The Jacobians of particle scattering properties with respect to aspect ratio and effective radius (r_e) using the invariant imbedding T-matrix method (IITM) are significantly improved by circumventing the divergence of derivatives. The effective radius r_e is used in three definitions, which are r_V, the equal-volume spherical radius, r_S, the equal-surface-area spherical radius, and r_VS, the ratio of three times the volume to surface area, respectively. The improved IITM and their Jacobians are applied to discuss the discrepancies between the three different definitions. The analytical Jacobians with respect to aspect ratios and effective radii associated with three effective radii are verified using the finite-difference method. Jacobians with respect to different effective radii are proportional since the ratios of different effective radii are only related to the aspect ratios. Jacobians with respect to aspect ratios are extremely sensitive to the definitions of effective radius.

通过规避导数的发散，使用不变嵌入 T 矩阵方法 (IITM)的粒子散射特性关于纵横比和有效半径 ( r _{e ) 的雅可比得到了显着改善。}有效半径r _e用于三个定义，即r _V、等体积球半径、r _S、等表面积球半径和r _VS，分别是体积与表面积的三倍之比。改进后的 IITM 及其雅可比矩阵用于讨论三种不同定义之间的差异。使用有限差分法验证了关于纵横比和与三个有效半径相关的有效半径的解析雅可比行列式。关于不同有效半径的雅可比矩阵是成比例的，因为不同有效半径的比率仅与纵横比有关。关于纵横比的雅可比对有效半径的定义极为敏感。