# Mathematical Physics Analysis of Nozzle Shaping at the Gas Outlet from the Aperture to the Differentially Pumped Chamber in Environmental Scanning Electron Microscopy (ESEM)

## TL;DR

This paper analyzes how nozzle shapes affect gas flow and electron beam dispersion in environmental scanning electron microscopy.

## Contribution

The study introduces a method combining experimental measurements and mathematical physics to evaluate nozzle shaping effects in ESEM.

## Key findings

- Nozzles causing under-expanded gas outflow worsen electron beam dispersion.
- Flow velocity control impacts static pressure and beam dispersion.
- Over-expanded gas outflow shapes reduce electron dispersion more effectively.

## Abstract

A combination of experimental measurement preparations using pressure and temperature sensors in conjunction with the theory of one-dimensional isentropic flow and mathematical physics analyses is presented as a tool for analysis in this paper. Furthermore, the subsequent development of a nozzle for use in environmental electron microscopy between the specimen chamber and the differentially pumped chamber is described. Based on experimental measurements, an analysis of the impact of the nozzle shaping located behind the aperture on the character of the supersonic flow and the resulting dispersion of the electron beam passing through the differential pumped chamber is carried out on the determined pressure ratio using a combination of theory and mathematical physics analyses. The results show that nozzle shapes causing under-expanded gas outflow from the aperture to the nozzle have a worse impact on the dispersion of the primary electron beam. This is due to the flow velocity control. The controlled reduction in the static pressure curve on the primary electron beam path thus causes a significantly higher course of electron dispersion values than variants with shapes causing over-expanded gas outflow.