Ga adsorption on the Si(112) surface results in the formation of pseudomorphic Ga atom chains. Compressive strain in these atom chains is relieved via creation of adatom vacancies and their selforganization into meandering vacancy lines. The average spacing between these line defects can be controlled, within limits, by adjusting the chemical potential of the Ga adatoms. We derive a lattice model that quantitatively connects density functional theory (DFT) calculations for perfectly ordered structures with the fluctuating disorder seen in experiment and the experimental control parameter . This hybrid approach of lattice modeling and DFT can be applied to other examples of line defects in heteroepitaxy. DOI: 10.1103/PhysRevLett.99.116102 PACS numbers: 68.35.ÿp, 05.65.+b, 68.37.Ef, 81.16.Dn Many physical, chemical, and biological phenomena are manifestations of self-organization of matter, such as crystal growth, protein folding, or formation of galaxy clusters. Well-known examples of self-organization in advanced materials systems include the formation of magic metal clusters, pyramid quantum dots, quantum dot superlattices in heterostructures [1], and the ordering of atom vacancies into line defects or vacancy line superstructures [2 -4]. For application purposes, it would be essential to control the size, uniformity, and spacing of such nano-objects. This goes against the odds of thermodynamic fluctuations that are especially profound in low-dimensional systems, and the stochastic nature of nucleation and growth.Here, we show that the average spacing between VLs in a monatomic Ga layer on Si(112) can be controlled and varied continuously, within limits, via the chemical potential of the adsorbate species, . Ga vacancies self-organize into a n 1 VL superstructure [ Fig. 1(a)-1(c)], similar to the formation of the well-known n 2 superstructures for Ge on Si(100) [2 -4]. Entropic fluctuations compete with this ordering process, however, resulting in meandering VLs where the average line spacing is fixed by the chemical potential. The meandering amplitude is limited by elastic repulsions between VLs. Such a two-dimensional (2D) interacting vacancy line system has been modeled as a 1D random walker trapped in a harmonic potential representing the collective mean field of all the other VLs [4]. For systems such as Ge on Si(100), this 1D model seems to capture the observed meandering quite well, allowing for a straightforward determination of the kink energy and line repulsion from statistical analysis of fluctuations in Scanning Tunneling Microscopy (STM) images.Step fluctuations on vicinal crystal surfaces have been analyzed in similar fashion, using continuum modeling [5].An interesting question is how the constraints imposed by the discreteness of a lattice modifies the elastic interactions that are driving the self-organization of vacancies, particularly for short VL spacings. Discrete VL spacings conceivably lead to significant vacancy-vacancy correlations that cannot be captured by the usual mean field or ...