Correlated Debye model for atomic motions in metal nanocrystals

Abstract

The Correlated Debye model for the mean square relative displacement of atoms in near-neighbour coordination shells has been extended to include the effect of finite crystal size. This correctly explains the increase in Debye–Waller coefficient observed for metal nanocrystals. A good match with Molecular Dynamics simulations of Pd nanocrystals is obtained if, in addition to the phonon confinement effect of the finite domain size, proper consideration is also given to the static disorder component caused by the undercoordination of surface atoms. The new model, which addresses the analysis of the Pair Distribution Function and powder diffraction data collected at different temperatures, was preliminarily tested on recently published experimental data on nanocrystalline Pt powders.

Keywords:

• Correlated Debye model
• mean square displacement
• Debye–Waller coefficient
• nanocrystalline materials
• X-ray scattering
• EXAFS

Notes

1. Analogous reasoning can be used to calculate the perpendicular MSRD component, MSRD_⊥. See Appendix 1.

2. An equivalent expression can be obtained by assigning atom j phase e⁰ = 1, and atom i phase $e^{i\underline{k}·{\underline{r}}_{\mathit{ij}}}$; this leads to the same result as in Equations (8(8) $$\begin{array}{cc}〈{\sigma }_{\mathit{ij}}^2〉\hfill & =\sum _{k,\lambda }{\left({\widehat{r}}_{\mathit{ij}}·{\widehat{e}}_{k\lambda }\right)}^2〈a_{k\lambda }^2〉\left\{1-〈cos\left[\underline{k}·\left({\underline{r}}_i-{\underline{r}}_j\right)\right]〉-〈cos\left[2{\omega }_{k\lambda }t-2\underline{k}·\left({\underline{r}}_i+{\underline{r}}_j\right)-2{\delta }_{k\lambda }\right]〉\right\}\hfill \\ \hfill & =\sum _{k,\lambda }{\left({\widehat{r}}_{\mathit{ij}}·{\widehat{e}}_{k\lambda }\right)}^2〈a_{k\lambda }^2〉\left\{1-cos\left[\underline{k}·\left({\underline{r}}_i-{\underline{r}}_j\right)\right]\right\}\hfill \end{array}$$(8) ) and (13(13) $$〈{\sigma }_{\mathit{ij}}^2〉=\frac{2ħ}{\mathit{Nm}}\sum _{k,\lambda }\frac{{\left({\widehat{r}}_{\mathit{ij}}·{\widehat{e}}_{k\lambda }\right)}^2}{{\omega }_{k\lambda }\left(\underline{k}\right)}\left[n\left({\omega }_{k\lambda }\left(\underline{k}\right)\right)+\frac{1}{2}\right]\left[1-cos\left(\underline{k}·{\underline{r}}_{\mathit{ij}}\right)\right]$$(13) ), given that $\frac{1}{2}{\left|e^{i\underline{k}·{\underline{r}}_{\mathit{ij}}}-1\right|}^2=1-cos\left(\underline{k}·{\underline{r}}_{\mathit{ij}}\right)$

3. The correct dispersion relation for a monoatomic crystal reads: ${\omega }_{k\lambda }\left(\underline{k}\right)=c\left(2k_D/\pi \right)sin\left(\pi k/2k_D\right)$

4. V is the volume of the primitive unit cell. For fcc Pd, V is ¼ of the volume of the conventional unit cell with unit cell parameter a₀ = 3.8907 Å.

5. In calculations: m is given by the molar weight divided by Avogadro’s number (e.g. 106.42 × 10⁻³/6.022 × 10²³ for Pd); h = 6.26606896 × 10⁻³⁴ ($ħ=h/2\pi$) and k_B = 1.3806504 × 10⁻²³; if lengths are given in Å units then A₁ and A₂ should be multiplied by a factor of 10²⁰.