Anharmonic Thermal Vibrations Observed by Surface X-Ray-Diffraction for Cs/Cu(001)
Surface Science 333: 1422-1429
Surface X-ray diffraction measurements at 190 and 303 K have been used to analyse the thermal disorder of an uniaxial incommensurate quasi-hexagonal Cs-monolayer on Cu(001). Within the harmonic vibration model a Debye-like thermal behaviour is found for the mean square vibration amplitudes U-ii (i = 1,2) of the Cs-atoms along the  (i = 1) and the  (i = 2) direction, where the adsorbate structure is commensurate and incommensurate with the substrate, respectively. At 190 K we determine U-11 = 0.10(2) Angstrom(2) and U-22 = 0.16(2) Angstrom(2), at 303 K U-11 = 0.18(2) Angstrom(2) and U-22 = 0.29(3) Angstrom(2). From this temperature dependence Debye temperatures of theta() = 45 K and theta() = 35 K are derived which are comparable with the bulk Cs-Debye temperature of 38 K. A significant better agreement between calculated and observed structure factors is obtained by taking account of anharmonic contributions to the thermal disorder using the Gram-Charlier series expansion of the harmonic (Gaussian) probability density function. Besides a larger width of the probability density function along  and  the derived anharmonic probability density function exhibits a stronger anisotropy as compared to its harmonic counterpart. The shape of the anharmonic probability density function can directly be correlated to the geometry of the Cs-overlayer structure. At 190 K this is most evident by a small width of the anharmonic probability density along , where the Cs-atoms are arranged in a densely packed chain. At 303 K the anisotropy is less pronounced indicating isotropisation close to the break down of the structure at 328 K.