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arXiv:cond-mat/0307626v2 [cond-mat.stat-mech] 1 Dec 2005 Step Position Distributions and the Pairwise Einstein Model for Steps on Crystal Surfaces Amber N. Benson,1, 2, ∗Howard L. Richards,2, † and T. L. Einstein3, ‡ 1Department of Physics and Astronomy, Mississippi State, MS 39762-5167 2Department of Physics, Texas A & M University–Commerce, Commerce, TX 75429-3011 3Department of Physics, University of Maryland, College Park, MD 20742-4111 (Dated: November 8, 2018) The Pairwise Einstein Model (PEM) of steps not only justifies the use of the Generalized Wigner Distribution (GWD) for Terrace Width Distributions (TWDs), it also predicts a specific form for the Step Position Distribution (SPD), i.e., the probability density function for the fluctuations of a step about its average position. The predicted form of the SPD is well approximated by a Gaussian with a finite variance. However, the variance of the SPD measured from either real surfaces or Monte Carlo simulations depends on ∆y, the length of step over which it is calculated, with the measured variance diverging in the limit ∆y →∞. As a result, a length scale LW can be defined as the value of ∆y at which the measured and theoretical SPDs agree. Monte Carlo simulations of the terrace-step-kink model indicate that LW ≈14.2ξQ, where ξQ is the correlation length in the direction parallel to the steps, independent of the strength of the step-step repulsion. LW can also be understood as the length over which a single terrace must be sampled for the TWD to bear a “reasonable” resemblence to the GWD. PACS numbers: PACS Number(s): 05.70.Np, 68.55.Jk, 68.35.Ct, 68.35.-p I. INTRODUCTION A key factor determining the equilibrium morphology of a vicinal crystal surface is the interaction between the steps on that surface. In many cases, the elastic and electronic contributions to the step-step interaction take the form V (L) = A L2 , (1) where A determines the strength of the step-step inter- action and L is the distance between steps. Because this is a typical step-step interaction, and because it has the remarkable property of yielding exact solutions to very plausible approximate theories1,2,3, we confine ourselves in this paper to interactions of the form given in Eq. (1). With this restriction, many of the quantities discussed in this paper depend only on a single dimensionless param- eter, ˜A ≡ ˜βA (kBT )2 , (2) where ˜β is the step stiffness, kB is Boltzmann’s constant, and T is the absolute temperature. One of the easiest methods4,5,6 for experimentally de- termining the interaction between steps on a vicinal crys- tal surface is through the observation of the Terrace Width Distribution (TWD). Typically, this has been done by fitting the TWD to a Gaussian, which is a good approximation and justified by the Gruber-Mullins approximation7,8 (analogous to the Einstein model9 of solids) if the steps strongly repel each other. The step- step interaction is then extracted from the variance of the Gaussian. Unfortunately, however, the Gaussian approx- imation is only good for strongly interacting steps, and there are conflicting theories7,8,10,11,12,13,14 regarding the relationship between the step-step interaction and the variance. Over the past decade4,5,6 it has become apparent that the so-called Generalized Wigner Distribution (GWD) provides a much better approximation to the TWD. The GWD exhibits the positive skew observed in TWDs from experiments and simulations, and it is a good fit quan- titatively to TWDs produced from Monte Carlo simula- tions of the terrace-step-kink (TSK) model. More signifi- cantly, the GWD can be justified on the basis of plausible approximations3,15, the most important of which is that the interaction and fluctuations of two adjacent steps are explicitly considered; the Gruber-Mullins approximation only explicitly considers one step. The two steps are kept close to each other by a harmonic well, which approxi- mates the interactions with all other steps. This model is refered to as the Pairwise Einstein Model (PEM). Both the Gruber-Mullins and pairwise Einstein models start by interpreting the steps as world-lines of spinless fermions, with the y-direcion (the average direction of the steps) corresponding to time. This paper considers a different statistical measure of the vicinal surface: the Step Position Distribution (SPD). In Sec. II, we show that the pairwise Einstein model predicts a Gaussian-like distribution for the position of steps. In Sec. III these predictions are shown to compare well with numerical results from simulations of the TSK model, at least for systems of the “right size”. The depen- dence of the SPD on the length of the steps is discussed in Sec. IV; for the purpose of comparison, the dependence of the TWD on the length of steps is likewise discussed in Sec. V. Finally, in Sec. VI we summarize and draw our conclusions.
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2 II. PREDICTIONS FROM THE PAIRWISE EINSTEIN MODEL As was shown in Ref. 3, the Generalized Wigner Dis- tribution can be derived from a phenomenological treat- ment in which only two steps are treated explicitly, the rest contributing a “confinement potential” related to the two-dimensional pressure and compressibility of the sys- tem of steps. We use the usual trick of mapping steps onto the worldlines of one-dimensional spinless fermions, which in this case have the Hamiltonian3,15 H = −1 2 ∂2 ∂x2 1
- ∂2 ∂x2 2
˜A (x2 −x1)2 + ω2 2 �x2 1 + x2 2 . (3) In this dimensionless formulation, we require that ⟨x2 −x1⟩= 1 ; (4) this fixes the value of ω to ω = 2b̺ , (5) where b̺ ≡ ” Γ � ̺+2 2 Γ � ̺+1 2 #2 , (6) and ̺ = 1 + p 1 + 4 ˜A . (7) After a change of variables3,15 to xcm = x1 + x2 2 (8) s = x2 −x1 , (9) this Hamiltonian becomes separable3,15, H = − ∂2 ∂s2 + 1 4 ∂2 ∂x2cm + ˜A s2 + b2 ̺ �s2 + 4x2 cm , (10) and it has the remarkable property that all of the eigen- states are known. The only eigenstate of interest to us at present, however, is the ground state, which can be written3,15 Ψ0,0(s, xcm) = a1/2 ̺ s̺/2 exp −b̺s2 2 × ” 1 2 p πb̺ exp �−4b̺x2 cm
, (11) where a̺ = 2b(̺+1)/2 ̺ Γ[(̺ + 1)/2] (12) is a constant of normalization. The probability of finding the combination a specific combination of relative separa- tion and “center of mass” is, of course, just Ψ2 0,0(s, xcm), which can be rewritten in terms of the original variables x1 and x2: P(x1, x2) = Ψ2 0,0(s, xcm)
a̺ p πb̺ (x2 −x1)̺ exp[−2b̺(x2 1 + x2 2)] ,(13) subject to the constraint x2 ≥x1. We can integrate out all possible values of x2 to find the probability density function for x1: Q1(x1) = Z ∞ x1 P(x1, x2) dx2
a̺ p πb̺ exp(−2b̺x2 1) × Z ∞ x1 (x2 −x1)̺ exp(−2b̺x2 2) dx2 . (14) As should be expected, the mean value of x1 is −1/2 and the mean value of x2 is +1/2, so we define the analytic SPD to be the calculated probability density function for x1 −⟨x1⟩: Q(x) ≡Q1 x + 1 2
a̺ p πb̺ exp ” −2b̺ x + 1 2 2# × Z ∞ x x2 −x + 1 2 ̺ exp(−2b̺x2 2) dx2 .(15) Although Q(x) can only be evaluated numerically (it can be rewritten as a complicated expression involving hypergeometric functions, but this does not seem to be genuinely helpful), it is straightforward, though tedious, to calculate its moments. The two most important are the mean, which is zero by definition, and the variance, which is given by σ2 Q,W = 1 4 ̺ + 2 2b̺ −1 (16) ∼3 8̺−1 . (17) These two moments would be enough to entirely spec- ify the SPD if it were a Gaussian distribution, which it should be approximately; the Gruber-Mullins approxi- mation for the TWD, since it concerns the fluctuations in position of only a single step, can be equally well inter- preted as an approximation for the SPD. In fact, both the coefficient of skewness16 and the kurtosis16 of the theoret- ical SPD vanish in the limit of strong step-step repulsion. The coefficient of skewness is given asymptotically by γ1 ≡⟨(x1 −⟨x1⟩)3⟩ σ3 Q,W ∼− √ 6 18 ̺−1/2 ; (18) note that the coefficient of skewness would have the op- posite sign if it had been defined as ⟨(x2 −⟨x2⟩)3⟩σ−3 Q,W.
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3 The kurtosis, which is the same regardless of which step is considered, is given asymptotically by γ2 ≡⟨(x1 −⟨x1⟩)4⟩ σ4 Q,W −3 ∼1 12̺−2 . (19) The fact that the kurtosis is not exactly zero is not in itself surprising; even within the Gruber-Mullins approx- imation, the Gaussian distribution is only obtained in the limit of large ˜A. The symmetry of our original prob- lem of an infinite number of steps on an infinite vicinal surface, on the other hand, means that the coefficient of skewness, by contrast, must be zero for the original problem. Any given step on the surface can be consid- ered “step 1”, with its downhill neighbor as “step 2”, or it can be considered “step 2”, with its uphill neighbor as “step 1”; calling it one or the other breaks the symmetry and permits a nonzero coefficient of skewness. III. COMPARISON WITH MONTE CARLO SIMULATIONS In order to test the applicability of Eq. (15), we have performed Monte Carlo simulations of the terrace-step- kink (TSK) model and measured the SPD for several values of ˜A. The geometry of the simulated systems was as follows. All simulations were for systems of 30 steps; the length of each of which was Ly = 1000a (where a is the lat- tice constant) in the average direction of the steps (the y-direction in “Maryland notation”). The mean step sep- aration was ⟨L⟩=10a, and periodic boundary conditions were applied. The dynamic used was a local Metropolis update. The temperature was set at kBT = 0.45ǫ, where ǫ is the kink energy; in a previous study, this was approximately the temperature at which TWDs from the restricted TSK model showed the best agreement with the Generalized Wigner Distribution. Each simulation was equilibrated for at least 500 000 Monte Carlo steps per site (MCSS) at the temperature and value of ˜A at which measurements were taken; the initial configurations, however, were not typically straight steps, but steps that had been equili- brated at some other value of ˜A. Data were taken from 1 000 “snapshots,” taken at intervals of 1 000 MCSS. Although the terrace width is always an integer multi- ple of a in the TSK model, the average step position can be any rational number, depending only on the size of the simulation. Since the step position x is always an in- teger, the histogram of positions for any given step need not be symmetric. In order to show concretely what this means, consider a situation in which a Gaussian distribution with mean µ and variance σ2 is binned into a histogram as follows. The weight assigned to each integer k is given by inte- -10 -5 0 5 10 k 0.00 0.05 0.10 0.15 0.20 W(k) µ = +0.452 µ = -0.279 µ = -0.131 FIG. 1: An illustration of problem that can be caused by the variability of the mean step position when the step posi- tion distribution (SPD) is calculted from numerical or exper- imental results. In this example, Gaussian distributions with identical variances (σ2 = 2.52) are binned into histograms by means of Eq. (20). The only differences between the three distributions are the values of µ: circles, µ = 0.452; squares, µ=−0.279; diamonds, µ=−0.131. grating the Gaussian between k−1/2 and k+1/2: W(k) = 1 σ √ 2π Z k+1/2 k−1/2 exp −(x −µ)2 2σ2 dx = 1 2 ( erf k −(1/2) −µ 2σ −erf k + (1/2) −µ 2σ ) . (20) For our example, we choose σ =2.5 and three “random” values of µ between -0.5 and +0.5. The results are shown in Fig. 1. Clearly none of the histograms is completely symmetric, and the differences between them are note- worthy. Something similar can and does happen when the SPDs are calculated from Monte Carlo simulations by binning the positions into histograms. As a result, the statistical uncertainties are considerably larger than they are for the corresponding TWDs, and the SPDs are not perfectly symmetric about their peaks, as can be seen in Figs. 2 and 3. Note the qualitative similarities between the Monte Carlo results (circles) in Figs. 2 and 3 and the values of W(k) for µ= −0.279 (squares) and µ= −0.131 (diamonds) in Fig. 1. This agreement suggests that dur- ing the process of equlibration, the majority of the steps moved slightly to the left (i.e., uphill). In spite of this, the agreement of the SPDs calcu- lated from simulations and the theoretical Q(x) calcu- lated from Eq. (15) is reasonably good. Even more im- pressive is the agreement between Q(x) and the Gaussian with zero mean and variance given by Eq. (16). Although Eqs. (18) and (19) suggest that the Gaussian approxima- tion will be increasingly good as ˜A becomes large, it is clear from the figures that the Gaussian approximation is good for even for ˜A=0.
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4 -1.5 -1 -0.5 0 0.5 1 1.5 x 0.00 0.25 0.50 0.75 1.00 1.25 Q(x) Gaussian Exact Monte Carlo FIG. 2: Comparison of the SPD for ˜A=0 given by Eq. (15) (solid curve) with a histogram SPD from a Monte Carlo sim- ulation (symbols). Also shown is a Gaussian (dotted curve) with a mean of zero and a variance given by Eq. (16). -1.5 -1 -0.5 0 0.5 1 1.5 x 0.00 0.25 0.50 0.75 1.00 1.25 1.50 1.75 Q(x) Gaussian Exact Monte Carlo FIG. 3: Comparison of the SPD for ˜A=8 given by Eq. (15) (solid curve) with a histogram SPD from a Monte Carlo sim- ulation (symbols). Also shown is a Gaussian (dotted curve) with a mean of zero and a variance given by Eq. (16). IV. SCALING OF THE SPD Although the agreement between Eq. (15) and the nu- merical SPDs discussed above is highly suggestive, it is clear that actual step position distributions must depend on the length ∆y of step over which they are averaged. This is best demonstrated by considering the variance of a measured SPD, which is given by σ2 Q(∆y) ≡(∆y)−1 *Z ∆y/2 −∆y/2 [x(y) −x]2dy + , (21) where x ≡(∆y)−1 Z ∆y/2 −∆y/2 x(y)dy . (22) Clearly, σ2 Q(∆y) is closely related to17 gx(∆y) ≡ D [x(∆y) −x(0)]2E , (23) which characterizes the wandering of an individual step17,18,19,20,21. 0 5 10 15 20 A~ 0 50 100 150 200 Correlation Length ξQ ξGM FIG. 4: Comparison of the correlation length calculated from the SPD (ξQ) and from the Gruber-Mullins approximation (ξGM), evaluated numerically for ⟨L⟩= 10 and kBT = 0.45ǫ. Although there is decent agreement for ˜A > 5, ξGM unphys- ically diverges as ˜A →0. In contrast, ξQ remains finite and reasonable for all nonegative values of ˜A. It is tempting to identify x, the average value of x for a particular conformation of a step, with x(0), the value of x at the average y-position. This leads to σ2 Q(∆y) ≈(∆y)−1 Z ∆y/2 −∆y/2 gx(y)dy . (24) For small ∆y, gx ≈c1|∆y|17,18,19,20,21; Eq. (24) implies σ2 Q ≈(c1/2)∆y. Likewise, for large ∆y,17,18,19,20,21 gx(∆y) ≈c2 + c3ln|∆y| (25) and Eq. (24) implies σ2 Q(∆y) σ2 Q,W ≈c4 + c5ln|∆y| . (26) The observation, made in the previous section, that Q(x) is to a very good approximation Gaussian is help- ful towards the calculation of the characteristic length for σ2 Q. In Ref. 8, the “TWD” was calculated within the Gruber-Mullins approximation; because the position of only one step was explicitly taken into account, though, it could equally be considered a SPD. In fact, the Gaus- sian solution is a more appropriate description of a SPD, which is symmetric, than a TWD, which is asymmetric. Substituting the variance of the SPD into Eq. (18) of Ref. 8, we find the correlation length to be ξQ = 2⟨L⟩2 ˜βσ2 Q,W kBT . (27) Figure 4 shows a comparison between ξQ and the corre- lation length from Ref. 8. Scaled by σ2 Q,W and ξQ, σ2 Q(∆y) appears to be inde- pendent of ˜A; although the PEM incorrectly predicts that σ2 Q(∆y) remains finite in the limit ∆y →∞, it neverthe- less provides the correct scaling factors. This is remark- able, since although gx(∆y) shows scaling with temper- ature, it does not exhibit scaling independent of ˜A17.
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5 10 -2 10 -1 10 0 10 1 10 2 ∆y / ξQ 10 -3 10 -2 10 -1 10 0 10 1 σ 2 Q (∆y) / σ 2 Q,W A~ = 0 A~ = 2 A~ = 4 A~ = 6 A~ = 8 power-law fit logarithmic fit FIG. 5: A power-law fit to all the Monte Carlo estimates of σ2 Q(∆y) for 0 ≤˜A ≤8, ∆y < ξQ, indicates an initial growth with an exponent of 0.797 ± 0.006. 10 -1 10 0 10 1 ∆y / ξQ 0.0 0.2 0.4 0.6 0.8 1.0 1.2 σ 2 Q (∆y) / σ 2 Q,W A~ = 0 A~ = 2 A~ = 4 A~ = 6 A~ = 8 power-law fit logarithmic fit FIG. 6: A fit to the Monte Carlo estimates of σ2 Q(∆y) for ˜A=8, 4ξQ <∆y<Ly/2, indicates an asymptotic growth given by σ2 Q(∆y)/σ2 Q,W ≈0.158+0.3175 ln(∆y/ξQ). The length LW, defined by Eq. (28), σ2 Q(LW) ≡σ2 Q,W, is consequently given by LW =(14.183 ± 0.035)ξQ. For ∆y < ξQ, a least-squares fit indicates power-law growth of σ2 Q(∆y) with an exponent of 0.797 ± 0.006 (Fig. 5). Equation (24) predicts power-law growth, but with an exponent of 1. Interestingly, the power-law be- havior of gx(∆y) extends only out to17 ∆y ≈0.1ycoll; since ycoll = ξQ/(π −2) (for ˜A = 0), power-law scaling extends farther for σ2 Q(∆y) than for gx(∆y). For large ∆y, σ2 Q(∆y) follows the logarithmic scaling of Eq. (26). A least-squares fit was performed on the ˜A=8 data, since this has the smallest value of ξQ among the available simulations, and hence the largest avail- able values of ∆y/ξQ. To avoid the crossover from the power-law regime, the fit was restricted to ∆y > 4ξQ; likewise, the fit was limited to ∆y <Ly/2 to limit finite- size effects. The resulting fit, shown in Fig. 6, is in good agreement with data from all values of ˜A except where finite-size effects become evident. The fitted parameters, c4 =0.1578 ± 0.0004 and c5 =0.3175 ± 0.0002, allow us to 0 5 10 15 20 25 L 0.00 0.05 0.10 0.15 P(L) GWD ∆y / ξQ = 4 ∆y / ξQ = 8 FIG. 7: Terrace Width Distributions calculated between a single pair of neighboring steps depend on ∆y, the length of step over which the distribution is averaged. Although in the limit ∆y →∞the TWD converges to the Generalized Wigner Distribution (to a very good approximation), when ∆y/ξQ is small the TWD is dominated by noise. These results are typical for ˜A=0. find a “Wigner length”, LW, defined by σ2 Q(LW) ≡σ2 Q,W , (28) to be LW = (14.183 ± 0.035)ξQ . (29) V. SCALING OF THE TWD It seems somewhat surprising that so many correlation lengths are necessary for the PEM to agree with the ob- served variance. To better understand this, it is helpful to consider the corresponding scaling of the TWD when it is calculated under the same restrictions as σ2 Q(∆y). Specifically, the TWD must be averaged over a given length, ∆y, of a single pair of adjacent steps in a sin- gle “snapshot”. This is very different from the analysis presented in Ref. 22, where, as in other previous work, averages were made over the entire length Ly of the sim- ulations, over all pairs of neighboring steps, and over all “snapshots”. Remembering that the y-direction corre- sponds to time in the worldline interpretation of steps, the averages we are about to calculate correspond to time averages in statistical mechanics, whereas the pre- vious averages have combined the time average with two kinds of ensemble average (over different pairs and dif- ferent “snapshots”). Only in the limits of long times and large ensembles should one expect these averages to be identical23. In the language of Ref. 17, ξQ is approximately the dis- tance between “collisions” of neighboring steps. In order to sample the distribution of terrace widths adequately, a step must “collide” several times with its neighbors. This is shown clearly in Figs. 7 and 8. For the simula- tion parameters given in Sec. III, LW > Ly for ˜A = 0; consequently, the TWDs shown in Fig. 7 are dominated
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6 0 5 10 15 20 L 0.00 0.05 0.10 0.15 0.20 0.25 P(L) GWD ∆y / ξQ = 4 ∆y / ξQ = 8 ∆y / ξQ = 12 ∆y / ξQ = 16 FIG. 8: For small ∆y/ξQ, typical TWDs are dominated by noise, but for ∆y > LW ≈14.2ξQ, the Generalized Wigner Distribution dominates. These results are for ˜A=8. 0 5 10 15 20 25 30 ∆y / ξQ 10 -2 10 -1 10 0 1 - σ 2 P,1(∆y) / σ 2 P,W A~ = 0 A~ = 8 FIG. 9: The variance of the TWD also approximately scales with ∆y/ξQ. At ∆y = LW ≈14.2ξQ, the average variance of TWDs generated from single pairs of neighboring steps is within about %10 of the variance given by the GWD. by noise. For ˜A = 8, on the other hand, LW < Ly, and we are able to see in Fig. 8 the crossover into the regime where the GWD, not noise, is dominant. The role of step collisions in equilibrating the TWD can also be seen from σ2 P (∆y), which is the variance of TWDs calculated from a length ∆y of neighboring steps, aver- aged over all pairs of neighboring steps, starting points, and “snapshots”. Like σ2 Q(∆y), lim∆y→0 σ2 P (∆y) = 0; unlike σ2 Q(∆y), lim∆y→∞σ2 P (∆y) is finite and given approximately22 by the PEM result24, σ2 P,W = ̺ + 1 2b̺ −1 . (30) This suggests plotting 1 −σ2 P (∆y)/σ2 P,W vs. ∆y/ξQ to determine whether the approach to the asymptotic limit is exponential or power-law. As shown in Fig. 9, the scaling appear to be neither a simple power law nor a simple exponential decay, but it is difficult to be certain since the TWD does not converge exactly to the GWD even in the limit ∆y →∞. Also, the scaling does not appear to be quite as precise as in Figs. 5 and 6. this is not surprising, since the correlation length ξP for the TWD is not identical to ξQ. More significantly, Fig. 9 indicates that σ2 P (LW) is within about %10 of the approximate asymptotic value, σ2 P,W. This is a very plausible threshold for statistics from the TWD. VI. CONCLUSION For the common case in which steps on a vicinal crys- tal surface interact according to Eq. (1), the generalized Wigner distribution (GWD) has been shown previously to be in excellent agreement with the terrace width dis- tribution (TWD). To fully appreciate the model which predicts the GWD, though, it is necessary to examine its predictions for other statistical properties and how well these predictions agree with actual measurements. This article has made such a comparison between the pre- dicted and measured step position distribution (SPD). The results demonstrate both the strength and limita- tions of the Pairwise Einstein Model (PEM). Since the SPD is so well approximated by a Gaussian, it is tempting to compare it directly with Gaussian theo- ries of the TWD. As can be seen in Table 1, in the limit of strongly interacting steps the variance of the SPD is slightly larger than that of the Gruber-Mullins approx- imation, but less than the variance of the TWD given by either the “Saclay” or “modified Grenoble” approxi- mations. This is reasonable; unlike the Gruber-Mullins Hamiltonian, Eq. (3) does not have fixed walls, so the steps can experience larger fluctuations. In spite of this, since the Gruber-Mullins approximation allows only one step to move, it can be regarded equally as an approxima- tion for the TWD or for the SPD. The fact that the SPD is smaller than the other approximations of the TWD is apparently due to the fact that correlations between fluctuations of adjacent steps are to some degree taken into account in all these approximations, so that they are specifically approximations for the TWD, not the SPD. Because the PEM confines both steps within a har- monic well, the theoretical asymptotic variance of the SPD must be finite. However, the vicinal surface is rough, and the variance of the SPD diverges logarithmically with the length of step ∆y from which it is calculated. At some finite length, LW, the prediction of the PEM is accurate. As Fig. 6 shows, LW ≈14.2ξQ. That so many “collisions” between neighboring steps are needed to adequately sam- ple the statistics resulting from their interactions is sup- ported by obeservations of the dependence of the TWD on ∆y, as shown in Figs. 7, 8, and 9. In principle, the SPD could be used to determine ˜A. However, the SPD is strongly affected by the random position of the average step position, and it depends far too strongly on ∆y. The TWD has neither of these two restrictions and is a more practical alternative for deter- mining ˜A. Instead, the utility of the SPD lies in clarifying
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7 TABLE I: Asymptotic variances in the limit of strong step- step repulsion. The Gaussian-like approximation for the step position distribution (SPD) given by Eq. (15) is compared with selected approximations for the TWD. Except for the Generalized Wigner Distribution, all approximate TWDs are Gaussian approximations. Note also that our approximation for the SPD and the Generalized Wigner Distribution are both independent of the number of interacting steps, whereas the Gaussian approximations are not. (See also Table 1 of Ref.6.) Distribution Reference Asymptotic Variance SPD Eq. (16) 0.375̺−1 Generalized Wigner 24,25,26 0.5̺−1 Gruber Mullins (all steps) 7 0.278̺−1 ” (nearest neighbors) ” 0.289̺−1 Modified Grenoble (all steps) 10,11,24 0.495̺−1 ” (nearest neighbors) ” 0.520̺−1 Saclay (all steps) 12,13,14 0.405̺−1 the Pairwise Einstein Model. The finite length LW in- troduced in this work emerges more naturally than the finite length V that was introduced in Refs. 3 and 15, but the two are obviously related. Both help describe a short-lived dynamic constraint that is roughly analagous to a reptation tube27 in polymer physics. Naturally, the remarkable success of the Pairwise Ein- stein Model suggests that a Debye model9 might lead to even better descriptions of vincinal crystal surfaces. Pre- liminary results28 from such studies correctly show that gx(∆y) diverges logarithmically. Acknowledgment This research was supported by an award from Re- search Corporation. The authors also thank Jeremy Yancey and April St. John for critical readings. ∗amber benson@yahoo.com † Corresponding author: Howard Richards@tamu-commerce.edu; http://faculty.tamu-commerce.edu/HRichards/index.html ‡ einstein@umd.edu; http://www2.physics.umd.edu/~einstein/ 1 F. Calogero, J. Math. Phys. 10, 2191, 2197 (1969). 2 B. Sutherland, J. Math. Phys. 12, 246 (1971). 3 H. L. Richards and T. L. Einstein, Phys. Rev. E 72, 016124 (2005). 4 M. Giesen and T. L. Einstein, Surf. Sci. 449, 191 (2000). 5 H. L. Richards, S. D. Cohen, T. L. Einstein, and M. Giesen, Surf. Sci. 453, 59 (2000). 6 T. L. Einstein, H. L. Richards, S. D. Cohen, and O. Pierre- Louis, Surf. Sci. 493, 460 (2001). 7 E. E. Gruber and W. W. Mullins, J. Phys. Chem. Solids 28, 875 (1967). 8 N. C. Bartelt, T. L. Einstein, and E. D. Williams, Surf. Sci. 240, L591 (1990). 9 See e.g. N. W. Ashcroft and N. D. Mermin, Solid State Physics (Saunders College, Philadelphia, 1976) pp. 457– 463. 10 O. Pierre-Louis and C. Misbah, Phys. Rev. B 58, 2259 (1998); 58 2276 (1998). 11 T. Ihle, C. Misbah, and O. Pierre-Louis, Phys. Rev. B 58, 2289 (1998). 12 L. Masson, L. Barbier, J. Cousty, and B. Salanon, Surf. Sci. 317, L1115 (1994). 13 L. Barbier, L. Masson, J. Cousty, and B. Salanon, Surf. Sci. 345, 197 (1996). 14 E. L. Goff, L. Barbier, L. Masson, and B. Salanon, Surf. Sci. 432, 139 (1999). 15 J. A. Yancey, H. L. Richards, and T. L. Einstein, Surf. Sci. (in press). 16 B. P. Roe, Probability and Statistics in Experimental Physics, 2nd ed. (Springer, New York, 2001), pp. 7,8. 17 N. C. Bartelt, T. L. Einstein, and E. D. Williams, Surf. Sci. 276 308 (1992). 18 J. Villain and P. Bak, J. Phys. (Paris) 42 657 (1981). 19 J. Villain, D. R. Grempel, and J. Lapujoulade, J. Phys. F 15 809 (1985). 20 W. F. Saam, Phys. Rev. Lett. 62 2636 (1989). 21 N. C. Bartelt, T. L. Einstein, and E. D. Williams, Surf. Sci. 224 149 (1991). 22 H. Gebremariam, S. D. Cohen, H. L. Richards, and T. L. Einstein, Phys. Rev. B 69 125404 (2004). 23 See e.g. R. K. Pathria, Statistical Mechanics 2nd Ed. (Butterworth-Heinemann, Boston, 1996) pp. 30–40. 24 T. L. Einstein and O. Pierre-Louis, Surf. Sci. 424, L299 (1999). 25 M. L. Mehta, Random Matrices, 2nd ed. (Academic, New York, 1991). 26 F. Haake, Quantum Signatures of Chaos (Springer, Berlin, 1991). 27 M. Doi and S. F. Edwards, The Theory of Polymer Dy- namics (Clarendon, Oxford, 1986) pp. 189–217. 28 C. A. Greene and H. L. Richards, in preparation.
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