The asymmetric or symmetric nature of the dimer adatom arrangement on reconstructed Si and Ge(001) surfaces is still controversial. There has not been a direct quantitative determination of the asymmetry. Using x-ray standing waves we have directly measured the asymmetry of Ge dimers on the Si(001) surface as 0.55 ±0.02 A. Our results unequivocally rule out the possibility of symmetric dimers. At elevated temperatures the dimer asymmetry decreases, raising the possibility of a high temperature asymmetric to symmetric phase transition on the (001) surfaces. PACS numbers: 68.55.Ce, 68.35.Bs, 81.10.Bk Since the earliest low-energy electron diffraction [l] (LEED) studies the detailed nature of the dimer adatom arrangement on the technologically important Si (001) surface has been studied intensively by all manner of surface probes [2][3][4][5] and techniques [6,7]. These studies have in turn stimulated sophisticated theoretical efforts [8-10] using ab initio total energy and molecular dynamics approaches [11,12] to locate the surface dimer positions and subsurface distortions. In spite of these efforts, ambiguities persist regarding the symmetric or asymmetric nature of surface dimers, particularly at room temperature.Scanning-tunneling-microscopy (STM) observations provide direct, though qualitative, information on the buckling of dimers on the Si(00l) surface. In earlier work, room temperature results showed [7] that roughly half the dimers were asymmetric, that is, the two dimer atoms were at different heights with respect to the surface. A recent low temperature STM study by Wolkow [13] shows that at room temperature asymmetric dimers are present near defects while a large part of the surface is covered with apparently symmetric dimers (both atoms at the same height). On cooling to low temperature (120 K) an increase of dimers in the asymmetric configuration is observed. It has been suggested that any symmetric appearance of dimers [13] is due to a rapid switching [14] of the dimer orientation in the up and down direction.On the experimental side, two questions need clarification: (a) What is the magnitude of the dimer asymmetry, if any, and (b) are the dimers at room temperature asymmetric or symmetric? In this Letter we describe a direct quantitative x-ray standing wave (XSW) measurement of the surface dimer displacement and asymmetry on the well-known Si(001)/Ge (2x1) reconstructed surface [15] at room temperature and, to our knowledge, we present the first characterization of the dimers at high temperatures. With suitable preparation, a monolayer of Ge atoms deposited by molecular beam epitaxy at high temperature onto a clean UHV Si(001) surface substitutes for the topmost Si adatoms, forming Ge dimers [16]. Although a Ge-substituted dimer surface is not a replica of either the bare Si(001) or Ge(001) surfaces, theoretical calculations [8,9,11,17] suggest that the adatom stability of both Si and Ge(001) surfaces is similar. Substituting Ge for the Si dimer atoms at the (001) surface does not perturb the obs...