Clusters of atoms coupled by long range interactions

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Clusters of atoms coupled by long range interactions

J.P. Gayda, H. Ottavi

To cite this version:

J.P. Gayda, H. Ottavi. Clusters of atoms coupled by long range interactions. Journal de Physique,

1974, 35 (5), pp.393-399. �10.1051/jphys:01974003505039300�. �jpa-00208162�

LE JOURNAL DE PHYSIQUE

CLUSTERS OF ATOMS COUPLED BY LONG RANGE INTERACTIONS

J. P. GAYDA and H. OTTAVI

Département d’Electronique (*),

Université de Provence, Centre de

Saint-Jérôme,

¹³³⁹⁷ Marseille Cedex

4,

^France

(Reçu

^{le 13 novembre}

1973)

Résumé. ²⁰¹⁴ Nous 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

- une 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. ²⁰¹⁴ We present ^a study ^on cluster size distribution for atoms randomly distributed in

a three-dimensional 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 statis-}

tical 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.

Classification Physics ^Abstracts

1.660

1. Introduction.. ^{- In}

preceding

papers

[1, 2, 3],

we

study

^{EPR line}

shapes

^{for ions diluted in a diama-}

gnetic

matrix. Accent is put ^{on the}

particular

^role

played by

clusters of ions

coupled by long

range

exchange

interactions. - We 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]).

We 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 http://dx.doi.org/10.1051/jphys:01974003505039300

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

expres- sions for

counting

the clusters of small size in a

large

range of concentration.

Notation. ^- We call the

probability

^{that a}

point belongs

^{to a} cluster of size N,

PN,

^{to a} cluster of size N or more,

P/,

^{to a cluster} of infinite

size, P 00

^{= lim}

P;-.

N --+- 00 N

2. The Monte Carlo calculations. ^- 2.1 THE ME - THOD. ^- The

principle

of cluster construction is 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 genera- tion. 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. ^- Construction of a cluster.

These rules are

pictured

ⁱⁿ

figure

^{1 where the cons-}

truction of a cluster is shown _up to the 2nd

generation.

The 1 st

generation

^{contains the two}

points ^1,

^2.

We 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 4 000

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. ^- The results are

presented

ⁱⁿ

figure 2,

under the form

This

representation

^must

put

in evidence the critical value

fl.

Indeed when W

Wc,

^{all the}

points

^are

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].

We 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.

FIG. 2. ^- PN ⁼ 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.

3.

Analogy

^{with a}

branching

process. ^- Our method 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 extinc-}

tion 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 branch-}

ing

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 :

- If

Wa

>

1,

^the

probability

of an infinite descen- dance is non null

(P 00 i= 0).

^{The mean}

population Gk

^{of the kth}

generation

^increases

indefinitely

^{with k.}

- If

Wa 1,

^{the mean}

population

tends toward

zero and

P 00

^{= 0.}

- If

Wa

⁼

1, P 00

^{is null or not}

depending

^on

whether the convergence of

Wk

^{is fast or not.}

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.

3 . 2 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

Then

on a _semi-log 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 asymp- totic direction of

log Gk

is less than or

equal

^{to zero}

so that

3. 3 Furthermore it is easy ^to show

that,

^when

Wa 1,

^the

probability P 00

^{that a} cluster grows

indefinitely

^{is zéro ;}

Gk

^{is an}

upperbound

^{to the}

proba- bility

that gk is ^non

null,

^this

probability

is itself an

upperbound

^to

Poo

^{and we can write}

for any value of k .

Now,

^if

Wa 1, lim

^{k- 00o}

^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}

396

FIG. 3. ^- GK : ^mean population ^{of each} generation according ^to experimental ^results.

generation Gk

up ^to the 20th

generation.

^{The half}

length

^{of error intervals is twice} the standard devia- tion. 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 computa- tion of the fertilities

Wk

^is

always possible

^{even in}

the nearest

vicinity

^of tc.

4. Validity

of the model of

branching

process with fertilities

depending

^{on the}

génération

^{number. -}

We 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.

For

that,

^we compute ^the theoretical

P;

^values

given by

this process where the values

Wk

^{are deduced}

from the

experimental Gk (Fig. 3) :

Then we compare with

PN experimental

^{values of}

the

figure

^2.

We 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

proba- bility PkN

^{that the}

family

descended from a man of the kth

generation (fertility Wk)

contains N men :

In

particular, k

^{= 0}

gives

^the

generating

^function

of the

probabilities P N :

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

the

PkN

coefficients

These relations have been used to compute

PN

N-1

and

PN -

¹

- L PN

^{with the}

^following

^values

k=l

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 W=3.}

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

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

^concen-

trations. In order to

clarify

^this

remark,

^{we consider}

in the

following

part ^a

branching

process where the

fertility

^{of a man}

depends

^on the number of men in its

generation.

We call this process ^a vital _space process.

5. Vital _space process. ^- We

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 experi-

mental

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.

FIG. 4. ^- W(n) : ^{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.

We 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. ^- PN ⁼ f(N). ^{Dots :} experimental ^{results. -} process with fertilities depending ^{on the} generation number. --- vital _space

process.

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. ^-

Figure

3 shows that in the _range k ⁼ 1 to

20, log Gk

^{is not} very far from a linear func- tion 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.

^We

follow a method similar to that used in

[11] :

The

expression (5)

^{is reduced to}

where

r is a

loop

enclosed in a circle of unit radius centered

on the

origin.

Then with the aid

of (4b) :

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}

FIG. 6. ^- Estimate of PN ^and PN given by ^the expression (8) of§6.

which is

The deviation is less than 3

%

^{for W} inside the interval

[0, 3.5].

Conclusion. ^- We 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].

We 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 3.018. All the

preced- ing

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

develop-

ment of a

family according

^{to a}

branching

process : for values of W less than the critical

valuesr

^the

experi-

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.

^{- We are}

grateful

^{to Professor}

M. Cadilhac and Professor J. Hervé for useful discus- sions 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), ^1963.

[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.