LE JOURNAL DE PHYSIQUE

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1 une In Nous Tome 35 N 5 MAI 1974 LE JOURNAL DE PHYSIQUE Classification Physics Abstracts CLUSTERS OF ATOMS COUPLED BY LONG RANGE INTERACTIONS J. P. GAYDA and H. OTTAVI Département d Electronique (*), Université de Provence, Centre de SaintJérôme, Marseille Cedex 4, France (Reçu le 13 novembre 1973) 2014 Résumé. présentons une étude statistique de la distribution de la taille des amas formés par des atomes répartis aléatoirement dans un milieu continu. Les résultats statistiques sont obtenus par une méthode de Monte Carlo. r0 mesurant la portée des interactions, nous trouvons qu un amas infini apparaît quand le nombre moyen d atomes, W, contenus dans une sphère de rayon r0, atteint valeur légèrement supérieure à 2,6. Pour rendre compte de ces résultats statistiques, nous avons élaboré des modèles de processus de ramification. Quelques propriétés essentielles sont dégagées, et on obtient une expression approchée pour une grande gamme de concentration du nombre d atomes appartenant à des amas de petites tailles Abstract. present a study on cluster size distribution for atoms randomly distributed in a threedimensional continuous medium. Statistical results are obtained by a Monte Carlo method. r0 being the range of interaction, it is found that the infinite cluster appears when the mean number of atoms, W, in a sphere of radius r0 reaches a value slightly higher than 2.6. Besides this, the statistical results are compared to the results deduced from branching process models : some essential features are then outlined and an approximate expression is proposed for the number of atoms belonging to clusters of small sizes, for a large range of concentration. 1. Introduction.. preceding papers [1, 2, 3], we study EPR line shapes for ions diluted in a diamagnetic matrix. Accent is put on the particular role played by clusters of ions coupled by long range exchange interactions. present here a statistical study on the distribution of the clusters, versus concentration. This problem is one aspect of the general percolation process (for a bibliography on percolation and physical applications, see [4]). consider atoms or points randomly distributed in a three dimensional continuous medium and coupled by an interaction of range ro. The statistical laws governing the cluster size distribution depend only on one parameter W, the mean number of points in a sphere of radius 0: where p is the number of points per unit volume. Holcomb, Iwasawa and Roberts (H. I. R.) [5] use the parameter t proportional to W : Clusters of infinite size appear above the critical value Wc. In the three dimensional continuous case, Domb and Dalton [6] propose the value 2.7 for Wc. This value is an asymptotic limit for critical values calculated by the series method [7] in lattices where ro is larger and larger. An approach to the continuous case is obtained by Holcomb and Rehr [8] who make use of a Monte Carlo calculation on a simple cubic lattice with ro equal to 3 times the lattice parameter. They obtain Wc 2.4 (tc 0.07). More recently, the same result if obtained by H. I. R. by a linear extrapolation of the mean cluster sizes obtained by a Monte Carlo method in the continuous case. The discrepancy with our value (Wc > 2.6) will be discussed in the following. Article published online by EDP Sciences and available at

2 The 2.1 Construction The 394 In the first part of this paper, we present a method of cluster construction which gives statistical results directly in the continuous medium. These results are called experimental results. In the other parts, we use the analogy between the experimental results and statistical laws governing a branching process to obtain on the one hand, an evaluation of Wc, on the other hand, analytic expressions for counting the clusters of small size in a large range of concentration. Notation. call the probability that a point belongs to a cluster of size N, PN, to a cluster of size N or more, P/, size, P 00 lim P;. N N + 00 to a cluster of infinite 2. The Monte Carlo calculations. THE ME principle of cluster construction is THOD. the following: we choose a point and examine the size of the cluster containing it. For that, we explore the sphere of radius ro centered on it. The random number n of neighbours in the sphere is determined according to a Poisson law of parameter W and their positions according to a statistical law with uniform density. (This needs utilization of random numbers which are generated by a multiplicative congruential method.) If n 0, the choosen point belongs to a cluster of size 1. If n # 0, we say we found n neighbours in the 1 st generation. Then we look to see if the points of the 1 st generation have neighbours themselves, by exploring the sphere of each one. However we note that the points found in a region previously explored must now be eliminated. The neighbours so obtained belong to the 2nd generation and the construction of the cluster is pursued until we find an empty generation. The chosen point belongs then to a cluster of size N, where N is the total number of points which have been found, plus the origin. In practice, an upper limit for N is imposed by the computer. In our calculations, this limit is 256. When the cluster size goes beyond this value, the first point is said to belong to a cluster of size 256 or more. In fact, in order to avoid the spherical coordinates, each sphere is provisionally replaced by a cube with edges of length 2 ro. The filling of the cube is performed in accordance to a Poisson law of parameter (6/n) W. But a point will be ignored if it fails to fall in the sphere. So, to be retained and belong to the generation number k, a point must be : a) inside the sphere centered on the point of the generation k 1 from whom it is descended, b) outside the spheres previously explored, centered on the points of the generation k 1, c) outside the spheres centered on the points of the generations numbered from 0 to k 2. FIG. 1. of a cluster. These rules are pictured in figure 1 where the construction of a cluster is shown up to the 2nd generation. The 1 st generation contains the two points 1, 2. have found for the point 1, the three possible neighbours 3, 4, 5 ; however, following the preceding criteria, 4 and 5 must be eliminated ; likewise the three neighbours 6, 7, 8 found for point 2 must be eliminated ; finally, the 2nd generation contains only the point 3. Our method seems less expensive than the method of H. I. R. Indeed, in this last case, in order to build a cluster containing 185 points, it is necessary to examine candidates ; in our method only about 400 candidates must be analysed. So, for W 2.3, we have been able to obtain five clusters containing 500 points or more, and for each studied value of W, at least 500 clusters have been built. 2.2 THE RESULTS. results are presented in figure 2, under the form This representation must put in evidence the critical Wc, all the points are value fl. Indeed when W in clusters of finite size and PN must tend toward zéro ; on the other hand, when W > Wc a non null proportion of points is in the infinite cluster and PN has a horizontal asymptote with ordinate P oo(w). Each value obtained for PN+ is represented with its error interval, the half length of which is twice the standard deviation. It appears from these results that the Wc value is certainly located between 2.3 and 3 ; this is compatible with the value 2.7 given in [6]. have verified that the values of ê (the mean cluster size) calculated from our results are in good agreement with the values given by H. I. R. It is a confirmation of the validity of the two cluster building methods.

3 If If If PN Our 395 In fact, for the problem of clusters, we can convince ourselves that the asymptotic value Wa (if it exists) cannot be greater than one even if W > Wc 3.1 Let gk be the number of points in the generation k for a given trial, and G, the mean value of gk. By definition and log Gk versus k has an asymptotic direction with a slope equal to log Wa On the other hand, for any cluster, the distance between a point of the kth generation and the origin is at most kro. The number of points (belonging or not to the cluster) inside the sphere of radius kro, N(kro), is an upper bound to the number of points in the generations 0 to k of the cluster, and a fortiori This remains true for the mean values FIG. 2. f(n). Dots : experimental results. results given by a branching process with fertilities depending on the generation number. The value W of the fertility of the ancestor is indicated on each curve. method 3. Analogy with a branching process. of cluster construction suggests the development of a family according to a branching process. In the simplest process, a man has r sons with probability pr (r 0, 1, 2...) and each of his sons has the same probabilities of having a given number of sons of his own. The fertility of every man is W, the mean number of sons. If W 1, the probability of extinction of the family is one; however if W > 1, the probability of an infinite descendance becomes non null [9]. In the construction of a cluster, the fertility is certainly not the same for all points. In particular the fertility of the first point is greater than the fertilities of the following points. Thus we consider a branching process more elaborate where the fertility Wk of a man depends on its generation number k. Let us assume that Wk has an asymptotic limit Wa. One can show that : Wa > 1, the probability of an infinite descendance is non null (P 00 i 0). The mean population Gk of the kth generation increases indefinitely with k. Wa 1, the mean population tends toward zero and P Wa 1, P 00 is null or not depending on whether the convergence of Wk is fast or not. Then on a semilog scale, log G, versus k is bounded by a function of k which has an asymptotic direction with a null slope. Thus the slope log Wa of the asymptotic direction of log Gk is less than or equal to zero so that 3. 3 Furthermore it is easy to show that, when that a cluster grows Wa 1, the probability P 00 indefinitely is zéro ; Gk is an upperbound to the probability that gk is non null, this probability is itself an upperbound to Poo and we can write Now, if Wa 1, k lim 00o for any value of k. Gk 0 so that All these results, i. e. : 1) Wa 1 for any value of W, 2) if Wa 1, P 00 0, or W W,, suggest that the critical value Wc is the smallest value of W for which Wa 1. Following this picture, we have represented in figure 3 the results of the Monte Carlo method under the form of the mean number of points in each

4 GK 396 For that, we compute the theoretical P; values are deduced given by this process where the values Wk from the experimental Gk (Fig. 3) : Then we compare with PN experimental values of the figure 2. failed to obtain the theoretical PN+ in the general case. However the calculus is easier if one supposes that Wk is constant above a given generation number 1 If 1 1 (two fertilities process) an exact expression has been obtained (see 6). If 1 > 1 recurrent formulae can be derived in this manner : Let Ak(s) be the generating function of the probability PkN that the family descended from a man of the kth generation (fertility Wk) contains N men : FIG. 3. : mean population of each generation according to experimental results. In particular, k 0 gives the generating function of the probabilities P N : generation Gk up to the 20th generation. The half length of error intervals is twice the standard deviation. The previous estimates of Wc are strongly confirmed by this representation. In particular, the critical value Wa 1 appears to the reached for a Wc value slightly higher than 2.6. This value is rather different from the value given by H. I. R. (Wc ~ 2.4). As previously noted by Domb [10] the linear extrapolation of the function 1 /ê used by H. I. R. must give an underestimate of Wc, because it neglects the curvature which is still apparent near the critical value. A better extrapolation would be attained if one could calculate ê for values of t closer to tc; unfortunately the value of ê is then essentially due to large clusters and the computation is limited by the computer capabilities. The situation is quite different in our method where the computation of the fertilities Wk is always possible the nearest vicinity of tc. even in 4. Validity of the model of branching process with fertilities depending on the génération number. want to examine whether the growing of clusters is well discribed by a branching process of this type i. e. ; a process where the number of sons of a man belonging to the kth generation is in accordance to a Poisson law of parameter Wk. Consider now a man of the kth génération ; he has Z, sons and the generating function of Z, is The total progeny of this man can be considered as the sum of Zk ensembles. With the aid of the theorems concerning the generating functions of sums of a random number of variables, we can write The recurrent formulae are obtained from these expressions: (4a) is written and by taking the derivative, we get Finally Ak(s) and A[(s) are replaced by their power series and we deduce the recurrent relations between

5 W(n) PN 397 the PkN coefficients These relations have been used to compute PN N1 and PN 1 L PN with the following values kl of Vfi evaluated from the results of figure 3 : W1 W2o 0.8, 0.9, 0.98 respectively for W 2, 2.3, 2.6, W1 Wg 1.07 for W3. According to a preceding remark, this last value is probably too high ; however the PN values are checked only for moderate values of N and we expect that the results depend essentially on the fertilities of the first generations. The standard error on PN calculated by this method, is given by fertility of a man depends on the number of men in its generation. call this process a vital space process. 5. Vital space process. proceed as in the preceding part, i. e. we compute the theoretical PN values according to the vital space process with fertilities deduced from the Monte Carlo results. For that, we calculate for each generation the experimental fertility of a point versus the number n of points in this generation. In fact, the precision of the results does not allow the effect of the generation number to be perceived and we take an average on the first generations. The values are given in figure 4. where (Jk is the standard deviation on Wk.oP; /owk was numerically computed. To obtain an estimate of 6k, we assume that the clusters grow according to a branching process with Poisson law. Then where M is the total number of constructed clusters. The values PN given by the branching process are presented in figure 2. The precision is shown by the error interval centered on P64+ (its half length is twice the standard error a ). The agreement with the expérimental values is very good up to W 2.3. For the upper values of W, the deviation is larger than the deviation allowed by the inacurracy of the results. This deviation may be understood if we note that in the growing of a cluster the fertility of a point depends strongly on the number of points of its génération : because of the possible overlap of the spheres centered dn these points, the volume to be explored per point (which is similar to a vital space) is certainly decreasing with their number. Now, the model considers only a mean fertility Wk. The results show that this is not allowed for large concentrations. In order to clarify this remark, we consider in the following part a branching process where the FIG. 4. : mean value on 19 generations for W 2.3, 18 générations for W 2.6,8 generations for W 3. ; : values used in the calculation of PN and PN with the vital space process. failed to obtain an analytic expression of PN or PN and we compute PN directly from its definition i. e., PN is the sum of the probabilities of having a family of N men characterised by a given set of the number of men in each successive generation. The number of such sets is 2N 2 and it is clear that PN can only be calculated for rather small values of N. The P+N values are shown in figure 5 up to N 20. In this range of values of N, they agree very well FIG. 5. f(n). Dots : experimental results. process with fertilities depending on the generation number. vital space process.

6 Figure Estimate 398 with the experimental results, even for large values of W. Thus it is clearly shown that for the large values of W, the concept of vital space of the points plays a fundamental role in the statistical laws governing the cluster size distribution. 6. Approximate analytical expression of PN for small N. 3 shows that in the range k 1 to 20, log Gk is not very far from a linear function of k and so Wk Gk 11 IGk is slowly varying in this range. This remark suggests that we consider a branching process with two fertilities as an approximate model giving the essential features of statistics on the clusters at least for values of N which are not too large. In this process, the fertility of the ancestor is W and all the descendants have the fertility W. The main interest of this model is in the possibility of obtaining an exact analytic expression for PN. follow a method similar to that used in [11] : The expression (5) is reduced to where FIG. 6. of PN and PN given by the expression (8) of 6. r is a loop enclosed in a circle of unit radius centered on the origin. Then with the aid of (4b) : which is The deviation is less than 3 % for W inside the interval [0, 3.5]. where F is a small closed loop encircling the origin and K W /W. The theorem of residue gives finally : A priori, the ratio K is a function of W. In fact, a good approximation is obtained for the constant value K 0.5. The results are presented in figure 6 under the form WP and WPN versus W for W 3 and N 1 to 4. They may be compared to the results given by the vital space process represented by dots (we have shown previously that this last process describes well the cluster construction for small values of N). For N 2, we have verified the agreement betwèen the approximate expression (8) and the exact one Conclusion. have presented a statistical study on cluster size distribution based on statistical results given by a Monte Carlo method. These results are in good agreement with those of Holcomb, Iwasawa and Roberts [5]. However, our extrapolation method is quite différent and gives a larger critical value of W, slighly higher than 2.6. This value is in accordance with the value 2.7 given by Domb and Dalton [6]. note that in a previous work [12], the authors attempted to obtain an estimate of Wc which was expected to be near the value for which the mean total curvature of the cluster frontier becomes nul. They obtained for this last value All the preceding results show that this value is certainly too high. In addition, we have examined the analogy between the statistical laws governing the construction of clusters by the Monte Carlo method and the development of a family according to a branching process : for values of W less than the critical valuesr the experi

7 399 mental results can be fitted by a process where the fertility of a man depends only on its generation number. For greater values of W, the effect of overlap of the spheres is described in a more correct manner by a process where the fertility of a man depends on the number of men in its generation. Finally, we derive an expression for PN from a two fertilities process and show that it can be used to count the number of atoms belonging to clusters of small size for a large range of concentrations. Acknowledgments. are grateful to Professor M. Cadilhac and Professor J. Hervé for useful discussions and criticism. References [1] GAYDA, J. P., BLANCHARD, C., J. Physique 30 (1969) 827. [2] GAYDA, J. P., J. Physique 32 (1971) 793. [3] GAYDA, J. P., DEVILLE, A., LENDWAY, E., J. Physique 33 (1972) 935. [4] SHANTE, V. K. S., KIRKPATRICK, S., Adv. Phys. 20 (1971) 325. [5] HOLCOMB, D. F., IWASAWA, M., ROBERTS, F. D. K., Biometrika 59 (1972) 207. [6] DOMB, C., DALTON, N. N., Proc. Phys. Soc. 89 (1966) 859. [7] SYKES, M. F., ESSAM, J. W., Phys. Rev. 133A (1964) 310. [8] HOLCOMB, D. F., REHR, J. J., Phys. Rev. 183 (1969) 773. [9] HARRIS, T. E., The theory of branching process (Springer Verlag, Berlin), [10] DOMB, C., Biometrika 59 (1972) 209. [11] FISHER, M. E., ESSAM, J. W., J. Math. Phys. 2 (1961) 609. [12] OTTAVI, H., GAYDA, J. P., J. Physique 34 (1973) 341.