# Application of perturbation theory to finding vibrational frequencies of a spheroid

**Authors:** M. O. Nestoklon, L. Saviot, S. V. Goupalov  
**arXiv:** [2509.09586](https://arxiv.org/abs/2509.09586)  
**Date:** 2025-09-11  
**Summary page:** https://www.summarizepaper.com/en/paper/2509.09586v1/

## Key points

- Study by Nestoklon, Saviot, and Goupalov explores vibrational frequencies of Raman-active spheroidal modes compared to spheres using perturbation theory of boundary conditions.
- Method developed by Migdal and Moszkowski for deformed atomic nuclei applied to analyze eigenfrequencies of CdSe and silver nanoparticles focusing on $j=2$ angular momentum modes.
- Perturbation method accurately predicts frequency splittings under rigid boundary conditions resembling a sphere's shape closely.
- Limitations arise when dealing with stress-free boundaries in predicting frequency splittings.
- Research offers valuable insights into vibrational frequencies in non-spherical structures and provides comparisons between theoretical predictions and numerical calculations.

## Summary

The study by Nestoklon, Saviot, and Goupalov explores the vibrational frequencies of Raman-active spheroidal modes in comparison to those of a sphere using perturbation theory of boundary conditions. This method was initially developed for deformed atomic nuclei by Migdal and Moszkowski and is applied to analyze the eigenfrequencies of CdSe and silver nanoparticles with a focus on modes characterized by a total angular momentum value of $j=2$. The results show that when the spheroidal shape closely resembles that of a sphere, the perturbation method accurately predicts frequency splittings under rigid boundary conditions where displacement is set to zero. However, limitations are encountered when dealing with stress-free boundaries. Despite this challenge, the study provides valuable insights into understanding vibrational frequencies in non-spherical structures and offers comparisons between theoretical predictions and numerical calculations. Overall, this research contributes to advancing our knowledge of vibrational modes in non-spherical geometries.