Question 7 of 7: Point Defects and Diffusion — Vacancy Concentration and the Zn-in-Cu Arrhenius Plot

Question 7: Point Defects and Diffusion — Vacancy Concentration and the Zn-in-Cu Arrhenius Plot (20 marks)

Question text not reproduced: the examination questions are © Engineers and Geoscientists BC. Open the official past paper (linked at the top of this page) to read the question, then follow the worked solution below.

Given. Vacancy formation energy for Na, $E_v=1.0\,\text{eV}$, target concentration $n/N=1/100{,}000=10^{-5}$. Figure P7: an Arrhenius plot of $\log_{10}D$ (cm$^2$/s) against $10^3/T$ (K$^{-1}$) for Zn diffusing in Cu.

Find. (a) definitions; (b) the temperature $T$; (c) $D_0$ and $E$ for Zn-in-Cu diffusion.

Approach. (a) recall the standard point-defect definitions; (b) invert the Boltzmann vacancy formula $n/N=e^{-E_v/k_BT}$; (c) fit the digitized Arrhenius data to $\log_{10}D=\log_{10}D_0-\dfrac{E}{2.303\,k_B}\cdot\dfrac1T$ and read off slope and intercept.

1. Part (a) — point-defect definitions. (i) Schottky defect: a PAIR of vacancies (in an ionic crystal, one cation vacancy and one anion vacancy) created by moving both ions to the crystal's surface, leaving the bulk vacant but electrically neutral and stoichiometric — typical in ionic crystals where cation and anion are similar in size (e.g. NaCl, KCl). (ii) Frenkel defect: a single ion is displaced from its regular lattice site into an INTERSTITIAL position, leaving behind a vacancy-interstitial pair at one site — typical when one ion (usually the cation) is much smaller than the other, small enough to fit interstitially (e.g. Ag$^+$ in AgBr/AgCl). (iii) Color center: a lattice vacancy (commonly an anion/halide vacancy in an alkali halide) that has trapped one or more electrons in place of the missing ion's negative charge; the trapped electron's discrete energy levels absorb visible light, giving the otherwise-transparent crystal a visible colour (the classic "F-center", from German Farbe, colour).
2. Part (b) — temperature for a given vacancy concentration. The equilibrium vacancy concentration follows the Boltzmann form $n/N=e^{-E_v/k_BT}$ (paper's eq. 24 style). Setting $n/N=10^{-5}$ and solving for $T$: $$T=\frac{E_v}{k_B\ln(10^5)}=\frac{1.0\,\text{eV}}{(8.617\times10^{-5}\,\text{eV/K})\ln(10^5)}$$ $$\boxed{T\approx1.01\times10^3\ \text{K}\ (1008\,\text{K})}$$ This is the answer to the stated Boltzmann relation. Note that it exceeds sodium's real melting point (about 371 K), so solid Na never actually reaches this vacancy fraction; the exam's 1.0 eV is an idealized formation energy chosen to exercise the formula.
3. Part (c) — $D_0$ and $E$ from the Zn-in-Cu Arrhenius plot. Diffusion follows the Arrhenius form $D=D_0\exp(-E/k_BT)$ (paper's eq. 25), i.e. in terms of the plotted axes ($x\equiv10^3/T$): $$\log_{10}D=\log_{10}D_0-\frac{E}{2.303\,k_B}\cdot\frac x{1000}$$ Reading the printed best-fit line against the axis ticks gives a slope of $-8.90$ decades per unit $x$, and the line (dashed in the source beyond the data) meets the $10^3/T=0$ axis at $\log_{10}D=-0.36$. A least-squares fit through the 21 printed data points gives the same values to within 1% in slope and 0.04 decade in intercept. Converting: $$E=-(\text{slope})\times2.303\,k_B\times1000=8.90\times2.303\times(8.617\times10^{-5}\,\text{eV/K})\times1000$$ $$\boxed{E\approx1.77\ \text{eV}}$$ $$D_0=10^{\text{intercept}}=10^{-0.36}$$ $$\boxed{D_0\approx0.43\ \text{cm}^2/\text{s}}$$ — both consistent with the graph's own reference marks (e.g. at $T=500\,\text{K}$, $10^3/T=2.0$, the line gives $D\approx7\times10^{-19}\,\text{cm}^2$/s, matching the lowest cluster of plotted points near the bottom-left of Figure P7).