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By using a low temperature scanning tunneling microscope we have probed the superconducting energy gap of epitaxially grown Pb films as a function of the layer thickness in an ultrathin regime (5-18 ML). The layer-dependent energy gap and transition temperature (Tc) show persistent quantum oscillations down to the lowest thickness without any sign of suppression. Moreover, by comparison with the quantum-well states measured above Tc and the theoretical calculations, we found that the Tc oscillation correlates directly with the density of states oscillation at E(F) . The oscillation is manifested by the phase matching of the Fermi wavelength and the layer thickness, resulting in a bilayer periodicity modulated by a longer wavelength quantum beat.

Process monitoring and fault diagnosis are crucial for efficient and optimal operation of a chemical plant. This paper proposes a reconstruction-based fault identification approach using a combined index for multidimensional fault reconstruction and identification. Fault detection is conducted using a new index that combines the squared prediction error (SPE) and T 2 . Necessary and sufficient conditions for fault detectability are derived. The combined index is used to reconstruct the fault along a given fault direction. Faults are identified by assuming that each fault in a candidate fault set is the true fault and comparing the reconstructed indices with the control limits. Fault reconstructability and identifiability on the basis of the combined index are discussed. A new method to extract fault directions from historical fault data is proposed. The dimension of the fault is determined on the basis of the fault detection indices after fault reconstruction. Several simulation examples and one practical case are presented. The method proposed here is compared with two existing methods in the literature for the identification single-sensor and multiple-sensor faults. We analyze the reasons that the other two methods lead to erroneous identification results. Finally, the proposed approach is applied to a rapid thermal annealing process for fault diagnosis. Fault subspaces of several typical process faults are extracted from the data and then used to identify new faults.