Expansions for the Error Function

Often in diffusion problems, it is convenient to be able to approximate the error function for very small or very large arguments. For such purposes, the following relations are useful  

Defects play a major role in the performance of materials, some wanted and some unwanted. Therefore, it is necessary to understand what they do, how they form, and how to control them. Defect are categorized by their dimensionality point defects (zero dimensional), line defects (1-D), planar defects (2-D), and volume defects (three dimensional [3-D]). 

Point defects include vacancies, impurity or substitutional, and interstitial. Vacancy defects are the result of a thermodynamic equilibrium and are unavoidable. Substitutional and interstitial defects may contain unwanted impurity atoms, or may be engineered to improve the material s properties. If impurity atoms are added that have different valences from the host atoms, the material is nonstoichiometric and vacancies must form to maintain charge neutrality. 

Any surface, whether it be the face of a crystal or a grain boimdary in a polycrystalline solid, is considered a defect because the atoms on the surface have unsatisfied bonds. 

Volume defects in the form of a second phase that is purposely added or precipitated from a supersaturated solid solution are often used to improve the mechanical properties. Voids due to the clustering of vacancies or from trapped gases are examples of unwanted 3-D defects. 

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As previously indicated, the only interest in this result is in its form for large values of the variable z. We expect that the asymptotic solution will give the correct form for the thermal flux in the case of the actual geometry for regions far from the source. We obtain the asymptotic form for (6.129) by introducing the expansion for the error function in the case of large x, namely,... 

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