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Distance between layers

Structural information, like an interlayer relaxation and surface thermal expansion can be calculated directly from the difference between the average heights of the $i$ and $i+1$ layers:
\begin{displaymath}
d_{i,i+1}=\left< \frac{1}{n_{i+1}} \sum_{j \in i+1} z_j- \frac{1}{n_{i}} \sum_{j \in i} z_j \right >
\end{displaymath} (6.12)

where $z_i$ is a $z$-coordinate of the atom $i$, the sum includes atoms in the layer $i$ and $i+1$, and the angular brackets denote averaging over time. Distances between the neighboring layers could be calculated at temperatures up to the temperatures where surface premelting is sets in. Above such temperature the very concept of distinct layers becomes somehow doubtful, even though some structure is still visible in the local density profiles.
Figure: Distance between the neighboring layers vs. temperature of Va(011). Note the inward relaxation of the topmost surface layers.
\begin{figure}\centerline{\epsfxsize=7.0 cm \epsfbox{/home/phsorkin/Diploma/Surface/Chap3/dist011.eps } }\end{figure}
At low temperatures the topmost surface layers exhibit an inward relaxation, i.e. $\Delta_{1,2} < 0$, where
\begin{displaymath}
\Delta_{1,2}=\frac{d_{1,2}-d_{bulk}}{d_{bulk}}
\end{displaymath} (6.13)

here $d_{1,2}$ is the distance in the $z$-direction between the first and the second surface layers, and $d_{bulk}$ is distance between the two neighboring layers in the bulk, as it is shown in Figs. 5.26. One of explanation of the inward relaxation is following [83]; the surface suffers a ``deficit'' of the electron density, relative to the electron density in the bulk, and therefore it compensates the ``deficit'' by contracting the distance between the first and the second layers. The second and third layers exhibit a downward relaxation.
Figure: Interlayer distances, normalized to the distance between bulk neighboring layers at T=$0$ K, of $Va(001)$ and $Va(011)$ at T=$2200$ K.
\begin{figure}\centerline{\epsfxsize=8.0 cm \epsfbox{/home/phsorkin/Diploma/Surface/Chap3/norm_dist.eps } }\end{figure}
The distance between neighboring layers increases with the temperature, and the thermal expansion of the surface layers is sufficiently larger than the thermal expansion of the layers close to the bulk. The observed ``anomalous'' thermal expansion of the surface layers is a direct manifestation of the broken symmetry of positional inversion at the crystal surface. Therefore, atoms rearrange their equilibrium positions in the surface layers and probe the more anharmonic region of the potential. The difference in the geometry of the faces is reflected in the thermal expansion at the surface region, a surface with the less packed layers expands more than a surface with close packed ones (See Fig. 5.27)

We also compare the averaged distances between the first and second surface layers of the Va(011),Va(111) and Va(001) samples (See Fig. 5.28). It is found that the Va(111) surface layers exhibit the largest thermal expansion in comparison with the other faces, and therefore the onset of ``anomalous expansion'' takes place at lower temperatures, than for the Va(011) and Va(001) systems. (Beyond the temperature $T=1500K$ the first crystal layers of the Va(111) are molten and therefore the concept of distinct layers becomes meaningless) The larger thermal expansion of the Va(001) surface layer in comparison with the Va(011) surface layer can be followed to much higher temperatures.

Figure: Interlayer distance between the first and second surface layers $d_{12}$ as a function of temperature.
\begin{figure}\centerline{\epsfxsize=7.0 cm \epsfbox{/home/phsorkin/Diploma/Surface/Chap3/Dist.eps } }\end{figure}


next up previous
Next: Layer occupation and energetics Up: Results: surface melting Previous: Diffusion coefficients
2003-01-15