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The vacancies at the Al, Cu, Ag, and Rh (111) surface are investigated using total-energy and charge-density calculations. For comparison, results of the bulk vacancies are presented as well. In both cases the (v3 x~3)R30' surface unit cell is used. The calculations apply density-functional theory together with the local-density approximation and the O,b initio full-potential linear-mufBntin-orbital method. The results compare well with known experimental data. In addition, the results are discussed in terms of a tight-binding model in the second moment approximation. It is found that among those metals which are studied here, Al has exceptionally small values for the vacancyformation energies in the bulk and at the surface. This is related to the formation of a sp bonding component in the Al bonds on the (111)surface for the case of a periodic vacancy structure with a (V 3 x v 3)R30' surface unit cell.

We calculate the conductance through a Gaussian impurity potential in a quantum wire using the Lippmann-Schwinger equation. The impurity has a decay length d along the propagation direction while it is localized along the transverse direction. In the case of a repulsive Gaussian impurity it is shown that the conductance quantization is strongly affected by the decay length. In particular, increasing d causes gradual suppression of backscattering and smaller contribution of evanescent modes, leading to progressively sharper conductance steps. The dependence of the conductance on the impurity position is also examined. In the case of an attractive Gaussian impurity it is shown that multiple quasibound states are formed due to the finite size of the impurity. By varying the size of the impurity these quasibound states may evolve into highly localized states with greatly enhanced lifetime. It is also shown ͑for a model impurity potential very similar to the Gaussian͒ that the transmission exhibits asymmetric Fano line shape. Under certain circumstances the Fano line shape may appear "inverted" or evolve into a Breit-Wigner dip. We consider also the effects of the cross-sectional shape of the wire on the quantum transmission. It is shown that varying the cross-sectional shape causes shifting of the positions of the conductance steps ͑which is due to the rearrangement of the transverse energy levels͒ and influences the character of conductance quantization.

In this work, we propose an interesting combination of the Atwood machine and Galileo’s inclined plane to study quantitatively kinematics with a smartphone and the phyphox app. For this purpose, we use the optical stopwatch function, based on the photosensor of the smartphone. The choice of phyphox app has some advantages for presenting the concepts of kinematics to lower high school students, as it directly and wirelessly provides analysed data to the screen of a laptop or a desktop computer. Thus, eliminating the need to export the raw data and necessitating the demanding step of data processing by the young pupils.

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