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The isotope effect E °D a ͞D b 2 1 ¢ ͑͞ p m b ͞m a 2 1͒ of cobalt diffusion in the deeply supercooled melt of the metallic alloy Zr 46.7 Ti 8.3 Cu 7.5 Ni 10 Be 27.5 has been measured employing the radiotracers 57 Co and 60 Co. The isotope effect is very small, E 0.09 6 0.03, and exhibits no significant temperature dependence in a range up to 120 K above the calorimetric glass transition temperature T g , encompassing almost 3 orders of magnitude in the diffusivity. This result suggests that long-range diffusion in the deeply supercooled melt is not mediated by viscous flow but rather proceeds by collective hopping processes involving about ten atoms. [S0031-9007(98)06275-9] PACS numbers: 66.10. Cb, 66.30.Fq, 64.70.Dv Atomic transport in liquids and glasses has been the subject of many theoretical and experimental investigations, particularly in connection with the glass transition [1,2]. Diffusion in ordinary liquids at high temperatures is well understood. In this hydrodynamic regime all atoms contribute continuously to the mean square atomic displacement, and diffusion takes place via viscous flow, as described by the Stokes-Einstein relation [3]. Microscopically, transport in the hydrodynamic regime is governed by uncorrelated binary collisions of atoms. Kinetic theories for a simple liquid [3,4] predict the following mass and temperature dependence of the diffusivity D:where m is the atomic mass and n is close to 2 according to molecular dynamics simulations [1] and experiments [5]. Upon supercooling a liquid or melt the viscosity increases markedly because, due to the increase in density, atoms are more and more trapped in their nearest-neighbor "cages" for times much longer than the vibration time.According to the mode coupling theory [6] this cage effect causes viscous flow to freeze in at a critical temperature T c . Below T c , which is typically some 20% above the caloric glass transition temperature T g [7], long-range diffusion in the supercooled liquid is expected to occur only via thermally activated hopping processes. Molecular dynamics simulations have shown the transition from viscous flow at high temperatures to hopping in the glassy state [8][9][10]. The coexistence of both processes was observed in a certain temperature range in the supercooled liquid state. Moreover, computer simulations as well as neutron scattering [11] have confirmed the existence of a critical temperature above T g , where the decay of density correlations slows down drastically. Whereas generally hopping in crystalline solids is a single-atom jump process [12], recent extensions of the mode coupling theory to the glassy state envision hop-ping in glasses as a highly cooperative medium-assisted process [13]. Highly collective hopping processes have indeed been observed in molecular dynamics simulations [10,14,15]. These simulations reveal chainlike displacements involving some ten atoms, which are suggested to be closely related to the well known low frequency excitations in glasses [14]. While, depending on the alloy ...

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A simple (2+1) dimensional discrete model is introduced to study the evolution of solid surface morphologies during ion beam sputtering. The model is based on the same assumptions about the erosion process as the existing analytic theories. Due to its simple structure, simulations of the model can be performed on time scales, where effects beyond the linearized theory become important. Whereas for short times we observe the formation of ripple structures in accordance with the linearized theory, we find a roughening surface for intermediate times. The long time behavior of the model strongly depends on the surface relaxation mechanism.Keywords (PACS-codes): 68.35. 79.20.Rf, 82.20.Wt Introduction During the last years, two features of surface morphologies created by ion beam sputtering attracted particular attention: ripple structures on sub-micrometer length scales and self-affine, rough surfaces [1].The formation of periodic ripple structures has been observed experimentally in amorphous materials [2], metallic crystals [3,4] and semiconductors amorphized by the ion beam [5,6]. Ripples are typically oriented perpendicular to the projection of the ion beam in the surface plane for small angles of incidence Θ (relative to the surface normal), whereas for larger angles Θ, the observed ripple pattern is rotated by 90• . Surfaces eroded by ion bombardment may also exhibit self-affine properties [7]. With increasing ion fluence, a crossover from ripple structures to self-affine, rough surfaces has recently been observed experimentally [4].Our present understanding of these features is based upon the work of Bradley and Harper (BH) [8], who found that Sigmund's sputtering theory [9] implies a curvature dependence of the sputtering yield. Based on BH a continuum theory of surface evolution by sputter erosion was formulated as an anisotropic Kuramoto-Sivashinsky (KS) equation [10] with additive noise [11,12,13]. However, there are strong indications from experiment [3,14] that surface relaxation processes, are also important during pattern formation. Such processes have not yet been adequately included in existing theories. Our results will show that the long-time behavior of patterns depends crucially on details of surface relaxation.To investigate the analytic theory beyond the linearized regime, numerical integrations of the KS equation have been performed [15] [16], which uncovered two markedly different long-time regimes, depending on the signs of the non-linear couplings.Computer simulations may be helpful in clarifying both the role of surface relaxation and of non-linear effects. Two types of simulations have been performed

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