^{2}+i\omega(v_{f}/l_{\infty}+Av_{f}/ L_{eff})}}, \tag{3}\]  
where \(n_{bulk}\) is the complex refractive index of bulk metals, \(\omega\) is the angle frequency of incident light, \(\omega_{p}\) is the plasma frequency, \(v_{f}\) denotes the Fermi velocity, \(l_{\infty}\) represents the mean free path of the free electrons, \(A\) is a dimensionless parameter, usually assumed to be equal to 1, and \(L_{eff}\) is the reduced effective mean free path of the free electrons which equals the particle effective radius (equal-volume-sphere radius) [17]. For gold, \(\hbar\omega_{p}=9.03\) eV, which is obtained from the literature report [18], \(v_{f}=1.40\times 10^{15}\) nm/s and \(l_{\infty}=42\) nm, which are acquired from the literature study [19]. The values of \(n_{bulk}\) at different wavelengths are obtained from the data published by [PERSON] and [PERSON] [20].