Ab initio calculations of rare-earth diffusion in magnesium

We have used density-functional theory to calculate activation energies for diffusion of Al, Ca, and several rare-earth (RE) elements in Mg. The predicted solute-vacancy binding and exchange energies increase and decrease, respectively, with increasing solute size. Using the eight-frequency model for hcp lattices we determined correlation factors for each solute. Both Ca and all RE elements exhibit significant correlation effects whose temperature dependence can be described by an Arrhenius behavior. The correlation effects increase the activation energy, which for these solutes still remains below that for self-diffusion.

Figure 1

Illustration of the eight possible vacancy (white) exchanges in the neighborhood of a solute (red) for vacancy and solute starting in (a) different and (b) the same basal plane(s).

Figure 2

hcp jump distances used in the eight-frequency model.

Figure 3

(a) Solute-vacancy binding energy $E_B$ and (b) solute-vacancy exchange barrier $E_X$ plotting the difference between solute volume and Mg volume. Interbasal values are shown as filled symbols and intrabasal values are open symbols. Al has values of $0.02\left(0.03\right)\mathrm{eV}$ and $0.57\left(0.53\right)\mathrm{eV}$ for inter- (intra)basal energies $E_B$ and $E_X$, respectively, but is much smaller than Mg ($\Delta V^{\mathrm{Al}}=-0.19{\AA }^3$) and is therefore outside the plot range.

Figure 4

Valence electron density contour maps (in units of $e/{\AA }^3$) on the intrabasal migration path for (a) Al, (b) Ca, (c) La, (d) Eu, and (e) Gd in equilibrium positions with vacancy adjacent, and (f) Al, (g) Ca, (h) La, (i) Eu, and (j) Gd in the transition state using a $\langle 001\rangle$ cut. Color is on the same scale for all solutes and low density is represented by blue and high density by red. Contour lines are logarithmically spaced to allow examination of both behavior in the high-density regions near the atomic centers and also the lower density regions between atoms. The density ranges over $0.02–2.54e/{\AA }^3$.

Figure 5

Natural log of the correlation factors, $f_{Az}$ ($◯$), $f_{Bx}$ ($+$), and $f_{Ab}$ ($\times$), plotted against $1/k_{B}T$. The values for Al all lie on the $f=1$ axis, and solid lines (blue) show fits of the values for Ca to Eq. (8). For comparison we also show $f$ values computed from the data of Ref. 17 (dashed lines, red); see Sec. 4.

Figure 6

Natural log of (a) $f_{Az}$, (b) $f_{Bx}$, and (c) $f_{Ab}$ plotted against inverse temperature for RE solutes La (red $\times$), Ce (yellow $+$), Pr (green $◃$), Nd (cyan $▵$), Sm (blue $▹$), Eu (black $◯$), and Gd (magenta $\diamond$).

Figure 7

Activation energy $Q_{\parallel }$ consisting of vacancy formation energy $E_f^{\prime }=E_f^0-E_B^{\prime }$ (blue “$\setminus$”), solute-vacancy exchange barrier $E_X^{\prime }$ (green “+”), and contribution from correlation effects $E_{f_{Az}}$ (red “/”). Solutes are ordered according to increasing size.

Figure 8

Activation energy $Q_{\perp }$ for diffusion perpendicular to the $c$ axis. Values of $Q_{\perp }^{f=\mathrm{const}}$ that would be obtained without correlation effects are shown in separate shading (“$\setminus$”). Solutes are ordered according to increasing size.