The results of an x-ray-scattering study of the (3×1)-to-disordered phase transformation of the Si(113) surface are reported. A continuous commensurate-solid to incommensurate-fluid transformation at ${\mathit{T}}_{\mathit{c}}$=950±40 K is observed. At the transformation, the reconstructed layer becomes unaxially incommensurate along the cubic (11¯0) direction (x direction). It remains commensurate along the (332¯) direction (y direction). Critical scattering shows power-law behavior over nearly two decades of reduced temperature [t=(T-${\mathit{T}}_{\mathit{c}}$)/${\mathit{T}}_{\mathit{c}}$] with exponents β¯=0.66±0.05 for the incommensurability (ε), ${\nu }_{\mathit{x}}$=0.65±0.07 for the inverse correlation length in the incommensurate direction (${\mathrm{\kappa }}_{\mathit{x}}$), ${\nu }_{\mathit{y}}$=1.06±0.07 for the inverse correlation length in the commensurate direction (${\mathrm{\kappa }}_{\mathit{y}}$), and γ=1.56±0.13 for the susceptibility (χ). Below ${\mathit{T}}_{\mathit{c}}$ the variation of the square of the order parameter, proportional to the peak intensity at the commensurate position (${\mathit{I}}_0$), varies with an exponent 2β=0.22±0.04.

It is noteworthy that the correlation lengths in the disordered phase scale anisotropically, that is, ${\nu }_{\mathit{x}}$≠${\nu }_{\mathit{y}}$, and that the collected exponents do not conform to those of any previously known universality class. In addition to the critical exponents of the transformation, two universal constants have been measured. The ratio of the incommensurability and the inverse correlation length along the incommensurate direction in the disordered phase is found to be independent of temerature, i.e., β¯=${\nu }_{\mathit{x}}$, consistent with predictions for a two-dimensional chiral melting universality class, and to have the value ${\mathit{w}}_0$=1.6±0.2. Also, the combination ${\mathit{R}}_{\mathit{s}}$=χ${\mathrm{\kappa }}_{\mathit{x}}$${\mathrm{\kappa }}_{\mathit{y}}$/${\mathit{I}}_0$${\mathit{V}}_{\mathit{r}}$, where ${\mathit{V}}_{\mathit{r}}$ is the two-dimensional resolution volume, is independent of the reduced temperature, consistent with the derived hyperscaling relationship ${\nu }_{\mathit{x}}$+${\nu }_{\mathit{y}}$=γ+2β. According to the hypothesis of two-scale-factor universality, ${\mathit{R}}_{\mathit{s}}$ is a universal constant, which we find takes the value ${\mathit{R}}_{\mathit{s}}$=0.07±0.03.

• Received 23 August 1993

DOI:https://doi.org/10.1103/PhysRevB.49.2691