Increase of the mean inner Coulomb potential in Au clusters induced by surface tension and its implication for electron scattering

Electron holography in a transmission electron microscope was applied to measure the phase shift $\Delta \phi$ induced by Au clusters as a function of the cluster size. Large $\Delta \phi$ observed for small Au clusters cannot be described by the well-known equation $\Delta \phi =C_E{V}_{0}t$ ($C_E$, interaction constant; $V_0$, mean inner Coulomb potential (MIP) of bulk gold; and $t$, cluster thickness). The rapid increase of the Au MIP with decreasing cluster size derived from $\Delta \phi$ can be explained by the compressive strain of surface atoms in the cluster.

Figure 1

(a) Grayscale coded [256 gray levels between black (no phase shift) and white (phase shift equal to $2\pi \mathrm{rad}$)] reconstructed phase shift obtained for an Au cluster with $D=6.9\mathrm{nm}$. The frame represents the integration domain (see text). The inset shows a typical HRTEM image of a Au cluster with M-Dh structure. (b) ○, the experimental phase shift integrated along the $y$ direction. The solid line indicates the linearly interpolated background. (c) Experimental (○) and calculated (solid line) integrated phase shifts of Au clusters $\Delta {\phi }_{\mathrm{Au}}^{\mathit{int}}\left(R\right)$. The inset shows an M-Dh Au cluster with our experimental shape according to Ref. 29.

Figure 2

Line profile of the phase shift through the middle of (a) a small cluster with $R=0.9\mathrm{nm}$ and $\epsilon =-19.0\%$, (b) a cluster with $R=2.1\mathrm{nm}$ and $\epsilon =-5.2\%$, and (c) a large cluster with $R=3.4\mathrm{nm}$ and $\epsilon =-2.7\%$: (○) experimentally measured and calculated phase shifts for (◇) M-Dh Au clusters and for comparison with (▵) truncated octahedral clusters with $h=0.4R$.

Figure 3

(a) Uniform compressive strain of surface atoms $\epsilon \left(R\right)$ for M-Dh Au clusters (●, ideal M-Dh clusters), (b) total number of Au atoms $N_T\left(R\right)$ (○, ideal M-Dh clusters), and number of surface atoms $N_S\left(R\right)$ (▵, ideal M-Dh clusters).

Figure 4

Experimental (○) and calculated (solid line) Au MIP values in M-Dh Au clusters $V_0^{\mathit{cl}}\left(R\right)$. The calculations were performed using $V_0=32.2\mathrm{V}$ for the MIP value of the cluster core. The hatched region represents the maximum error of the $V_0^{\mathit{cl}}$ values induced by the upper and lower strain estimates.