We describe x-ray-scattering studies of the orientational epitaxy displayed by the hexagonally reconstructed Pt(OOl) surface. At 1685 K, the reconstructed surface layer undergoes a continuous rotational transformation with respect to the underlying bulk and exhibits mean-field behavior. A second, discontinuous rotational transformation is observed at 1580 K. These results are inconsistent with the predictions of current theories and lead us to propose a mean-field description of rotational transformations.PACS numbers: 68.35.Rh, 64.70.Dv An important aspect of the growth of one crystal upon another involves their relative orientation: How are the high-symmetry directions of the ad-lattice and of the substrate lattice aligned? Our present understanding of this question has been guided by the calculations of Novaco and McTague [1], who explicitly considered incommensurate monolayers of rare gases adsorbed on a rigid, basal plane substrate of graphite. They showed that, even though there is no translational registry at the interface, there may exist a preferred orientation. This is called orientational epitaxy. Their calculations further revealed that the high-symmetry directions of an incommensurate overlayer may be rotated away from the high-symmetry directions of the substrate and that the rotation angle is determined by the incommensurability of the overlayer and by its phonon dispersion relations. In many instances, most notably for the rare gases on graphite [2] and for alkali metals on transition-metal surfaces [3], the predictions of Novaco and McTague describe the orientational epitaxy of an adsorbed layer. Other systems for which the rotational epitaxy has been characterized include layers of alkali metals intercalated into graphite [4], in situ deposited monolayers of Pb on Ag(lll) and Au(lll) electrodes [5], crystallites of Pb deposited on Si(l 11) and Ge(lll) substrates [6], and clean Au (001) surfaces held in vacuum [7] and under electrochemical conditions [8].In this paper, we describe the rotational behavior exhibited by the clean (001) surface of Pt between 300 and 1820 K. We have found that between 7 = 1685 and 1820 K, the surface reconstructs to form a buckled, hexagonal monolayer [9] on top of the bulk planes of square symmetry lying immediately beneath. The top layer is incommensurate in both directions within the surface plane, and the average area per atom is contracted by 8% relative to its value in a bulk (111) plane. In this phase, a hexagonal high-symmetry direction is aligned with the cubic (1,1,0) direction. However, as the temperature is reduced below T c = 1685 K, there is a structural transformation in which domains of the hexagonal overlayer rotate continuously to angles ±0.9° away from the bulk (1,1,0) direction. Remarkably, the rotational transformation exhibits mean-field behavior: That is, the dependence of the rotation angle on temperature follows a onehalf power law versus reduced temperature e = (T c -T)/ T c . At 1580 K, there is a second, discontinuous transformation, lea...