Undergrad Defect concentration formula w/o Stirling approximation

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SUMMARY

The discussion focuses on the derivation of the defect concentration formula without utilizing the Stirling approximation (SA). It highlights that when the number of vacancies (n_v) is expected to be less than 1, the SA becomes inapplicable. The author explores the use of the gamma function to determine defect concentration but notes its erratic behavior in regions where n_v is less than 1. This indicates a need for alternative methods to accurately calculate defect concentrations in such scenarios.

PREREQUISITES
  • Understanding of defect concentration in materials science
  • Familiarity with the Stirling approximation
  • Knowledge of the gamma function and its applications
  • Basic concepts of Avogadro's number and its significance
NEXT STEPS
  • Research alternative methods for calculating defect concentrations in materials
  • Study the properties and applications of the gamma function in statistical mechanics
  • Explore the implications of Avogadro's number in defect concentration calculations
  • Investigate scenarios where the Stirling approximation fails and suitable alternatives
USEFUL FOR

Materials scientists, physicists, and researchers focused on defect analysis in crystalline structures will benefit from this discussion.

alwaystiredmechgrad
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TL;DR
The defect concentration is normally expressed by using Stirling approximation (SA) for very nice simplicity. However, in the case of wide bandgap materials, it is common to see the concentrations of electrons or defects are too small to use SA. Could you give me some nice ideas to express the low concentration of species, which can be lower than 1 cm^-3.
In many cases, the concentrations of defects or charges are quite big enough to use SA, due to a big number of Avogadro's number.
The derivation for the well-known formula of a defect concentration is followed.
그림1.png

If the n_v is expected to be lower than 1, then it would be impossible to use SA.
Then, how can we know the exact concentration of the defect?
I tried to use the gamma function, however, it behaves wield at the region lower than 1.
Thank you for reading this post.
 
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n_v >>1, e.g., denotes the number of vacancies, not the concentration of vacancies
 
Last edited:

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