Atomic Models for Diffusion (III): Ideal Crystals
2026-02-04
After this lecture, you will be able to:
Substitutional diffusion - atom replaces a lattice site - usually vacancy-mediated
Interstitial diffusion - atom moves between host sites - solute squeezes through interstitial positions
Does it make sense? A few implications: - Larger \(L_A\) gives longer crossing time - Higher \(T\) gives shorter crossing time - Heavier particles cross more slowly
Crossing rate is number of activated particles divided by crossing time
\[\begin{align} R_{\mathrm{cross}} = \frac{N^\#}{\tau_{\mathrm{cross}}} \end{align}\]where the total number of crossing \(N^\#\) follows
\[\begin{align} N^\# \approx N \frac{\tau_A}{\tau_W} \end{align}\]Ratio of residence times scales with probability of occupying each region For a simple well model, we have
\[\begin{align} \frac{\tau_A}{\tau_W} = \frac{L_A}{L_W}\exp\!\left(-\frac{E_a}{k_B T}\right) \end{align}\]Substituting the crossing time removes explicit dependence on \(L_A\), so that the jump frequency \(\Gamma'\) follows a Boltzmann distribution!
\[\begin{align} \Gamma' &= \sqrt{\frac{k_B T}{2\pi m}}\frac{1}{L_W} \exp\!\left(-\frac{E_a}{k_B T}\right) \end{align}\]If we define the attempt frequency \(\nu\) (frequency dependent on thermal kinetic energy and well geometry)
\[\begin{align} \nu = \sqrt{\frac{k_B T}{2\pi m}}\frac{1}{L_W} \end{align}\]Final form for the jump frequency:
\[\begin{align} \Gamma' = \nu \exp\!\left(-\frac{E_a}{k_B T}\right) \end{align}\]In 1D, the random-walk expression becomes
\[\begin{align} D = \frac{\Gamma' \langle r^2 \rangle}{2} \end{align}\]What is \(\langle r^2 \rangle\)? For this well, \(r \sim L_W + L_A\):
\[\begin{align} D = \sqrt{\frac{k_B T}{8\pi m}} \frac{(L_W+L_A)^2}{L_W} \exp\!\left(-\frac{E_a}{k_B T}\right) \end{align}\]The jump frequency in a parabolic well follows:
\[\begin{align} \Gamma' = \frac{1}{2\pi}\sqrt{\frac{\beta}{m}} \exp\!\left(-\frac{E_a}{k_B T}\right) \end{align}\]Analog: angular momenta of springs near the stationary point.
Using our previous analysis, it can be shown that plugging the jump frequency into the random-walk diffusivity equation is basically the famous Arrhenius form of diffusivity
\[\begin{align} D = D_0 \exp\!\left(-\frac{H^m}{k_B T}\right) \end{align}\]
For real materials, the Einstein relation / random-walk analog still holds, while a few more factors need to be considered:
\[\begin{align} D = \frac{\Gamma \langle r^2 \rangle}{6}\,f = \frac{z\,\Gamma'\langle r^2 \rangle}{6}\,f \end{align}\]Interstitial jumps are often uncorrelated - After one interstitial jump, the next jump is not strongly biased - No vacancy left behind to pull the atom back - Successive jumps are approximately independent
\[\begin{align} D_I = \frac{z\,\nu \langle r^2 \rangle}{6} \exp\!\left(\frac{S_I^m}{k_B}\right) \exp\!\left(-\frac{H_I^m}{k_B T}\right) \end{align}\]After combining vacancy formation and migration, \(D_A\) becomes dependent on both vacancy formation enthalpy and vacancy migration barrier!
\[\begin{align} D_A = \frac{z\,\nu \langle r^2 \rangle}{6} \exp\!\left(\frac{S_V^f + S_m}{k_B}\right) \exp\!\left(-\frac{H_V^f + H_m}{k_B T}\right)\,f \end{align}\]Simple estimate with \(z\)
\[\begin{align} f \simeq \frac{1-\frac{1}{z}}{1+\frac{1}{z}} \end{align}\]After today’s lecture, you should be able to - recognize diffusion in solids as an process associated with activation energy - link diffusivity with potential well picture - analyze the enthalpy dependency in the Arrhenius-type diffusivity equations
In next lecture, we will discuss diffusion processes with varied energy barriers, in particular: