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Dynamical Scaling Implications of Ferrari, Prähofer, and Spohn’s Remarkable Spatial Scaling Results for Facet-Edge Fluctuations

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Abstract

Spurred by theoretical predictions from Ferrari et al. (Phys Rev E 69:035102(R), 2004), we rederived and extended their result heuristically. With experimental colleagues we used STM line scans to corroborate their prediction that the fluctuations of the step bounding a facet exhibit scaling properties distinct from those of isolated steps or steps on vicinal surfaces. The correlation functions was shown to go as \(t^{0.15(3)}\), decidedly different from the \(t^{0.26(2)}\) behavior for fluctuations of isolated steps.

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Notes

  1. This paper encountered unfortunate refereeing difficulties regarding expanded expositions of some material in the three refereed publications [21, 22, 27]. Since a great deal of time has already been spent on this project, with much more likely needed to resolve disagreements, and because our primary objective is to highlight in this celebratory volume—which will soon be sent to press—the importance of Prof. Spohn’s insights in our work, we have followed the editor’s directive to delete sections 3.2 and 5 from Ref. [26]. Section 5 describes a toy model; it was included in Ref. [26] because it had been mentioned in Refs. [21, 23, 27] rather than for its import. On the other hand, Section 3.2 contains scaling relations and critical exponents for curved geometries and asymmetric potentials that may be of interest, especially to those who do not dismiss results making use of [28]. Curious readers are invited to view the deleted material on the arXiv [26] and judge for themselves.

  2. Physically, steps–in contrast to fermions–actually can touch, just not cross, leading to a finite-size correction to the standard fermion results, see Ref. [29].

  3. In Ref. [15] \(A\) is defined as \(b_\infty ^{\prime \prime }\); writing \(\mathcal A\) = (1/2)A simplifies Eq. (6).

  4. Direct experimental observation, e.g. of the spatial correlation function on a quenched crystallite, would be needed to obtain \(\alpha \) for facet-edge fluctuations.

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Acknowledgments

Work at University of Maryland has been supported by the UMD-NSF MRSEC under grant DMR 05-20471; TLE is now supported partially by NSF-CHE 07-50334 and 13-05892. Much of this paper is based on extensive collaboration with the experimental surface physics group at UMD, led by Ellen D. Williams until 2010, with ongoing guidance by Janice Reutt-Robey and William G. Cullen, in particular with Masashi Degawa, whose dissertation research accounts for much of the content of this paper. We also benefited from interactions with theory postdoc Ferenc Szalma and students Hailu Gebremarian and Timothy J. Stasevich.

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Correspondence to T. L. Einstein.

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To Prof. Dr. Herbert Spohn, with appreciation.

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Einstein, T.L., Pimpinelli, A. Dynamical Scaling Implications of Ferrari, Prähofer, and Spohn’s Remarkable Spatial Scaling Results for Facet-Edge Fluctuations. J Stat Phys 155, 1178–1190 (2014). https://doi.org/10.1007/s10955-014-0981-3

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