Application of perturbation theory to finding vibrational frequencies of a spheroid

TL;DR

The paper addresses vibrational frequencies of Raman-active spheroidal modes in nanospheroids by applying the Migdal–Moszkowski perturbation theory of boundary conditions to map a spheroid to a sphere, yielding a linear perturbation in the shape parameter $\mu_z$ with $\mu_z=2\frac{c-b}{c+b}$ for $|\mu_z|\ll 1$. The authors derive the $j=2$ spheroidal displacement field in a sphere, establish the transcendental equations linking frequency $\omega_n$ to wave numbers $q_n$ and $Q_n$ via $q_n=\omega_n/c_l$, $Q_n=\omega_n/c_t$, and determine the frequency corrections under spheroidal deformation through a perturbation operator $\Delta \hat{\Lambda}$ that yields a matrix of the form $C_2(J_z^2-2)$ with $C_2\propto R^{-2}$. Numerical validation using Rayleigh–Ritz and finite-element methods for CdSe and Ag nanoparticles shows good agreement with the perturbative predictions under rigid boundary conditions and small $|\mu_z|$, but the approach fails for stress-free boundary conditions, due to the transformation not preserving angles. These results provide analytical checks for numerical spectra and clarify boundary-condition effects on low-frequency Raman-active vibrations in anisotropic nanocrystals.

Abstract

We apply perturbation theory of boundary conditions, originally developed by A.B. Migdal and independently by S.A. Moszkowski for deformed atomic nuclei, to finding eigenfrequencies of Raman-active spheroidal modes of a spheroid from these of a sphere and compare the outcomes with the results of numerical calculations for CdSe and silver nanoparticles. The modes are characterized by the total angular momentum $j=2$ and are five-fold degenerate for a sphere but split into three distinct modes, characterized by the absolute value of the total angular momentum projection onto the spheroidal axis, in case of a spheroid. The perturbation method works well in case of the rigid boundary conditions, with the displacement field set to zero at the boundary, and accurately predicts the splittings when the spheroidal shape is close to a sphere, but fails in case of the stress-free boundary conditions.

Figures (2)

• Figure 1: (a): Dimensionless vibration frequencies ($q_0 R$) of the fundamental spheroidal mode with $j=2$ and $|m|=0,1,2$ as functions of the spheroid aspect ratio, $c/b$, for a restrained CdSe nanocrystal of mean radius $R=c^{1/3} \, b^{2/3}$. The results of the perturbation theory (solid lines) are compared to numerical calculations (dashed lines). The rigid boundary conditions are used in both cases. Numerical calculations were performed for $R=3$ nm. (b): Same as (a) but for a silver nanocrystal.
• Figure 2: Same as Fig. \ref{['fig1']} but for the free boundary conditions.