Quantum statistical effects in the mass transport of interstitial solutes in a crystalline solid

The impact of quantum statistics on the many-body dynamics of a crystalline solid at finite temperatures containing an interstitial solute atom (ISA) is investigated. The Mori-Zwanzig theory allows the many-body dynamics of the crystal to be formulated and solved analytically within a pseudo-one-particle approach using the Langevin equation with a quantum fluctuation-dissipation relation (FDR) based on the Debye model. At the same time, the many-body dynamics is also directly solved numerically via the molecular dynamics approach with a Langevin heat bath based on the quantum FDR. Both the analytical and numerical results consistently show that below the Debye temperature of the host lattice, quantum statistics significantly impacts the ISA transport properties, resulting in major departures from both the Arrhenius law of diffusion and the Einstein-Smoluchowski relation between the mobility and diffusivity. Indeed, we found that below one-third of the Debye temperature, effects of vibrations on the quantum mobility and diffusivity are both orders-of-magnitude larger and practically temperature independent. We have shown that both effects have their physical origin in the athermal lattice vibrations derived from the phonon ground state. The foregoing theory is tested in quantum molecular dynamics calculation of mobility and diffusivity of interstitial helium in bcc W. In this case, the Arrhenius law is only valid in a narrow range between ∼300 and ∼700 K. The diffusivity becomes temperature independent on the low-temperature side while increasing linearly with temperature on the high-temperature side.

Figure 1

Sawtooth model potential $U\left(x\right)$ for a Brownian particle in Eq. (1).

Figure 2

(a) The ratio of diffusivity of the impurity in a periodic potential and the diffusivity of free Brownian motion, $D/D_E$, against the ratio $\xi$ of the height of migratory energy barrier $E_m$ and the fluctuation-dissipation ratio $\eta ={\sigma }^2/2m\gamma$; i.e., $\xi =E_m/\eta$. (b) $\xi$ as function of temperature based on classical fluctuation-dissipation relation (CFDR), ${\xi }_C=E_m/k_{\mathrm{B}}T$, and quantum fluctuation-dissipation relation (QFDR) ${\xi }_Q=E_m/\eta \left(T\right)$, where $\eta \left(T\right)$ is given in Eq. (4) and $E_m=k_{\mathrm{B}}{\mathrm{\Theta }}_D=0.032\mathrm{eV}$.

Figure 3

The mobility of helium in bcc W ($700\mathrm{K}<T<3300\mathrm{K}$) calculated using drag dynamics simulations. The average $\mu ={\left(m\gamma \right)}^{-1}=8.4\times {10}^{-8}{\mathrm{m}}^2/\mathrm{s}/\mathrm{eV}$. The solid line is to guide the eye.

Figure 4

(a) The diffusivity of He in bcc W at temperatures from 10 to 3300 K, calculated using the Einstein formula $D={lim}_{t\to \infty }\langle {\left[\mathbf{r}\left(t\right)-\mathbf{r}\left(0\right)\right]}^2\rangle /6t$, where the He trajectory $\mathbf{r}\left(t\right)$ is collected from the many-body dynamical simulation [Eq. (6)] with the QFDR used to control the temperatures of bcc W, as well as the comparisons with the estimation of diffusivity $D_Q$ and $D_C$, derived from the analytical model [Eq. (2)] based on QFDR and CFDR, respectively. Here, in the analytical expression [Eq. (2)], $E_m=0.12\mathrm{eV}$ is obtained using the modified conjugated-gradient method (MCG) [36, 39]; the mobility $\mu =8.4\times {10}^{-8}{\mathrm{m}}^2{\mathrm{s}}^{-1}\mathrm{e}{\mathrm{V}}^{-1}$ is calculated from the drag simulation following Eq. (7). The simulation details were documented in Ref. [39]. (b) The relative error of $D_C$ with respect to $D_Q$, i.e., $\varepsilon =|D_C/D_Q-1|$ at temperatures ranging from 10 to 1200 K.