Surface structure of liquid Bi and Sn: An x-ray reflectivity study

X-ray reflectivity measurements of the liquid Bi surface are presented and analyzed together with previous liquid Sn results. Published measurements on liquid Ga, In, and K all exhibit a single strong maximum at a wave-vector transfer of the order of the reciprocal of an atomic-diameter, due to surface-induced layering. In contrast, both Sn and Bi exhibit—in addition—a weak broad peak at much smaller wave-vector transfers. This feature is an unambiguous signature of an enhanced electron density in the near-surface region. Possible ways of modeling this enhancement are presented. Once the different surface-roughening effects of thermal capillary waves are accounted for, the surface structure factors of Sn and Bi are remarkably similar. The principal difference between the two is that the depth of the layering below the surface is more than $\sim 40\%$ larger for Bi than for Sn. This is considerably larger than the ratio of their covalent radii which is only $\sim 10\%$. No theoretical explanation can be offered at this time for the surface structure difference between Sn and Bi and other elemental liquid metals studied to date: Ga, In, and K.

Figure 1

(Color online) Measured Fresnel-normalized x-ray reflectivity from the surface of liquid Sn (◻) previously published in Ref. 8. The solid and broken lines are the best fits of the present model discussed in the text for different ${\sigma }_A$. The dotted line is the best fit of the distorted crystal model (Refs. 7, 8).

Figure 2

(Color online) The electron densities corresponding to the two fits of the present model to the measured Sn reflectivity are shown in Fig. 1.

Figure 3

Parameters derived from the best-fit models for the Sn data as a function of the different fixed values of ${\sigma }_A$.

Figure 4

(Color online) Measured Fresnel-normalized x-ray reflectivity of Bi (◻) and Sn (▼). The model fits discussed in the text are shown as lines. The two fits of the present model for different ${\sigma }_A$ are distinguishable only at the largest $q_z$. The dotted line is the best fit of the distorted crystal model (Ref. 7). The dash-dotted-dash line shows the reflectivity that would be expected if for a simple interface with no layering $\left(\Phi =1\right)$ and the same surface tension as liquid Bi.

Figure 5

(Color online) Measured scattered intensity (◻) from the liquid Bi surface along the $q_y$ axis at a fixed $q_z\approx 1.0{\text{Å}}^{-1}$. The solid line is the best fit of Eq. (1), yielding a surface tension of 385 mN/m.

Figure 6

(Color online) The electron densities $⟨\rho \left(z\right)⟩/{\rho }_{\infty }$ corresponding to the two fits of the present model to the measured Bi reflectivity are shown in Fig. 4.

Figure 7

Parameters derived from the best-fit models for the Bi data as a function of the different fixed values of ${\sigma }_A$.

Figure 8

Comparison of the measured liquid surface structure factors for Sn (●) and Bi (◻). Inset: the same data plotted vs $q_z/q_{\mathrm{peak}}$ where $q_{\mathrm{peak}}=2.07{\text{Å}}^{-1}$ for Bi and $2.29{\text{Å}}^{-1}$ for Sn. There are only slight differences between the two curves.

Figure 9

(Color online) Comparison between the surface electron-density models for Sn and Bi fitted assuming an equal width of $\sim 0.5\text{Å}$ for the top adlayer and the adjacent first DCM layer.

Figure 10

Gaussian decay parameter, $\Lambda =1/\overline{\sigma }$, for Sn and Bi. The differences are larger than could be accounted for by the atomic sizes alone.

Figure 11

(Color online) Plot of the squares, ${\left|{\Phi }^A\left(q_z\right)\right|}^2$ (- - -) and ${\left|{\Phi }^{\text{DCM}}\left(q_z\right)\right|}^2$ (——), of the two additive terms of the surface structure factor $\Phi \left(q_z\right)$ [see Eq. (13)] for the best-fit parameters of Bi for ${\sigma }_A=0.2\text{Å}$. Inset: polar plot of $\text{Re}\mathbf{\left(}\Phi \right(q_z\left)\mathbf{\right)}+i\mathrm{Im}\mathbf{\left(}\Phi \right(q_z\left)\mathbf{\right)}$ in the range $3\pi /2d>q_z>0$.

Figure 12

(Color online) The surface structure factor squared for ${\rho }_A/{\rho }_{\infty }=0.8$, 0.92, 1.0, 1.08, and 1.2. The values for the other parameter are those of the best fits for Bi at ${\sigma }_A=0.2\text{Å}$.

Figure 13

(Color online) Schematic illustrating the shape of the bulk diffuse scattering underlying the specular peak in the $q_{xy}\text{−}{q}_z$ plane. The inset illustrates the shape of the measured intensity as a function of $q_x$ at $q_z$ smaller than the peak in the bulk diffuse scattering (1), at the peak (2), and larger than the peak (3).

Figure 14

(Color online) Measured intensity as a function of $q_x$ for $q_z=2.0{\text{Å}}^{-1}$ and $2.3{\text{Å}}^{-1}$ (●). The solid and dashed lines indicate, respectively, the theoretically expected shape of the diffuse scattering from the bulk liquid and the specularly reflected beam from the surface.