I Defect concentration formula w/o Stirling approximation

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The discussion focuses on the limitations of using the Stirling approximation (SA) for calculating defect concentrations when the expected vacancy concentration (n_v) is less than 1. It highlights that while SA is applicable in cases with high concentrations due to Avogadro's number, it fails for lower values of n_v. The author explores alternative methods, such as the gamma function, but notes its unreliable behavior in this low concentration region. The conversation emphasizes the need for accurate calculations of defect concentrations without relying on SA. Understanding these nuances is crucial for precise defect concentration analysis in materials science.
alwaystiredmechgrad
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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
 
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