Optimum vacancy concentration in a crystal

The importance of the presence of a small fraction of vacancies in a crystal structure is demonstrated from considerations of thermodynamic stability. We include in the density functional theory the effects due to the distortion of the lattice structure surrounding the vacancy and show that the free energy is less when vacancies are present. Near freezing point, our theoretical model obtains the equilibrium vacancy fraction in the hard sphere crystal to be $\sim {10}^{-5}$ and it decreases with increase of the density.

Figure 1

Constant free energy contours at $n_0{\sigma }^3=1.0$ on the $n_D$ (in units of ${\sigma }^3$) and ${\alpha }_D$ plane with fixed $\alpha =97.50$ (in units of ${\sigma }^2$) corresponding to the minimum. Inset shows variation of $\Delta f_{\alpha }∕n_D$ (see text) vs $\alpha$ at $n_D=3.95\times {10}^{-5}$ and ${\alpha }_D=76.46$ corresponding to the minimum. $\Delta f_{\alpha }$ is expressed in units of ${\beta }^{-1}$.

Figure 2

(Color online) $\beta \Delta f_D$ (see text) vs $n_0$ (in units of ${\sigma }^3$) and the corresponding estimate $-\ln (N_D∕N)$ (see text) for defect interaction energy from simple lattice model shown as squares. $\Delta f_D$ is expressed in units of ${\beta }^{-1}$.

Figure 3

Variation of defect concentration $n_D$ with respect to $n_0$ (both in units of ${\sigma }^3$). Inset shows the variation of free energy of crystal (solid line) and liquid (dashed line) with density. Free energy is expressed in units of ${\beta }^{-1}$.

Figure 4

(Color online) Variation of $n_D$ (in units of ${\sigma }_2^3$) in binary system with $n_0$ (in units of ${\sigma }_2^3$) at fixed concentration $x=0.5$ and fixed value of diameter ratio $\kappa =0.95$. Inset shows variation $n_D$ with $\kappa$ at fixed density $n_0=1.1$ and relative concentration $x=0.5$. At this packing fraction, the value of $n_D$ in the one-component limit is indicated with an arrow.