We derive analytical expressions for the surface‐ and orientation‐averaged surface enhanced Raman scattering factor (SERS EF) for small plasmonic spheroids. These expressions are derived taking into account the Stokes shift and are shown to result in very different EFs to those in the |E|⁴‐approximation for experimentally relevant parameters. The optical properties of spheroids are treated in terms of the common and improved electrostatic approximations (EA and IEA). The latter is newly introduced here and includes radiative damping, depolarization effects, and interactions with higher order multipoles, thus giving perfect agreement with exact T‐matrix calculations of far‐field cross sections and near‐field SERS EFs over a much wider range of sizes. These accurate analytical expressions will be particularly relevant to SERS experiments in colloidal solutions where surface averaging (because of random molecular adsorption) and particle orientation averaging must both be taken into account. Comparison with recent experimental data and simulations for cigar‐like particles show that the spheroid model tends to overestimate the SERS EF because of the higher curvature at the tips. Possible ways to generalize the analytic solution to other axially symmetrical particles are discussed.

中文翻译：

---

小等离激元粒子的表面和方向平均SERS增强因子的解析解

我们推导出小等离激元球体的表面和方向平均表面增强拉曼散射因子（SERS EF）的解析表达式。这些表达式是在考虑了Stokes位移的基础上得出的，并显示出与| E |中的EF截然不同的EF。⁴-实验相关参数的近似值。根据常见和改进的静电近似（EA和IEA）来处理球体的光学特性。后者是在此处新引入的，包括辐射阻尼，去极化效应以及与高阶多极子的相互作用，因此与在更大范围的尺寸范围内远场横截面和近场SERS EF的精确T矩阵计算完全吻合。这些准确的分析表达式与胶体溶液中的SERS实验特别相关，在胶体溶液中，必须同时考虑表面平均（由于随机分子吸附）和颗粒取向平均。与最新的实验数据和类似雪茄颗粒的仿真结果进行比较后，发现球体模型倾向于高估SERS EF，因为尖端的曲率更高。讨论了将解析解推广到其他轴向对称粒子的可能方法。