www.actamat-journals.com

Acta Materialia 54 (2006) 1957–1963

Nearest-neighbor distributions in thin films, sheets, and plates

A. Tewari, A.M. Gokhale *

School of Materials Science and Engineering, Georgia Institute of Technology, 771 Ferst Drive, E. Love Building, Atlanta, GA 30332-0245, USA

Received 13 March 2005; received in revised form 11 August 2005; accepted 14 December 2005Available online 17 February 2006

Abstract

Uniform random spatial distance distribution of point particles in thin films, sheets, and plates is modeled and an exact analyticalexpression for the nearest-neighbor distance distribution of such point particles is derived. The results can be applied to model spatialdistributions of nano-clusters in thin films as well as for modeling spatial arrangements of brittle inclusions in sheets and plates. It isshown that the nearest-neighbor distance distribution is not only a function of the number density of point particles but also a functionof film/sheet thickness and the distance of point particles of interest from the film/sheet surface. The overall mean nearest-neighbor dis-tance (averaged over all distances from the surface of a film/sheet) is also a function of film/sheet thickness such that for films thinnerthan a certain critical value, the mean approaches the classic two-dimensional planar solution; and for films thicker than another criticalvalue, the mean approaches the classic three-dimensional bulk solution. The critical thickness for approach to the three-dimensional bulksolution is very small, and consequently many thin films lie above this value and exhibit mean nearest-neighbor distances closer to thosepredicted by the classic three-dimensional bulk solution.� 2006 Acta Materialia Inc. Published by Elsevier Ltd. All rights reserved.

Keywords: Microstructure; Modeling; Thin films; Nearest-neighbor distance

1. Introduction

The geometry of many material products/components issuch that one of their dimensions (thickness) is significantlysmaller that the other two. Such product shapes includethin films, coatings, sheets, and plates. In numerous cases,the length scales of some of the microstructural featurespresent in such products are also of the order of the thick-ness of the component. The spatial arrangement of suchnano- and microstructural features in thin films, sheets,and plates is an important facet of the microstructuralgeometry that affects their properties and performance. Ingeneral, the nearest-neighbor distribution (NND) is animportant descriptor of the spatial arrangement of featuresin microstructures [1–16]. In thin films, the average nearest-neighbor distance of dopants has been correlated with theoptical and magnetic responses of the films [17,18]. Micro-structural damage evolution in bulk materials depends on

1359-6454/$30.00 � 2006 Acta Materialia Inc. Published by Elsevier Ltd. All

doi:10.1016/j.actamat.2005.12.017

* Corresponding author. Tel.: +1 404 894 2887; fax: +1 404 894 9140.E-mail address: arun.gokhale@mse.gatech.edu (A.M. Gokhale).

the nearest-neighbor distances between the inclusions[19]. Damage evolution in thin films and sheets is alsoexpected to depend on the NNDs of the microstructuralfeatures such as inclusions and precipitates present in suchdomains. Therefore, modeling of NNDs in thin films,sheets, and plates is of interest.

It is important to recognize that the spatial microstruc-tural domain of thin films/sheets is neither strictly two-dimensional (i.e., planar) nor three-dimensional (such asthat in bulk materials). Thin films/sheets have finite thick-ness, and consequently they have a finite extent of micro-structural domain along the thickness direction and apractically infinite domain in the other two directions.The descriptors of the spatial arrangement of particle cen-ters (such as NND) involve distances (and therefore lengthscales) ranging from zero to infinity, and consequently formodeling the spatial order (or disorder) in microstructurescontained in films, sheets, and plates, the microstructuraldomain cannot be assumed strictly as two- (2D) or three-dimensional (3D). It is anticipated that thin films wouldbehave as 2D planar structures as well 3D bulk materials,

rights reserved.

i ii iii

Δ

Δ/2+x

x

Δ/2-x O

(a)

(b)

Fig. 1. (a) Schematic of an infinite microstructural domain bound by finitethickness D in one direction. (b) Three distinct radial distance domains fora sphere inside the film: (i) the sphere is completely inside the film; (ii) thesphere is cut by one edge of the film; and (iii) the sphere is cut from boththe edges of the film.

1958 A. Tewari, A.M. Gokhale / Acta Materialia 54 (2006) 1957–1963

depending on their thickness: 2D when the thickness tendsto zero and 3D when the thickness is very large, with atransitional behavior for intermediate thickness values.Therefore, there is a need to develop a theoretical frame-work for the representation of microstructures in suchmicrostructural domains. The objective of this paper is topresent a rigorous derivation of the NND function of ran-domly distributed point particles where the microstructuraldomain is limited along one direction (e.g., films, sheets,and plates); to determine the conditions under which theclassic 3D and 2D solutions are applicable; and to presenta theoretical solution in the length-scale regimes where nei-ther the 2D nor the 3D solution is applicable. The NND ofuniform random points in a thin film can serve as a baselinefor a comparison of the experimental data on NNDs inthin films and can be used to gauge the deviation of spatialarrangements of microstructural features in thin films fromspatial randomness. As the NND for randomly distributedpoint particles is a function of film thickness (to be shownlater in the discussion), it can also serve as a useful gaugefor determining when to consider a film as bulk and whenas planar structure. The results can also be used to quantifythe edge effects in experimental determination of NND. Abrief background on the classic solution of NND functionsfor 3D microstructures of infinite extent and for 2D micro-structures in a plane is presented in the next section. This isfollowed by the theoretical development and derivation ofNND in thin films and its consequences and a discussion ofthe results.

2. Background

For a microstructure where the particles or inclusionscan be modeled as a set of points having uniform randomspatial distribution in 3D space of an infinite extent alongall directions (i.e., no edges and edge effects), the nearest-neighbor distance probability density function W3 is givenas follows [21]:

W3 ¼ 4pr2NV exp ð�4pr3NV =3Þ ð1Þwhere NV is the mean number of point particles per unitvolume (i.e., the intensity of the Poisson process). Thefunction W3(r) is the probability density function such thatW3(r)dr is the probability of having a nearest neighbor inthe distance range r and (r + dr). Therefore, the dimensionsof the probability density W3(r) are the reciprocal of length,whereas the probability W3(r)dr is dimensionless, as itshould be. The probability density function W3(r) can beused to derive an expression for the average nearest-neigh-bor distance Ær3æ given as follows [21]:

hr3i ¼Z 1

0

rW3ðrÞdr ¼ 0:554ðNV Þ�1=3 ð2Þ

Similarly, the equation for the nearest-neighbor probabilitydensity distribution function W2(r) and average nearest-neighbor distance Ær2æ of randomly distributed points in aplane of infinite extent [20] can be derived as

W2 ¼ 2prN A exp ð�pr2N AÞ ð3Þ

hr2i ¼Z 1

0

rW2ðrÞdr ¼ 0:5ðNAÞ�1=2 ð4Þ

In Eqs. (3) and (4) NA is the number of point particles perunit area of the plane, i.e., the intensity of the point process.

3. Theoretical development

Consider an infinite volume bound between two parallelinfinite planes separated by distance D as shown in Fig. 1(a).This represents the spatial domain of a thin film/sheet/platewhose thickness is D. The particles of interest are assumedto be uniform point objects randomly distributed in thisspace with an intensity of NV (the analysis is equally appli-cable to atoms, molecules, clusters of atoms, inclusions, pre-cipitates, etc., in a dilute structure). The assumption ofuniform random point objects (also referred to as the Pois-son process) implies the following two assumptions:

� the probability of finding a point object in an infinitesi-mal volume element dV of this structure is NV dV, and� the probability is same at all the locations, independent

of the probability in any other volume element.

The nearest-neighbor distance probability density func-tion W(r) for these points is defined such that W(r)dr isthe probability of finding a nearest neighbor to a point ina distance between r and r + dr. Let us consider a pointO (as shown in Fig. 1(a)) at a distance x from the centralplane. For subsequent analysis, it is useful to group thevariable radial distance r from this point in the followingthree distinct ranges (Fig. 1(b)):

A. Tewari, A.M. Gokhale / Acta Materialia 54 (2006) 1957–1963 1959

RA ¼ fr : 0 6 r 6 D=2� xg ð5aÞRB ¼ fr : D=2� x 6 r 6 D=2þ xg ð5bÞRC ¼ fr : D=2� x 6 r 61g ð5cÞThe radial distances r in the first domain RA are on the sur-face of a complete sphere about the point O, whereas theradial distances in domains RB and RC lie on incompletecut spheres (cut by the top plane for RB and cut by bothtop and bottom planes for RC as shown in Fig. 1(b)).The probability W(r)dr can be described as a joint proba-bility of two simple events. The first event being that thereis no point object in a radial distance r around a typicalpoint object (with probability P1), and the second thatthere is exactly one point object in a shell in the distancerange r and r + dr (with probability dP2). Since the twoevents are independent (due to the assumption of random-ness), their joint probability is equal to the product of theirindividual probabilities. Hence, the general equation forthe NND function would be as follows:

WðrÞdr ¼ P 1 dP 2 ð6ÞThe probability of the first event, that there is no point objectin a radial distance r around a typical point object, is equal to1 minus the probability that there is at least one point in theradial distance r around the point object [20,21], i.e.

P 1ðrÞ ¼ 1�Z r

0

WðrÞdr ð7Þ

Similarly, the probability of the second event is NV timesthe volume of the shell in the distance range r and r + dr:

dP 2ðrÞ ¼ N V dV ð8ÞSince this shell has different functionality for the volumefor different shell geometry (complete sphere or cut sphere)depending on the radial distance r and the domain (RA, RB,or RC), the function W(r) has three different functionalforms, one for each radial distance domain:

WðrÞ ¼WAðrÞ for r 2 RA ðaÞWBðrÞ for r 2 RB ðbÞWCðrÞ for r 2 RC ðcÞ

8><>: ð9Þ

The shell volumes are given by the following functions:

dV ¼4pr2 dr for r 2 RA ðaÞ2prðr þ D=2� xÞdr for r 2 RB ðbÞ2prDdr for r 2 RC ðcÞ

8><>: ð10Þ

W ¼WA ¼ 4pN V r2 exp ð�4=3pr3NV Þ 0 6 r 6 1=2D� x

WB ¼ pN V exp ½pðD� 2xÞ3NV =24�rð2r þ D� 2xÞ exp ½�pr2ð2=3r þ D=2� xÞN V � 1=2D� x 6 r 6 1=2Dþ x

WC ¼ 2pN V D exp ½pDNV ðD2 þ 12x2Þ=12�r exp ½�pr2DN V � 1=2Dþ x 6 r 61

8><>: ð23Þ

The above equations can be combined to get the followingthree integral equations:

WAðrÞ ¼ 1�Z r

0

WAðrÞdr� �

4pr2NV ð11Þ

WBðrÞ ¼ 1�Z D=2�x

0

WAðrÞdr �Z r

D=2�xWBðrÞdr

!

� 2prðr þ D=2� xÞN V ð12Þ

WCðrÞ ¼ 1�Z D=2�x

0

WAðrÞdr �Z D=2þx

D=2�xWBðrÞdr

�Z r

D=2þxWBðrÞdr

!2prDN V ð13Þ

The above integral equations can be solved to obtain thefollowing solution:

WðrÞA ¼ C1r2 exp ð�4=3pr3NV Þ ð14ÞWðrÞB ¼ C2rð2r þ D� 2xÞ exp ½�pr2ð2=3r þ D=2� xÞNV �

ð15ÞWðrÞC ¼ C3r exp ð�pr2DNV Þ ð16Þ

where C1, C2, and C3 are constants (independent of r).Since these functions represent the probability densityfunction, the sum of their integrals over their appropriatedomain limits should be equal to one. Further, the proba-bility density must be continuous at the boundaries of thethree radial distance domains RA, RB, and RC. These con-strains can be written in the form of the followingequations:Z D=2�x

0

WAðrÞdr þZ D=2þx

D=2�xWBðrÞdr þ

Z 1

D=2þxWCðrÞdr ¼ 1

ð17ÞWAðD=2� xÞ ¼ WBðD=2� xÞ ð18ÞWBðD=2þ xÞ ¼ WCðD=2þ xÞ ð19Þ

The above three equations are used to determine the valuesof the three constants C1, C2, and C3 as given below:

C1 ¼ 4pNV ð20ÞC2 ¼ pNV exp ½pðD� 2xÞ3NV =24� ð21ÞC3 ¼ 2pNV D exp ½pDN V ðD2 þ 12x2Þ=12� ð22Þ

On substituting the value of the constants, we get theexpression for the nearest-neighbor probability densityfunction for thin films:

This is a piecewise continuous function of D and x, in addi-tion to being a function of r and NV. The first function WA

of W is exactly the same as the bulk 3D solution (see Eq.(1)), except for the limited domain of applicability:0 6 r 6 (D/2) � x. This is because at short distances the

1960 A. Tewari, A.M. Gokhale / Acta Materialia 54 (2006) 1957–1963

particles do not ‘‘see’’ the edges of the film and behave ex-actly like a point in an infinite 3D domain. The third func-tion WC is similar (though not exactly the same) to the 2Dsolution (see Eq. (3)). Nonetheless, in the limiting case,when D tends to zero, WC is exactly same as the 2D solutionwhen the (NVD) term is substituted by NA. This is also ex-pected since the limiting case of D tending to zero yields a2D planar structure. The most interesting term in theexpression for W is the second function WB, which givesthe transitional behavior from the 3D solution to the 2Dsolution.

To get the mean nearest-neighbor distance, the productrW(r) is integrated over the complete range of r (from zeroto infinity). It turns out that this integral cannot be evalu-ated in an analytical form, due to complex functionality ofthe function W(r). Therefore, in the next section, the meannearest-neighbor distance values have been numericallycomputed and plotted as a function of different variables.To better understand the functionality of W(r), the inde-pendent variables as well as W(r) are made dimensionlessusing the following transformations:

q ¼ r=D ð24aÞx ¼ x=D ð24bÞh ¼ D3NV ð24cÞu ¼ DW ð24dÞ

This results in the reduction of the number of variablesfrom five (r, x, D, NV, W) to four (q, x, h, u), thus facilitat-ing the analysis of the behavior of the function. Moreover,the dimensionless results need to be computed only onceand can be used for any specific values of the four variables(r, x, D, NV) without requiring computations for each spe-cific case.

It should be noted that this analysis can be easilyextended to two-layer (film and substrate) or sandwichedstructures. This can be done by assuming a second distribu-tion of point particles with a different number density forthe space outside the film. If we assume that such pointparticles are present only on one side of the film then it willmimic the case of film and substrate. Otherwise, if weassume such point particles to be present on both the sidesthen it will be a model for the sandwiched structure. Thiswill add extra terms to the integral Eqs. (12) and (13).The results of such an analysis would provide not onlythe NND function, but also the fraction of points in thefilm having dissimilar nearest neighbors as a function offilm thickness and distance from the interface. However,the complete analytical solution of the above problem isquite complex and is beyond the scope of the presentinvestigation.

4. Numerical computations and analysis

The variation in the dimensionless probability densityfunction u (see Eq. (24d)) with the dimensionless radial

distance q as a function of different x and h is shownin Figs. 2(a)–(f). It can be seen that the probability den-sity function shifts to higher distances with the increase indistance from the centerline of the microstructuredomain. This is further illustrated in Fig. 3, which plotsmean nearest-neighbor distance as a function of x fordifferent values of h. The interesting point to be notedin Fig. 2 is that the three distribution functions forx = 0, 0.25, and 0.45 come close together at very lowand very high values of h. This is attributed to the factthat at very low or high values of h the resulting micro-structure can be considered as 2D planar structure or 3Dbulk, respectively. To further illustrate this point, Figs.2(a) and (f) also show the corresponding 2D and 3D dis-tribution curves plotted along with the function. At verylow values of h the function behaves like the NND func-tion for 2D planar structure, and at very large values of hit behaves like the NND function for 3D bulk. Based onthe numerical computations of the probability densityfunction, it is found that at h values less than 0.17, uis a weak function of x and can be approximated tothe corresponding 2D NND function (Fig. 2(a)). Further,at h values greater than 1000 it shows weak functionalitywith x, and can be approximated by the 3D bulk NNDfunction (Fig. 2(f)). Between these values of h it variessignificantly with x and cannot be approximated byeither the 2D or 3D solutions.

In some applications, the mean nearest-neighbor dis-tance is of interest and not the whole probability distribu-tion. For this purpose, values of the dimensionless meannearest-neighbor distance Æqæ have been calculated. Notethat Æqæ is a function of both x and h. Fig. 3 shows the var-iation of Æqæh1/3 as a function of x for different values of h(the factor h1/3 is used to normalize Æqæ for different valuesof h so that they can all be shown on the same plot). It isapparent that the mean nearest-neighbor distance is astrong function of x (i.e., the distance from the centerline)as also seen by the shift in the distribution functions inFig. 2, with the higher nearest-neighbor distances closer tothe edges. This is because as one approaches the edge, fewerparticles are seen (because there are no particles on theother side of the edge) resulting in higher distances beforea nearest neighbor is encountered. This indicates thatwhenever nearest-neighbor distance measurements are per-formed near an edge, the distances are an over estimate ofthe global mean nearest-neighbor distance in the film.

The global mean nearest-neighbor distance averagedover the whole thickness is shown in Fig. 4. The plot showsthe mean nearest-neighbor distance for a 2D planar struc-ture (Ær2æ, substituting NVD for NA), 3D bulk structure(Ær3æ), and the finite thickness structure of this analysis(i.e., thin films) as a function of thickness. Both the axesare normalized by the mean nearest-neighbor distance for3D bulk (infinite in all directions) system; this is an easyway of comparing the three results. It is seen that at smallvalues of D/Ær3æ (much less than 1) the mean nearest-neigh-bor distance in thin films approaches that of the 2D planar

Fig. 2. Variation of u (dimensionless W, i.e., W/D) as a function of q for different values of x. (a) and (f) also show the corresponding dimensionless infinite2D (W2) and infinite 3D (W3) probability distribution functions.

A. Tewari, A.M. Gokhale / Acta Materialia 54 (2006) 1957–1963 1961

structures. This corresponds to a h value less than 0.17(note that h = 0.17(D/Ær3æ)3), and can be easily derived fromthe definition of h and Eq. (2). This is a special case of theprevious result given in Fig. 2(a), in which the complete ufunction for different x values is similar to the 2D function(W2) for a h value of 0.17. This result is understandable,since, if the thickness of the film (D) is less than the meannearest-neighbor distance in bulk 3D (Ær3æ), the film isexpected to behave like a 2D planar structure. In contrast,at D/Ær3æ values larger than 2 (which corresponds to h val-ues greater than 1.36), the mean nearest-neighbor distanceapproaches the 3D bulk solution. This approximately cor-

responds to films with thickness of two monolayers (2 ML)and higher. This is counterintuitive, since one would thinkthat the thickness value of 2 ML is still very thin to be con-sidered as bulk, although it turns out to be otherwise.Hence for the purpose of mean nearest-neighbor distances,one can consider a film with h values less than 0.17 as a 2Dplanar structure, and that with h values greater than 1.36 asa bulk 3D structure. For intermediate values of h, the meannearest-neighbor distance shows a transition from 2Dplanar to 3D bulk solution. It should be noted that themean nearest-neighbor distance of a thin film rapidlyapproaches the 3D bulk solution at h values greater than

0

0.5

1

1.5

2

2.5

3

0 0.5 1 1.5 2 2.5 3

2D surface

Thin film

3D bulk

<r>

/<r 3

>

Δ/<r3>

←

←

Fig. 4. Mean nearest-neighbor distance in thin film (averaged over thecomplete thickness of the film) as well as the corresponding 2Dplanar structure (Ær2æ) and 3D bulk mean nearest-neighbor distances(Ær3æ) as a function of film thickness. Both axes have been normalizedby Ær3æ.

Fig. 3. Variation of mean dimensionless nearest-neighbor distance Æqæ(normalized by h1/3, to make all the plots for different h fit in one figure)plotted as a function of x. It is seen that the mean dimensionless nearest-neighbor distance is significantly higher at x = 0.54, implying an edgeeffect.

1962 A. Tewari, A.M. Gokhale / Acta Materialia 54 (2006) 1957–1963

1.36, although the nearest-neighbor distance distribution(Fig. 2(f)) approaches the 3D bulk solution only at h valuesof 1000 and higher (corresponding to D/Ær3æ greater than18).

5. Results and discussion

The present analysis provides limits on the applicabilityof 2D and 3D solutions for nearest-neighbor distance dis-tributions of uniform randomly distributed point particlesin thin films. These limits would not be the same if the

spatial arrangements of the points deviate from random-ness (as is the case in clustered or ordered structures); how-ever, their behavior is expected to show similar trends.Hence, the analysis serves as a basis for the evaluation ofnearest-neighbor distances for non-random particles. Thisis used in the following discussion to understand the exper-imental observations of other researchers.

Fons et al. [3] studied the nucleation and growth of ZnOfilms on ð1�120Þ sapphire substrate using molecular beamepitaxy. They used X-ray absorption fine-structure spec-troscopy to measure nearest-neighbor distances and foundthat there is no discernible change in the values of the meannearest-neighbor distance with the film thickness. The hvalues for their films varied from 600 to 1010 (3–600 nm),which are well above the required limiting value of 1.36for it to behave like 3D bulk. This explains why their exper-imental data on the mean nearest-neighbor distance are notsensitive to the film thickness. Similarly, Thiele et al. [7]studied nearest-neighbor distances in Co/Pt(111) thin filmshaving h values from 1.36 to 170 (2–10 ML) and found themto be identical to the bulk values, which is justified based onour analytical results. The crystallographic structure of Cofilms on Cu(001) was studied by Fevre et al. [10]. Theyobserved that films thicker than a h value of 4.6 (3 ML)showed bulk Co arrangements, which is in agreement withour analysis; more interestingly, they found that films thin-ner than this value showed significant surface interactionsso much so that the Co arrangements were similar to theCu substrate. This is evidence in support of our analyticalresult that films thinner than a critical value would have sig-nificant surface effects. Observation by Andryushechkinet al. [12] of chlorinated copper surface films with h valuesof about 170 (10–12 atomic layers) revealed bulk-likebehavior. Andryushechkin et al. [12] attributed this result(without justification) to the fact that their films were thickenough to be considered as bulk. Similarly, Craig et al. [15]assumed (again without justification) that a slab of siliconlayers with h value greater than 500 (14 ML) would be suf-ficiently thick to produce bulk-like behavior. The presentanalytical results provide the much-needed justification tothe above explanations. They give a complete picture asto when a thin film should be considered as bulk and whenit should be considered as 2D planar structure, and as towhen the influence of edges (surfaces) becomes significant.Thus, the present analytical results provide an explanationfor the experimental results in the literature for thin filmsand add to the basic understanding of the transition of afilm from 2D planar structure to 3D bulk.

6. Summary and conclusions

An exact analytical expression for the nearest-neighbordistance distribution of randomly distributed point particlesin a thin film is derived. It is shown that the mean nearest-neighbor distance is a function of the film thickness andthe distance from the edge. The mean nearest-neighbordistance increases on approach to the edge, resulting in edge

A. Tewari, A.M. Gokhale / Acta Materialia 54 (2006) 1957–1963 1963

(surface) effects. The global mean nearest-neighbor dis-tances and the distributions can be approximated to thenearest-neighbor distances and distributions for 2D planarstructures at h (=D3NV) values smaller than 0.17. For h val-ues greater than 1.36, the mean nearest-neighbor distancesof the films can be approximated to the bulk 3D mean near-est-neighbor distances. Interestingly, this value of thicknessfor approach to the 3D bulk solution is so small that manythin films studied lie above this value, and therefore showbulk behavior. This result is supported by the experimentalobservations of thin films of other researchers [5,7,11,12,15].The NNDs, however, cannot be approximated to the bulk3D distributions until the h values are greater than 1000.This provides an understanding of the behavior of thin filmsand the transition from 2D planar structure to 3D bulk.

Acknowledgement

The Division of Materials Research, US National Sci-ence Foundation supported this work through researchGrant DMR-0404668. The financial support is gratefullyacknowledged.

References

[1] Tewari A, Gokhale AM. Acta Mater 2004;52:5165.[2] Kuo CH, Gupta P. Acta Metall 1995;43:397.

[3] Lacerda RG, Tessler LR, Santos MC, Hammer P, Alvarez F,Marques FC. J Non-Cryst Solids 2002;200:299.

[4] Tewari A, Gokhale AM, German RM. Acta Mater 1999;47:3721.[5] Fons P, Iwata K, Yamada A, Matsubara K, Niki S, Nakahara K,

et al. J Cryst Growth 2001;227–228:911.[6] DeHoff RT. Acta Metall 1991;39:976.[7] Thiele J, Belkhou R, Bulou H, Heckmann O, Magnan H, Le Fevre P,

et al. Surf Sci 1997;384:120.[8] Segurado J, Gonzalez C, Llorca J. Acta Mater 2003;51:2355.[9] Kolb DM, Holzle MH, Wandlowski TH. Surf Sci 1995;335:

281.[10] Le Fevre P, Magnan H, Heckmann O, Chandesris D. Physica B

1995;208–209:401.[11] Heckmann O, Magnan H, Le Fevre P, Chandesris D, Rehr JJ. Surf

Sci 1994;312:62.[12] Andryushechkin BV, Eitsov KN. Surf Sci 1992;265:L245.[13] Lozzi L, Passacantando M, Picozzi P, Santucci S, De Crescenzi M.

Surf Sci 1991;251–252:579.[14] Magnan H, Chandesris D, Villette B, Heckmann O, Lecante J. Surf

Sci 1991;251–252:597.[15] Craig BL, Smith PV. Surf Sci 1990;226:L55.[16] Mehra M, Johnson WL, Thakoor AP, Khanna SK. Solid State

Commun 1983;47:859.[17] Purans J, Kuzmin A, Parent P, Laffon C. Electrochim Acta

2001;46:1973.[18] Kronmuller H, Mohan CV, Seeger M. J Magn Magn Mater

1997;174:89.[19] Horstemeyer MF, Matalis M, Siebel AM, Botos ML. Int J Plasticity

2000;16:979.[20] Gurland J. In: Dehoff RT, Rhines FN, editors. Quantitative

microscopy. New York (NY): McGraw-Hill; 1968. p. 278.[21] Chandrasekhar S. Rev Mod Phys 1943;15:86.