# Defect concentration formula w/o Stirling approximation

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In summary, the concentration of defects or charges can often be large enough to use SA, thanks to Avogadro's number. However, if the expected concentration is lower than 1, SA cannot be used. Instead, the gamma function can be used to determine the exact concentration, although it may behave strangely for values lower than 1.
TL;DR Summary
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.

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.

n_v >>1, e.g., denotes the number of vacancies, not the concentration of vacancies

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## What is the formula for calculating defect concentration without using Stirling approximation?

The formula for calculating defect concentration without using Stirling approximation is given by:
C = N / N0 = exp(-Ef/kT), where C is the defect concentration, N is the number of defects, N0 is the total number of lattice sites, Ef is the formation energy of the defect, k is the Boltzmann constant, and T is the temperature.

## Why is Stirling approximation not used in the defect concentration formula?

Stirling approximation is not used in the defect concentration formula because it is only accurate for large values of N. In the case of defects, N is usually a small number, making Stirling approximation inaccurate.

## What is the significance of the formation energy in the defect concentration formula?

The formation energy in the defect concentration formula represents the energy required to form a defect in the crystal lattice. It is a crucial factor in determining the concentration of defects in a material.

## How does temperature affect defect concentration according to the formula?

According to the formula, defect concentration decreases as temperature increases. This is because at higher temperatures, more thermal energy is available to promote the movement of atoms, reducing the likelihood of defects forming or remaining in the lattice.

## Can the defect concentration formula be applied to all types of defects?

Yes, the defect concentration formula can be applied to all types of defects, including point defects, line defects, and surface defects. However, the formation energy and other parameters may vary depending on the type of defect and the material in which it occurs.

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