X-Git-Url: https://hackdaworld.org/gitweb/?p=lectures%2Flatex.git;a=blobdiff_plain;f=posic%2Fthesis%2Fbasics.tex;h=243d262f8e4b93d670100675d34fbb6a86fd92ea;hp=f6a8eda37e41458fa71099fe61ac63fe811dcbe8;hb=cbed3c8589a70f445c4f2c7ae6532261afb65351;hpb=e9b904cd12213f534e64f904e48071337f1f6cb6 diff --git a/posic/thesis/basics.tex b/posic/thesis/basics.tex index f6a8eda..243d262 100644 --- a/posic/thesis/basics.tex +++ b/posic/thesis/basics.tex @@ -600,6 +600,22 @@ E_{\text{f}}=\left(E_{\text{coh}}^{\text{defect}} where $N$ and $E_{\text{coh}}^{\text{defect}}$ are the number of atoms and the cohesive energy per atom in the defect configuration and $E_{\text{coh}}^{\text{defect-free}}$ is the cohesive energy per atom of the defect-free structure. Clearly, for a single atom species equation \eqref{eq:basics:ef2} is equivalent to equation \eqref{eq:basics:ef1} since $NE_{\text{coh}}^{\text{defect}}$ is equal to the total energy of the defect structure and $NE_{\text{coh}}^{\text{defect-free}}$ corresponds to $N\mu$, provided the structure is fully relaxed at zero temperature. +However, there is hardly ever only one defect in a crystal, not even only one kind of defect. +Again, energetic considerations can be used to investigate the existing interaction of two defects. +The binding energy $E_{\text{b}}$ of a defect pair is given by the difference of the formation energy of the defect combination $E_{\text{f}}^{\text{comb}} $ and the sum of the two separated defect configurations $E_{\text{f}}^{1^{\text{st}}}$ and $E_{\text{f}}^{2^{\text{nd}}}$. +This can be expressed by +\begin{equation} +E_{\text{b}}= +E_{\text{f}}^{\text{comb}}- +E_{\text{f}}^{1^{\text{st}}}- +E_{\text{f}}^{2^{\text{nd}}} +\label{eq:basics:e_bind} +\end{equation} +where the formation energies $E_{\text{f}}^{\text{comb}}$, $E_{\text{f}}^{1^{\text{st}}}$ and $E_{\text{f}}^{2^{\text{nd}}}$ are determined as discussed above. +Accordingly, energetically favorable configurations result in binding energies below zero while unfavorable configurations show positive values for the binding energy. +The interaction strength, i.e. the absolute value of the binding energy, approaches zero for increasingly non-interacting isolated defects. +Thus, $E_{\text{b}}$ indeed can be best thought of a binding energy, which is required to bring the defects to infinite separation. + The methods presented in the last two chapters can be used to investigate defect structures and energetics. Therefore, a supercell containing the perfect crystal is generated in an initial process. If not by construction, the system should be fully relaxed.