l3ynamical properties of fractal networks: Scaling,
numerical simulations, and physical realizations
Tsuneyoshi

Nakayama

and Kousuke Yakubo

Department of Applied Physics, Hokkaido Uni versi ty, Sapporo 080, Japan

Raymond L. Orbach
Department of Physics, University of California, Riverside, California 9ZSZ1This article describes the advances that have been made over the past ten years on the problem of fracton
excitations in fractal structures. The relevant systems to this subject are so numerous that focus is limited
to a specific structure, the percolating network. Recent progress has followed three directions: scaling,
numerical simulations, and experiment.
In a happy coincidence, large-scale computations, especiallythose involving array processors, have become possible in recent years. Experimental techniques such as
light- and neutron-scatteringexperiments have also been developed. Together, they form the basis for areview article useful as a guide to understanding
these developments and for charting future research
directions. In addition, new numerical simulation results for the dynamical properties of diluted antiferromagnets are presented and interpreted in terms of scaling arguments. The authors hope this article willbring the major advances and future issues facing this field into clearer focus, and will stimulate further
research on the dynamical properties of random systems.

a. Shape of core regionb. Asymptotic behavior

CONTENTS

I. Introduction
II. Percolating Networks

382
as an Example

of a Random

FractalA. Nodes-links-blobs model
B. Critical exponents
C. Correlation length and fractal dimension
III. Random Walks on Fractal Networks
A. Anomalous diffusion
B. Fracton dimension and the Alexander-Orbach conjecture
IV. Scaling Theories for Dynamics of Fractal Networks
A. Vibrational density of states and fracton dimension

1. Alexander and Orbach's original version
2. Finite-size scaling
B. Characteristics of fractons
1. Localization
2. Dispersion relation
3. Crossover from phonons to fractons: Characteristic frequency
V. Large-Scale Simulations and Physical Realizations
A. Vibrations of a percolating network
1 Density of states: Site percolation with scalarelasticity
a. Two-dimensional caseb. Three-dimensional case
2. Density of states: Bond percolation with scalarelasticity
3. Density of states: Vector elasticity
a. Theoretical prediction based on the nodes-

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~

links-blobs modelb. Simulated results for vector displacements
4. Missing modes in the density of statesa. Missing modes at low frequencies: scaling ar-

B.

interpretation
Localized properties of fractons
1. Mode patterns of fractons
2. Ensemble-averaged fractons

Reviews of Modern Physics, Vol. 66, No. 2, April 1994

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402

gumentsb. The hump at high frequencies:

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geometrical

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405

c. Multifractal behavior
C. Observation of fractons in real materials:

Neutronscattering experiments
1. Fractality of silica aerogels and other disorderedsystems

2. Observed density of states
VI. Scaling Behavior of the Dynamical Structure Factor
A. Theoretical treatments of the dynamical structure
factor S(q, co)
1. Expression for the intensity of inelastic scattering

2. Scaling arguments on S(q, co)
B. Numerical simulations of S(q, co)
C. Inelastic light scattering for fractal materials
1. Raman-scattering experiments
2. Analysis of inelastic light-scattering results forsilica aerogels

VII. Magnons and Fractons in Percolating Magnets
A. Fractons on percolating ferromagnets
B. Fractons on percolating antiferromagnets
1. Scaling arguments
2. Simulated results of the density of states
C. Dynamical structure factor of antiferromagnets
1. Theories and numerical simulations
2. Experiments
VIII. Transport on a Vibrating Fractal Network
A. Anharrnonicity
1. Phonon-assisted fracton hoppinga. Characteristic hopping distanceb. Contribution to the thermal conductivity
2. Temperature dependence of the sound velocity
B. Thermal conductivity of the aerogels
C. Transport properties of glassy and amorphous materials
1. Thermal conductivity
2. Sound velocity
D. Magnitude of the anharmonic
IX. Summary and Conclusions
Acknowledgments
References

0034-6861 /94/66(2) /381 (63)/$1 1.30

coupling constant1994 The American Physical Society

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382

Nakayama,

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I. INTRODUCTION

Over the decade a great deal of activity has been concentrated on understanding the nature of quantized excitations in fractal networks, such as their spatial anddynamical properties and their relationship to physicalobservables. In addition, the dynamics of fractal networks has been used as a model to aid the understandingof thermodynamic and transport properties of random
physical systems. Progress has followed three directions:scaling theories, numerical simulations,and physicalrealizations. The first has proven remarkably useful forthe physical interpretation of the dynamics of random
systems, the second for dealing with the complexity of
random systems, and the third for application of theideas developed through scaling and numerical simulations. A happy coincidence arose with the possibility oflarge-scale computations, especially those involving arrayprocessors, together with the developmentof lightscattering and low-energy inelastic neutron-scattering experimental techniques to the point where measurementsthat elucidate microscopic mechanisms have become possible.
The purpose of this review is to describe the advancesthat have been made over the past ten years in this field.
The relevant systems that have been investigated are so
numerous that we shall focus on a specific structure forclarity: the percolating network. This is the random
fractal most thoroughly studied, and one for which themost information is currently available. Although someof the physical systems that have been experimentally investigated most certainly do not map onto percolating
networks (e.g. , the aerogels), some certainly do (e.g. , sitediluted antiferromagnets).
However, even where themapping is not technically correct, the insights into thephysical properties of random systems afforded by examination of the properties of the percolation network will
prove extraordinarily useful.
Work on the dynamics of a percolating network received its most important promotion in the pioneering
paper of de Gennes (1976b). He formulated the problem
as follows: an ant parachutes down onto a site on the percolation network and executes random walk. 8%at is themean square distance -the ant traverses as a function oftime? By applying the scaling theory to this problem,
Gefen, Aharony, and Alexander (1983) developed theconcept of a range-dependent difFusion constant on the
percolation network, introducing an exponent 0 so thatthe diffusion constant, for length scales r less than the
percolation correlation length, scales as r
This insight, together with the realization that solving
the problem of diffusion was equivalent to solving the(scalar) elastic vibration problem (Montroll and Potts,
19S5; Alexander et al. , 1981), led Alexander and Orbach(1982) and Rammal and Toulouse (1983) to evaluate thedensity of states (DOS) for vibrations of a percolation
network with the introduction of the fracton dimension
d. This quantity was a specific combination of 0 and theRev. Mod. Phys. , Vol. 66, No. 2, April 1994

properties of fractal networks

fractal dimension Df. It should be noted that the DOS
had in fact already been calculated for magnetic systems
by Shender (1976a, 1976b) and, independently for deterministic fractals, by Dhar (1977). In particular, they
showed that the DOSs are characterized by a new exponent. Alexander and Orbach (1982) determined the
DOS and the dispersion relation for vibrational excitations of a fractal lattice, which they termed fractons, interms of the fracton and fractal dimensions. The conceptof crossover was introduced: at long length scales, the vibrational excitations were (softened) phonons with alinear dispersion law; whereas at length scales less than acrossover length scale (the percolation correlation lengthfor percolating networks), corresponding to frequencies
greater than an analogous crossover frequency, the vibrational excitations were fractons with their own vibrational dispersion law.
In addition, Alexander and Orbach (1982) noted thatthe fracton dimension for percolation networks was numerically remarkably close to the mean-field value, 4/3(exact in Euclidean dimension d=6), for all dimensions
greater than one, even though 0 and the fractal dimension Df were by no means constant as a function of dimension. This numerical evidence led them to speculatethat the fracton dimension might be exactly 4/3 for percolation networks for d ~2. This has come to be known
as the Alexander-Orbach conjecture. It is important because, if true, it would allow the conductivity exponent tobe evaluated in terms of static exponents. Recent numerical simulations, considerably more powerful, show thatthe conjecture is only approximate, though the values
found for the fracton dimension continue to be remarkably close to 4/3 for d ~ 2.
Immediately after this work appeared, Rammal andToulouse (1983) developed an analogous scaling method
to calculate the vibrational density of states and the mean
number of sites visited by de Gennes's "ant" during therandom walk on a fractal network. From this result,they were able to use the scaling theory of localization toestablish that fractons are localized for fracton dimensions less than two (see Sec. IV.H. 1).
Scaling has proven to be a very valuable tool for obtaining insight into the dynamical properties of fractal
structures. However, it has its limitations. In particular,
it is useful for ensemble averages of a physical quantity.
However, when products or matrix elements of physicalquantities are involved, it is by no means obvious (and, ingeneral, it is untrue) that the product or matrix elementof the individual ensemble average has any relationshipto the ensemble average of the product or matrix eleInent. This sharply limits the utility of scaling assumptions, though of course one might hope that any final result would exhibit scaling properties. The problem withrandom systems is that, without the use of average quantities, fIuctuations in general preclude the computation ofobservables obtainable experimentally, this being the ultimate goal for any microscopic theory. Instead, one mustresort to ensemble averages of numerical simulations.

Nakayama,

Yakubo, and Orbach:Dynamical

Fortunately, results obtained from such methods do appear to obey scaling. There are two types of scaling arguments: one actually computes critical exponents, and theother can be used to find relations between different exponents. The latter perspective is the one mainly used in
the subject related to this review. Again, one must resort
to numerical simulations to obtain the values of the critical exponents.
The scaling predictions of Alexander and Orbach(1982) for the vibrational density of states were first subjected to scrutiny numerically by Grest and Webman(1984). They simulated the vibrations of a percolating
network for a relatively small number of sites (N & 2200).
Nonetheless, they were able to establish that the predicted crossover between the phonon and fracton density ofstates was, in fact, present, and that the latter was, towithin their numerical accuracy, in accord with theAlexander-Orbach conjecture. The insights gained fromthis early work were very important for the direction offurther research.
As the size of supercomputers
increased, especiallywith the advent of array processors, the size of the percolating networks that could be simulated grew enormously (Yakubo and Nakayama, 1987a, 1987b). Morethan 10 sites are now possible, yielding a wealth of new
results (and insights). Examples are the calculation of thedensity of states for fractons; the asymptotic form of thefracton wave function; matrix elements for inelastic lightscattering in vibrating percolation networks; and the
dynamical structure factor (see Secs. V.A and V. B).
Numerical simulations do more than simply verify
physical assumptions: they point the way to a new qualitative understandingof the nature of excitations ofstrongly random structures, ultimately enabling the investigator to develop phenomenologicalexpressions forphysical quantities.
Soon, these simulations will shedconsiderable light on the debate over the use of scaling
for the interpretationof light-scattering experiments(Alexander,
1989; Alexander, Courtens, and Vacher,1993), as well as be useful for the calculation of matrix
elements for vibrational (hopping) transport processes. A
quite natural question to ask is this: are there physicalrealizations that justify extensiue numerical eQorts? Fortunately, there are a number of physical systems exhibiting measurable dynamical properties that exhibit fractal
geometry. Examples are site-diluted magnetic structures(spin-wave or magnon excitations crossing over to fracton magnetic excitations); silica aerogels (phonons, crossing over to fracton vibrational excitations); and glasses
and amorphous materials (which, though certainly not
fractals, appear to exhibit thermal transport properties
that coincide with predictions from phonon-fracton dynamics).
The dynamical properties of these materials have been
investigated by the measurement of the thermal conductivity and sound velocity, and by inelastic neutron- andlight-scattering experiments. In general, where the connection with a theoretical model has been direct (e.g. , theRev. Mod. Phys. , Vol. 66, No. 2, April 1994

properties of fractal networks

383

site-diluted antiferromagnet that maps onto the percolation network), detailed numerical agreement between
theory and experiment has been found. Where the mapping is less direct (e.g. , the silica aerogels), but where thefractal nature of the structure has been unequivocallyverified (see Secs. V.C and VI.C), scaling theory seems
adequate to represent the experimental data, though, ofcourse, unable to predict the value of the exponents. In
the case where the materials are certainly not fractals(e.g. , glasses and amorphous materials), the relevance of ascaling model is less direct. Remarkably, a fractal model
does appear to generate predictions for the higher temperature thermal transport and velocity of sound whichappear to be consistent with experimental observations.
All three of these features
scaling, numerical computations, and physical realizations
are coming together
at the present time. They provide a basis for a review article useful as a guide for understanding these developments and for charting future research directions. Ourreview begins in Sec. II with a detailed introduction to
the properties of a percolating network. We study the
dynamics of such a structure because it is a random fractal, and because so much analysis and understanding of
its properties have been developed over the past few
years. An excellent introduction to percolation theory isavailable (Bunde and Havlin, 1991; Stauffer and Aharony, 1992); so we shall only outline the most importantfeatures of this system. In Sec. III, we describe the solution of Gefen, Aharony, and Alexander (1983) for anomalous diffusion on a fractal network, with specific application to percolation. We then introduce the transform ofAlexander and Orbach (1982) and develop the concept
and application of the fracton dimension. The power andlimitations of scaling are developed in Sec. IV, including
the characteristic crossover length scale and frequency
for sound waves and fractons, the dispersion relation,and the vibrational density of states. Physical realizations for the density of states are introduced in Sec. V.
The dynamical structure factor S(q, co) gives rich information on the dynamics of fractal structures. We describe some features of S(q, co) for lattice vibrations offractal networks in Sec. VI, where the results of scattering experiments, scaling arguments, and simulations are
given by illustrating silica aerogels. The density of states
and the dynamical structure factor for diluted antiferromagnets are exhibited in Sec. VII. Inelastic neutronscattering measurements of S(q, co) by Uemura and Birgeneau (1986, 1987) allow a detailed comparison to bemade between experiment
Vibrational
and theory.
anharmonicity is introduced in Sec. VIII as a means for
fracton hopping. The associated contribution to thermaltransport and to the sound velocity is calculated. Commeasurements
parison is made with thermal-conductivity
for the silica aerogels that exhibit fractal geometry, andfor glasses and amorphous materials that do not. It is argued that, above a crossover frequency, all vibrational excitations of glasses or amorphous materials are localized,
allowing the use of the hopping model. Comparison is——Nakayama,

Yakubo, and Orbach:Dynamical

also made with sound velocity measurements on thesematerials. A detailed discussion is given of the magnitude of the anharmonic coupling constant extracted fromthese measurements.
Very recent modeling experimentsin accord with the apshow enhanced anharmonicity
proach taken here.
Our summary and conclusions constitute Sec. IX. Section IX also outlines some important research opportunities which we feel are available in this field.
We believe that the unusual conjunction of scaling
theory, numerical simulations, and physical realizations
has created an exciting climate for theoretical and experimental investigation of the dynamics of random structures. We hope that this review will bring the major advances and issues facing this field into clearer focus, andthat it will foster further research in the fascinatingworld of random systems.
II. PERCOLATING NETWORKS AS AN
EXAMPI E OF A RANDOM FRACTAL

That the percolating network is a fundamental modelfor describing geometrical features of random systems
and takes fractal (self-similar) structure was first noticed
by Stanley (1977). The theory of percolation was formulated by Broadbent and Hammersley (1957) in connection
with the diffusion of gases through porous media (Hammersley, 1983). ' They developed the geometrical andprobabilistic aspects of percolation. Soon after the paper
by Broadbent and Hammersley (1957), Anderson (1958)
and de Gennes, Lafore, and Millot (1959a, 1959b) pointed out the physical implications of percolation theory,and Domb and Sykes (1960) provided arguments supporting its critical behaviors. Since these works, it has been
widely accepted that percolation theory can be used tointerpret an exceptionally wide variety of physical andchemical phenomena. The concept of a fractal (Mandelbrot, 1975, 1977) has contributed significantly to ourpresent understanding of percolation. Percolation theory
describes satisfactorily a large number of physical andchemical phenomena,such as gelation processes (deGennes, 1979), transport in amorphous materials (Zallen,1983), hopping conduction in doped semiconductors(Shklovskii and Efros, 1984), and many other applications (Harder, Bunde, and Dietrich, 1986; Ingram, 1987).
In addition, it forms the basis for studies of the Aow ofliquids or gases through porous media.
The physical implications of percolation theory havebeen described in many review articles or books, enlightening both the static and dynamic properties of percolating networks (see, for example, Shante and Kirkpatrick,
1971; Kirkpatrick, 1973a, 1979; Stauffer, 1979, 1985;
Thouless, 1979; Essam, 1980; Farmer, Qtt, and Yorke,

The concept of percolation was introduced rather earlier by
Flory '1941a, 1941b, 1941c) and Stockmayer (1943) during theirstudy of the gelation process.
t,Rev. Mod. Phys. , Vol. 66, No. 2, April 1994

properties of fractal networks

1983; Stanley and Coniglio, 1983; Z allen, 1983;
Shklovskii and Efros, 1984; Aharony, 1986; Efros, 1986;
Sokolov, 1986; Deutscher,
1987; Havlin and BenAvraham, 1987; Feder, 1988; Bouchaud and Georges,
1990; Clerc et al. , 1990; Isichenko, 1992; Odagaki, 1993;
Sahimi, 1993), and in many conference proceedings (deGennes, 1983; Deutscher, Zallen, and Adler, 1983; Goldman and Wolf, 1983; Pynn and Skjeltorp, 1985; Pynn and
Riste, 1987). In particular, recent reviews by Bunde andHavlin (1991; see also Havlin and Bunde, 1991) and
Stauffer and Aharony (1992) bring the subject to its
current state of understanding.
There are two main kinds of percolating networks: siteand bond
To. create a site-percolating (SP) network,
deach intersection(site) of an initially prepared
dimensional lattice is occupied at random with probability p. Sites are connected if they are adjacent along a
principal direction. In a bond-percolation (BP) network,
all sites are initially occupied and bonds are occupied
randomly with probability p. At a critical (different) concentration p =p„both site and bond percolation exhibit
a single, infinite cluster spanning
all space. The
difference between the geometrical structure of SP and
BP networks is a short-range one, namely, a bond has
more nearest neighbors than a site. For example, in a
d=2 square lattice, a given bond is connected to sixnearest-neighborbonds, whereas a site has only fournearest-neighborsites. This is the reason why BPs always have smaller p, than those of SPs (see Table I in
which we list the values of p, for various percolationgeometries and dimensions).
A. Nodes-links-blobs

model

A very useful model for an infinite network near p, wasfirst proposed by Skal and Shklovskii (1974) and deGennes (1976a) and refined by Stanley (1977), Pike andStanley (1981), and Coniglio (1982b). This model, called
the nodes-links-blobs model, is quite useful for describing
the geometrical features of percolating networks, though
it was not evident a priori that such a simple picture wasmathematically valid.
The nodes-links-blobs model is based on the fact thatan infinite cluster contains a backbone network with acharacteristic length scale g(p), defined in Euclidean
space, and dead ends attached to the backbone. The
backbone comprises the networks made of links (quasi-ld
strings) and nodes (intersection of links). The length oflinks, i.e., the linear spacing g(p) between nearestneighbor nodes, corresponds to the correlation length ofthe percolating network. This is the original version of
the model proposed by Skal and Shklovskii (1974) and deGennes (1976a), which is called the SSdG model or thenodes-links model. This model neglects the strongly
bonded (multiconnected) regions in the links. The SSdG
model was refined by incorporating these parts, called
blobs. One imagines for a network at p p, in this pic-)

ture the percolating backbone consisting of a network of

Nakayama,

Yakubo, and Orbach:Dynamical

properties of fractal networks

385

TABLE I. Percolation thresholds p, for several lattices and the Cayley tree.
DimensionLattice

Sites

SquareTriangle
Honeycomb
KagomePenrose
Simple cubic (1st nn)

0.5927460+0. 0000005'

Simple cubic (2nd nn)
Simple cubic (3rd nn)
Body-centered-cubic

0. 137'
0.097'
0.245'

Bonds

1/2 fb2 sin(m/18) ( =0. 34729)
1—
2 sin(m. /18) ( =0.65271)

1/2~b0.6962'0.652 704
0.5837+0.0003
0.311 61'0.524 430
0.477 0+0.0002
0.248 65+0.0001 3'
0.2488+0. 0001'

0. 198'0.428'
0. 197+0.001'

Face-centered-cubic
Diamond

0. 141+0.001'
Simple cubic

0. 108'0.085'
1/(2d —I )1/(z —1)

Cayley tree

0. 18025+0.00015~
0. 1795+0.0003"
0. 119'0.388'
0. 161+0.0015"
0. 160 13+0.000 12'
0. 160 05+0. 000 15g
0. 118+0.01'
0. 1182+0.0002"
0. 118 19+0.000 04
0.094 075+0. 0001"
0.094 20+0. 0001
0.078 62+0. 000 03
0.078 685+0. 000 03g'Ziff (1992).
Sykes and Essam (1963, 1964).
'Stauffer (1985) and Stauffer and Aharony (1992)
Sakamoto, Yonezawa, and Hori (1989) and Sakamoto, Yonezawa, Aoki, et al. (1989).
'Grassberger (1986).
'Domb (1966).
IAdler et al. (1990).
"Gaunt and Sykes (1993).
'Jan, Hong, and Stanley (1985).
'Gaunt and Ruskin (1978).
Adler, Aharony, and Harris (1984).
'Nakanishi and Stanley (1980).
~(links), tying together a set ofstrongly bonded regions (blobs) whose typical separation
is of the order of the correlation length g(p) [see Fig.

quasi-ld string segments

1(a)].
Stanley (1977) called the links red bonds and gave thefollowing definition: Consider the situation in which a
voltage is applied between two sites at opposite edges of ametallic percolating network at p, . %'hen the singly connected region (red bonds) is cut, the current fiow stops.
This bond carries the total current. In this connection,
Stanley and Coniglio (1983) introduced the terms blue
and ye/low bonds. Blue bonds carry current. But when ablue bond is cut, the resistance of the system increases.
Yellow bonds belong to dead ends and can be cut out
without changing the resistance.

B. Critical exponents
The occupation probability p of sites or bonds in percolation theory plays the same role as the temperature in
Rev. Mod. Phys. , Vof. 66, No. 2, April 1994

thermal critical phenomena. There exists a critical concentration p, below which (p (p, ) only finite clusters exist and above which (p &p, ) an infinite cluster is present
as well as 6nite clusters.
Let us define n, (p) as the average number (per site) offinite clusters with the size (site number) s. The quantityn, (p) is related to various physical quantities characterizing the network. The scaling theory of percolation isbased on the idea that there exists a certain "parameter"
characterizing the system, which diverges at p =p, . This
allows one to express the scaling Ansatz for n, (p) bydefining a parameter s (p):n, (p) =s

I' [s/s (p)where I' (x) is an unknowns (p) close to p, as],function.

Taking the form of(2. 1)

one has an alternative equation for p ~p, and s~ ~,Nakayama,

Yakubo, and Orbach:Dynamical

properties of fractal networks

Substituting

P(p)=

node
J

I

blob

Eq. (2.2) into the above equation, one finds

f s' '[F(0)—F(x)]ds+(p —p, ),where x =(p —p, )s". Changing the variable s to x gives

P (p) =Po(p —p, ) (pp, )~+(p —

)p,

)

(2.3)

where Po is the constant prefactor and the critical exponent P is defined asdead end'n(2.4)
Because /3 is always less than unity, except for the case of
the Cayley tree (see Table II), the first term of Eq. (2.3)
dominates. Note that P= 1 for the Cayley tree, and thetwo terms in Eq. (2.3) are of the same order in (p —p, ).
The quantity P(p) was first introduced by Broadbent(1954), corresponding to the order parameter for thermal
critical phenomena.
The exponent o.. This is the exponent for the total
number of finite clusters given by

M(p)= g n, (p) .

(2.5)

The function M(p) consists of the analytic part M'(p) attop =p, and a singular part M"(p) proportional—p,
as seen from the same procedure as that in~p
the case of Eq. (2.3). The critical exponent a is given by

~,

(2 6)

For a thermal phase transition, the exponent e corresponds to that of specific heat, and M(p) to the free enerFIG. 1. Schematic of the nodes-links-blobs model. (a) A percolating network above p, . The solid circles indicate nodes)

forming the homogeneous network at length scales I. g. (b)Links-blobs model at p =p, . The hierarchical structure ofblobs is stressed here.

n, (p)=sF[(p —p, )s"] .

(2.2)

Equation (2.2) allows scaling forms to be obtained forvarious physical quantities around p, in terms of a set ofcritical exponents.
The exponent P. The probability that a site belongs to
the infinite network, P(p), is associated near p, with theexponent P. Since an occupied site must be either in a
finite cluster or in the infinite cluster, one has the exact
relation,

P(p)+ g n, (p)s =p .
For p &p„only finite clusters exist and g, n, (p)s =p.
When p approaches p, (p & p, ), the above relation be-gy.
The exponent y. The average mass S(p) (number
sites or bonds) of finite clusters is related to n, (p) by&(p)=g n, (p)sg n, (p)sNote here that the factor n, (p) /sg, n, (p)s is the probabiIity of an occupied site belonging to a cluster of s sites.
Becausen, (p)s =p when p (p„one has the equationin the limit of p —+p,g,g n, (p)s
&(p)=

(2.7)
Pc

Substituting Eq. (2.2) into (2.7), the critical exponent for
the average mass can be given by

S(p) =&o~p —p,~where So is the constant prefactor and

(2.g)

comes

P(p)= g [n, (p, ) —n, (p)]s+(p —p, ) .
Rev. Mod. Phys. , Vol. 66, No. 2, April 1994

of

For thermal critical phenomena,is the susceptibility.

the analogous

quantity

Nakayama,

Yakubo, and Orbach: Dynamical properties of fractal networks

387

TABLE II. Percolation exponents for d=2, 3, 4, 5, and d «6. Rational numbers give exact results, whereas those with a decimal
fraction are numerical estimates.

d=3

Exponents

0 463"

5/36'

45/18'

Df

91/48'

0.454+0. 008
0.474+0. 014'
0.435+0.0350
0.43+0.04
0.405+0. 025'
1.73+0.03
1.79'
1.71+0.06"
1.82+0. 04"
1.74+0. 015'
1.77+0.02
1.805+0. 02'
0.88+0.02
0.89+0.01"
0.88+0.05
0.90+0.02"
0.83+0.01
0.81'
0.875+0.008
0.872+0. 007'
2.48+0. 09

0.665+0. 15'
0.64'
0.65+0.04
0.64+0. 02"
0.639+0.020'

0.83+0. 1'
0.84'
0.835+0.005"
0.835+0.005'

1.41+0.25'

1.25+0. 15"
1.20+0. 03'
1.18"
1.185+0.005'

1.44+0. 05'
1.44"
1.435+0. 015'

0.68+0.03
0.68"
0.678+0. 050'

0.57"
0.571+0.003'

3.21+0.07'
3. 12+0.02P

3.74+0. 4'
3.69+0.02"

Nienhuis (1982), den Nijs (1979), Nienhuis, Riedel, and Schick (1980), Pearson (1980).
Gaunt, Whittington, and Sykes (1981).
'Adler, Aharony, and Harris (1984) and Adler et al. (1986a, 1986b).
Gaunt and Sykes (1983).
'Adler et al. (1986a, 1986b).

Grassberger (1986).
gAdler (1984).

"de Alcantara Bonfim, Kirkham, and McKane (1980, 1991).
'Adler et al. (1990).
'Gaunt, Sykes, and Ruskin (1976).
"Saleur and Derrida (1985).
'Reeve (1982), Reeve, Guttmann, and Keck (1982).
Strenski et al. (1988).
"Margolina, Herrmann, and Stauffer (1982), Heermann and Stauffer (1981).
'Stanley (1977).
PJan, Hong, and Stanley (1985).

C. Correlation length and fractat dimension
The lower cutofF scale characterizing the percolating
network is the length a that forms the lattice spacing of
the original network. There exists another characteristic
length g(p), called the correlation length, which wasmentioned in Sec. II.A. This 1ength scale exhibits critica1
behavior in the vicinity of p, .
The exponent v. The diameter of finite clusters below
p, is characterized by the correlation length g(p), definedas the root-mean-square distance between two sites i andin the same cluster, averaged over all finite clusters.
The average distance between two sites in a given s cluster is written byjRev. Mod. Phys. , Vol. 66, No. 2, April 1994'=, g — f'.

R,

1

/r,

S

r,

E~

The number of ways of connecting two sites in a given
cluster of size s is
so that s n, becomes the number ofways of connecting two sites in clusters with the same
size s. The correlation length g(p) is R, averaged by theprobability s n, /g, s n, This is expres. sed bys,g~(p) =

QR, s n,
s n

(2.9)

S

We assume the same scaling form for R, as in Eq. (2.2),

R, =s M[(p —p, )s"],(2. 10)Nakayama,

Yakubo, and Orbach: Dynamical properties ot tractal networKs

where m is the new critical exponent. Substituting(2.2) and (2. 10) into Eq. (2.9), one has, close to—p,Wp) ==-Ol»

p„

CO

Ioo

(2. 1 1)

I

100

1000

Eqs.
C)

30

where =o is the constant prefactor and

LD

0.1

0]

0)
C3

The correlation length g(p) represents the characteristic
size of the voids in the percolating system when p & p„
and the characteristic size of a finite cluster when p & p, .
The fractal dimension DI Sta. nley (1977) was the first
to notice that percolating networks exhibited selfsimilarity and could be characterized by a noninteger
Mandelmass dimension, i.e., that they were "fractal.
brot gave a simple definition of fractals: A fractal is ashape made of parts similar to the whole in some way(Feder, 1988). Fractals can be classified as deterministic
or random, depending on whether the self-similarity isexact or considered as the average. Percolating networks
are a typical example of a random fractal.
From Eq. (2. 10), one sees R, ~s~ at p =p, . This isrewritten as"s(R, ) ~R'(2. 12)It should be noted that s (R, ) is a measure of the system;
i.e. , s (R, ) corresponds to the "mass" M(R, ) in our prob-

lem. The exponent D& = 1/m is called the fractal dimension or Hausdor6' dimension. We can interpret g(p) as alength scale up to which ihe cluster can be regarded as
fractal. For the percolating network for p & p„ thestructure can be regarded as homogeneous for 1engthscales larger than g(p). Kapitulnik et al. (1983) demonstrated through Monte Carlo simulations that the networks for p p, are homogeneous on length scales L g
and fractal on scales L g. Summarizing,

)

~,
'L

(

L«g,

M(L) ~ L", L»)(2.13)

There are many reviews and books concerning the properties and uses of fractals (see, for example, Mandelbrot,
1975, 1977, 1982, 1989; Family and Landau, 1984; Stan1ey and Ostrowsky, 1986; Barnsley, 1988; Vicsek, 1989;
Feder and Aharony, 1990; Sapoval, 1990; Takayasu,
1990; Feder, 1988; Peitgen and Saupe, 1988; Pietronero
and Tosatti, 1988; Bunde and Havlin, 1991; Family andVicsek, 1991).
The shorter limiting length scale l„characterizing theunderlying components making up the fractal, depends
on the type of percolation, i.e., site (SP) or bond (BP).
This is because the random filling produces a relatively
small number of neighboring sites around an occupied
site in SP networks, whereas BP networks have many
masses on neighbors that are not directly connected.
This severely inAuences the di6'erence in short-range
character of these networks. Figure 2 presents results ofa calculation by Stoll, Kolb, and Courtens (1992), which
Rev. Mod. Phys. , Vol. 66, No. 2, April 1994

0

0 o
o 0

0, 01

~ ooo~

001
0.03
Reduced wave vectorqFIG. 2. Structure factor S(q) for SP and BP networks

d=2. The dots are for L=12 systems

withover 20000
realizations), and the full curves for L=6800. The dotted lines
have slope D& =91/48. After Stoll, Kolb, and Courtens (1992}.
(averaged

indicates that the lower limiting length scale l, of BP networks is shorter than that of SP networks by one order of
1Tlagn1tude at' + pc
The criticality of li nks T'he . critical behavior of thelinks (red bonds) was derived by Coniglio (1982a, 1982b)
using renormalizationarguments.
Coniglio (1982a,
1982b) verified that the number of red bonds varies withpas
L i ~ (p —p, )' ~ g(p)'~' .

(2. 14)This is rigorous in all dimensions. This relation also indicates that the fractal dimension of the links is 1/v beis the measure in this case.
cause
The chemical distance. %'e should mention that there
are exponents, besides those described so far, that are
useful for the description of the purely static geometricalproperties of fractal networks. An example is the shortest path A, along the percolating network from one site(i) to another ( j). We can define the shortest path R, between the site i and as the minimum number of steps bywhich we can reach from i, with restriction to existing
paths between connected sites. This is termed the chemical distance R, (Alexandrowicz, 1980; Middlemiss, Whittingston, and Gaunt, 1980; Pike and Stanley, 1981; Havlin and Nossal, 1984; Cardey and Grassberger, 1985). It
is not the same as the linear length A, measured between
the two points (referred to as the "Euclidean distance").
The chemical (or topological) dimension d, is defined

I,

j
j

from

M(R, ) ~R,',

where M(R, ) is the mass within the chemical distanceR, . From Eqs. (2. 13) and (2.14), the chemical distance

between two sites that are separated by the Euclidean distance L, is given byC

cc' L,

fC

minimum
Herrmann=0;„is the fractal dimension for the1984;
two sites (Stanley,
path between
and Stanley, 1988). The chemical dimension

The ratio of D&/d,Nakayarna,

Yakubo, and Orbach:Dynamica1 properties of fractal networks

d, has been determined numerically for d=2 and d=3
networks,
and they take the values
percolatingd, = 1.678+0.005 and d, = 1. 885+0.015, respectively(Havlin and Nossal, 1984; Havlin et al. , 1985, Herrmann
and Stanley, 1988; Neumann and Havlin, 1988).
Scaling relations. The relation between D& and the exponents p and v can be derived from a simple argument.
For percolating networks with p
the correlation
length g(p) is finite and is the unique length scale describConsider theing geometrical features on a scale I.
number of sites M(g) within a box of size g. From thedefinition of P (p), iM'(g) is given by

)p„

))a.

M(g) ~g P(p) ~g"—

Thus the fractal dimen-——(2. 15)where P(p) ~(p p, )~ is used.
sion D& is given by

Df =dThis relation is called the hyperscaling relation, since it
depends on the Euclidean dimension d. The exponents p
and v are universal, so that D& is also universal. They
depend neither on the lattice structure nor on the type of
percolation (SP or BP), but are a function only of thedimensionality d.
It is important to note that the critical exponents
defined in Eqs. (2.4) —(2.8) are related to each other. Using Eqs. (2.4) —(2.8), one has the scaling relations

a=2 —dv=2 —2P —y,1

p+y(2. 16)(2. 17)(2. 18)389

field theory is valid is d=6. Mean-field percolation canbe modeled by percolation on a Cayley tree (Bethe lattice). Hyperscaling relations (2. 15) and (2. 19) are valid
for space dimensions less than the critical value d=6. In
the classical range d ~ 6, the values for all the exponents
are given by the relation at d=6, namely, p=l and
v= 1/2, independent of d.
III. RANDOM WALKS ON FRACTAL NETWORKS

The percolationtransition

has been characterized,

inSec. II, by quantities such as the cluster size n, (p), themean size of a finite cluster S(p), the correlation lengthg(p), and the order parameter P(p). These quantities describe the static (geometrical) properties of percolating
networks. In this section, we describe the dynamic properties of percolating networks.
The first example isdiffusion of randon1 walkers on a percolating network.
In uniform systems, the mean-square displacement of arandom walker, (R (t) ), is proportional to the time t,

for any Euclidean dimension d. In percolating systems,
for a length scale (g', difFusion is anomalous (Gefen,
Aharony, and Alexander, 1983). The mean-square displacement is described by the form

( R 2(t) ) c t2/(2+8)(3. 1)

0)with
0. This slowing down of the diffusion is causedby the delay of a diffusing particle because of hierarchically intricate structure and the presence of dead ends.
A. Anomalous

diffusion

The diffusion coeKcient of random walkers is defined

and

by

dv=2P+y .

(2. 19)All the exponents given above can be found if the valuesof two of them are known.
The values of the exponents have been calculated byvarious methods:
the renormalization-groupmethod(Bernasconi, 1978a, 1978b; Fucito and Marinari, 1981;
Derrida and de Seze, 1982; Lobb and Karasek, 1982;
Sahimi, 1984), the cluster expansion methods (Sykes,
Gaunt, and Glen, 1976a, 1976b; Gaunt, 1977; Nakanishiand Stanley, 1980; Gaunt, Whittington, and Sykes, 1981;
Adler et al. , 1986a, 1986b; Sykes and Wilkinson, 1986;
Harris, Meir, and Aharony,
1987; Takayasu andTakayasu, 1988; Adler et al. , 1990; Wada, Watanabe,and Uchida, 1991), Monte Carlo calculations (Margolina
et al. , 1984; Rapaport, 1985; Kim et ai. , 1987), and thefinite-size scaling method (Chayes et al. , 1986; Sakamoto,
Yonezawa, Aoki, et al. , 1989; Sakamoto, Yonezawa, andHori, 1989). Table II shows the values of these criticalexponents. For d=2, the critical exponents are known

exactly.
The upper critical dimension
Rev. Mod. Phys. , Vol. 66, No. 2, April 1994

above which the mean-1d(R (t))dt
where (R2(t) ) is the mean-square displacement after ttime
For p p, and the length
steps.
scale(R (t))'~ ))g(p), the system looks homogeneous andnormal diffusion holds,
2

)

(3.2)
where the diffusion coe%cient D on this cluster is related to the dc conductivity o. z, through the Einstein relation
e nD

kB T
where e denotes the carrier charge, n their density, kB
the Boltzmann constant, and T the temperature.
The subscript ~ of D rejects the fact that, on thesescales, the dc conductivity occurs only through theinfinite cluster (p )p, ). Empirical evidence for powerlaw behavior was given by Last and Thouless (1971) and390

Nakayama,

Yakubo, and Orbach: Dynamical properties of fractal networks

by Watson and Leath (1974). Last and Thouless (1971)
measured the current through sheets of graphite paperwith randomly punched holes. The exponent p is defineding to the form G(z),finds the asymptotic form„o-z

~ fromEq. (3.5). One(3.8)

through the relationo (p) =x,(p —p, )~ .

From Eqs. (3.7) and (3.8), we have the relations

Noting that the carrier density n is proportionalto P(p),

VD„~(p —p, )" ~.

(3.3)

We should emphasize that p —/3 is always positive bewith p+ —cause Dfor
Euclidean dimension
any
+p,
d.
The mean squared distance (R (r) ) should be linearly
proportional to the time t for p )p, and (R (t))'~ ))g.
One has from Eqs. (3.2) and (3.3)~0(R'(r) ) ~ t(p —p, )" ~ .

—p,
" is used.
g~ Ip —
p,
Ip

I

(3.5)

Here the relation
All of theseequations can be derived from the dynamic scaling form,I(R'(r) )'"= r "G (p p, )r'j . —
I

(3.6)

)

When Eq. (3.4) holds for p p„Eq. (3.6) leads to thez'" P'~ . One can thenasymptotic form G (z),derive the relation

„~

2

For p

3'

'
~ +x= —

(p„(R t) ) becomes independent of time t, lead(

1(3.10)

2v+ p —P
these into Eq. (3.6), one finds at p =p,(R'(t)) ~ t(3.1 1)

where

d. =2+"(3. 12)This is of the same form as Eq. (3.1), if we define
8=(p —P)/v. The slow process described by Eq. (3.11) iscalled anomalous diffusion, and d is called the diffusion
exponent.
From Eq. (3.6) one can estimate the characteristic time
~ of the crossover from anomalous diffusion to normal
diffusion.
Anomalous
diffusion occurs when tw,whereas normal diffusion occurs when t~. Thecharacteristic time ~ can be obtained by setting the argument in the function G (z) in Eq. (3.6) to be of the orderof unity:

»

~(

(3.7)

2

=

Substituting(3.4)

For finite-size clusters, (R (t) ) becomes independent oftime t after a sufficiently long time. One has for p & p,

(3.9)

2v+p —/3 'we find that)—(2v+p —P)(3. 13)The value of d is obtained by direct numerical calculations of (R (r) ), or from the numbers of sites visited by

TABLE III. Dynamic exponents of percolating networks.

d~6

d=4

Exponents

2.69+0.04'
2.871+0.001

3.45+0. 1'
4.00+0.OS"
3.755

297+ 0. 007e

1.264+0. 054g
] 303 + 0. 004h
1.315+0.008'
1.334+0.007'
1.323+0.004'
1.322+0. 003"
1.33+0.01'
1.325+0. 006"
'Hong et al. (1984).
Havlin and Bunde (1991).
'Movshovitz and Havlin (1988).
dRoman (1990).
'Lobb and Frank (1984}'Adler (1985).
~Sahimi et al. (1983).
"Frank and Lobb (1988).
~

Rev. Mod. Phys. , Vol. 66, No. 2, April 1994

«

2.04'
1.876+0. 035g

1.14+0.02'
1.32+0. 06'
1.328+0. 01
1.31+0.02'
1.317+0.03~

2.39'

2.72'

39m

44II1

1.36"

1.36"

1.31+0.03'
1.279+0. 07"'Octavio and Lobb (1991}.
'Essam and Bhatti (1985).
"Zabolitzky (1984).
'Argyrakis and Kopelman (1984).
Alexander and Orbach (1982).
"Daoud (1983).
See Fig. 5 in Sec. V of this review.
"See Fig. 6 in Sec. V of this review.

4/3

Nakayama,

Yakubo, and Orbach: Dynamical properties of fractal networks(Benrandom walkers within the time t, V(t) ~ t Dff
Avraham and Havlin, 1982, 1983; Argyrakis and Kopelman, 1984; Hong et al. , 1984; Pandey et al. , 1984; Pandey, Stauffer, and Zabolitzky, 1987; Wagner and Balberg,
1987; Movshovitz and Havlin, 1988; Bunde, Havlin, andRoman, 1990; Roman, 1990; Duering and Roman, 1991;
Havlin and Bunde, 1991). When calculating these quantities, one must note that Eq. (3. 11) represents the
diffusion for t
on the single infinite cluster. Valuesof
as well as other dynamic exponents, are given in
Table III. From Table III, we see that, for any Euclidean
dimension d, d is larger than the value 2, the value fornormal di6'usion.
/d

«r

d,

B. Fracton dimension and theAlexander-Orbach conjecture
The linear size of the region of sites visited by the"ant'* afterr-timefromsteps
Eq. (3.11) is(R (t))' ~t' ' + '. Therefore the number of visited
sites V(t) becomesV(t) ~R f ~r

(3. 14)

where the fracton (or spectral) dimension d is defined by' 2+8 d.
2Df

2Df

(3. 15)Alexander and Orbach (1982) tabulated the thenknown values for the quantities Df, 0, and d for percolating networks on d-dimensional Euclidean lattices. They
pointed out that while Df and 0 change dramaticallywith d (below d=6), d does not. They conjectured fromthe numerical values for d that for percolationd =4/3,for 2 & d & 6 .

1
p= —[(3d —4)v —P] .

(3. 17)2the dynamical

exponents, such as d,d, orp, has been a challenge in the past decade, and many conjectures have been proposed.

exponents are not known, except for the case d ~ 6 (footnote 3). Values for p are usually estimated by numericalmethods (Fogelholm, 1980; Derrida and Vannimenus,
1982; Li and Strieder, 1982; Mitescu and Musolf, 1983;
Sahimi et al. , 1983; Herrman et al. , 1984; Lobb andFrank, 1984; Zabolitzky, 1984; Sarychev et al. , 1985;
Seaton and Glandt, 1987; Frank and Lobb, 1988; Normand et a/. , 1988; Gingold and Lobb, 1990; Octavio andLobb, 1991); by analytical approximations such as series
expansions (Fisch and Harris, 1978; Adler, 1985; Adler
techet al. , 1990), small cell real-space renormalizationnique (Bernasconi, 1978a, 1978b), and the c.-expansion
method (Harris et al. , 1984; Harris and Lubensky, 1984,
1987; Wang and Lubensky, 1986; Harris, 1987); and fromexperiment (Song et al, 1986; Wodzki, 1986; Domeset a/. , 1987; Careri et al. , 1988; Lin, 1991).
The Alexander-Orbach conjecture leads to a value ofthe ratio p/v equal to 91/96=0. 948 for d=2 percolating
networks. This exponent p/v describes how the resistivity R diverges with the linear size I. of the system,
R ~ L "~ . The value of p/v has been studied numericallywith the finite-size scaling technique (Hong et al. , 1984;
Lobb and Frank, 1984; Zabolitzky, 1984; Normand
et al. , 1988). Gordon and Goldman (1988a, 1988b) tried
to obtain the value of p/v experimentally for Al thinfilms of 50 nm thickness. They prepared an 800X800
square lattice by exposing Al thin films to an electron
beam. These values for p/v previously reported areslightly smaller than the conjectured value 91/96, whichmeans that d is less than 4/3.
The exponent d is also related to d by Eq. (3. 15). If
the Alexander-Orbach conjecture holds, d should takethe value of 91/32 = 2. 844 for 0= 2 percolating networks.

(3. 16)The expression "hyper-universal*'or "super-universal"
was coined to express the possibility that an exponent
could be independent not only of the details of the lattice
but also of the dimensionality d itself (see Leyvraz andStanley, 1983). This conjecture is crucial, because, if exact, the dynamic exponent p can be related to the staticones through the relation

Determining

391

The exact values for these

2The term "fracton" denotes a localized mode peculiar to fractal structures, coined by Alexander and Orbach (1982). These
excitations exist not only for vibrational systems, but also fordilute magnets. This dimension (d ) plays an important role indescribing the dynamical features of percolating networks, suchas the density of states, dispersion relation, and localization.
This subject is discussed in detail in Sec. IV.
Rev. Mod. Phys. , Vol. 66, No. 2, April 1994

3The fracton (or spectral) dimension can be obtained exactly
for deterministic fractals (Rammal and Toulouse, 1983; Rammal, 1983, 1984a; Given, 1984; Southern and Douchant, 1985;
Yu, 1986; Ashraff and Southern, 1988). In the case of a ddimensional Sierpinski gasket {Rammal and Toulouse, 1983;
Rammal, 1983, 1984a), the fracton dimension isd =2

ln(d+1)ln(d +3)

We see from this that the upper bound for a Sierpinski gasket isd=2. Hattori, Hattori, and Watanabe (1986) have noted that
this upper bound is a general result for fractal systems in whichcoarse-graining treatments are applicable. It is interesting tonote that the fracton dimension for a d=3 Sierpinski carpet
takes the value d= 2.894 (Hattori, Hattori, and Watanabe, 1985;
Hattori and Hattori, 1988). This is because the coarse-graining
treatment is not valid for this carpet. Bourbonnais, Maynard,
and Benoit (1989) have shown the existence of channelingmodes in the d=2 Sierpinksi carpet which have large amplitudes along the line of dense matter in the network. BenAvraham and Havlin (1983) and Havlin, Ben-Avraham, andMovshovitz {1984) have presented a family of exact fractals
with a wide range of fracton dimensions, including the case of

d=2.

392

Nakayama,

Yakubo, and Orbach:Dynamical

The Monte Carlo method for random walk is useful for
estimating the exponent d„ through Eq. (3. 11) (BenAvraham and Havlin, 1982; Havlin and Ben-Avraham,
1983; Havlin et al. , 1983, 1984; Hong et al. , 1984; Majid
et al. , 1984; Pandey et al. , 1984, 1987; McCarthy, 1988;
Roman, 1990; Duering and Roman, 1991). The calculated values of d are somewhat larger than 91/32, whichcorresponds to d ~ 4/3.
The fracton dimension d can be evaluated directly by
where S& is the number of disthe relation
tinct sites visited during an X-step random walk on an
infinite percolating network (Rammal and Toulouse,1983). Using this relation, the value of d has been numerically calculated by random-walk simulations (BenAvraham and Havlin, 1982, 1983; Havlin and BenAvraham, 1983; Argyrakis et al. , 1984; Rammal et al. ,1984; Keramiotis et al. , 1985). The results indicate thatthe values of d are quite close, but not equal, to 4/3. The
exponent d has also been calculated by series expansion
andBhatti (1985). Their result
isby Essamd = 1.334+0.007 for d= 2 percolating networks.
Although the value of fracton dimension d is different
depending on the method used, it is now accepted that
the Alexander-Orbach conjecture is not correct and has
really no analytical basis apart from numerical coincidences; the true value of d is slightly smaller than 4/3
for d&6 (see Table III). It remains, nevertheless, a remarkably accurate estimate of d for all d & 2.

5&-X",

IV. SCALING THEORIES FOR DYNAMICS

properties of fractal networks

nection with the results of computer simulations on the
DOS of percolating networks incorporating the vector
nature of atomic displacement. In this section we review
the works of both Alexander and Orbach (1982) andRammal and Toulouse (1983).

1. Alexander and Orbach's original version
As noted in Sec. III, de Gennes (1976b) posed the following problem. "An ant is dropped onto an occupied siteof the infinite cluster of a percolating network. The ant at
every time unit makes one attempt tojump to one of its adthat site is occupied, it moves to that site
jacent sites.
it is empty, the ant stays at its original site Wha. t is the(ensemble) averaged square distance that the ant travels ina time t?" The scaling argument to this problem waspresented by Gefen, Aharony, and Alexander (1983) as
described in Sec. III.A. The solution opened the way fora nearly complete description of the dynamics of fractal

If

If

networks.
The structure of the diffusion equation is the special
case of a master equation, which in turn has the same
form as the equation of motion for mechanical vibrationsor the linearized equation of motion for ferromagnetic
spins (Alexander et al. , 1981). This allows the vibrational problem to be mapped onto the diffusion problem.
The master equation of diffusion on a lattice is written
dI';

=gW;P

OF FRACTAL NETWORKS
A. Vibrational density of statesand fracton dimension

The fracton dimension d is a key dimension for
describing the dynamical properties of fractal networks
in addition to the fractal dimension Df. Df describes
how the mass of the geometrical object depends on its
length scale, whereas the fracton dimension d characterizes anomalous diffusion on the fractal system. Alexander and Orbach (1982) mapped the problem of anomalous diffusion onto the vibrational problem with scalarelasticity. They showed that the basic properties of vibrations on fractal networks, such as the density of states(DOS), the dispersion relation, and localization, arecharacterized by the fracton dimension d. Rammal andToulouse (1983) derived the fracton dimension d via ascaling argument.
They showed that various randomwalk properties, such as the probability of closed walks
and the mean number of visited sites, were governed by
the fracton dimension d. Feng (1985a, 1985b) included
vector displacements in terms of the nodes-/inks-blobs
model described in Sec. II. He claimed that the rotationally invariant Hamiltonian having a stretching force constant and an angular force constant required an additional dimensionality db, which we call the bending-fracton
dimension. His theory is discussed in Sec. V.A. 3 in conRev. Mod. Phys. , Vol. 66, No. 2, April 1994

(4. 1)

where I'; is the occupation probability of the diffusing
on site i, and 8' is the probability that adiffusing particle hops from site i toIf the diagonal
elements
are de6ned to satisfy the condition-,
gi W; =0, Eq. (4. 1) reduces to the conventional maser
equation. The equation of motion for lattice vibrations
with scalar nearest-neighbor interactions is expressed byparticle

8,

u

m;

dt 2

j.

-

= QXtuj(4.2)

where m; and u; are the mass and the displacement of
the atom at site i, and K, is the spring constant connecting two atoms at sites i and j, respectively. The diagonal
elements K;, satisfy the condition
K; =0, due to theuniform-translational
invariance of the system; i.e., theuniform translation gives rise to no additional energy[this can be derived by putting u. =const in Eq. (4.2) for
The only difFerence between the two equations isany
the order of the time derivatives (Alexander et al. , 1981).
The spectral DOS can be obtained from the single-site
Careen's function for the vibrational problem bygj].

X)(E) = —— lim Im(PO(1

7T

where ( .

a~0

8+i0+) ),—(4.3)) means the ensemble average. The function

Nakayama,

Yakubo, and Orbach:f—st)(PO(t))dt,

exp(

(4.4)

where Po(t) is the autocorrelation function.
The physical meaning of (Po(t) ) in the corresponding
diffusion problem is the probability of finding the particle
at the origin at time t if it were initially at the origin attime t=O. For compact difFusion (d &2) (Alexander,
1983; Rammal and Toulouse, 1983), we have the relation(4.5)2)(bee) ~bee /(4. 12)The explicit expression for the exponent a of the dispersion relation is obtained from the exponent of anomalous
difFusion [Eq. (3. 11)]; i.e., the corresponding mappings oft~l/henand (R

L ~EN)' ~L yield—2/d

The comparison of the assumed dispersion relation (4. 11)
with this equation leads toV(t) is the number of visited sites within time t
From Eq. (3. 14), this quantity is written as

where

d

Df

2

d

(4.6)
where d( & 2) is defined by Eq. (3.15) in the context of the
diffusion problem as

2Df

2+ 6I
The substitution

where Eq. (3. 12) has been used.
Since the structure is fractal (self-similar), he@ can be
replaced by an arbitrary frequency cu. Namely, one has
the DOS from Eq. (4. 13),2)(co) 0- co"of Eq. (4.6) into Eq. (4.4) leads to(Po(E)) =constXE/'

J 0 exp( —x)x

/ dx

.

(4.7)

Note that the integration takes a real positive value. By+ in Eq. (4.7) and substituting intoe~ co +iO—co

(4. 8)It should be noted that d must be smaller than 2 due tothe condition for convergence of the integral in Eq. (4.7).
This conclusion is valid under some conditions (see footnote 3). The DOS is obtained as2)(co) ~ co

~ [L (co ) ](4. 14)

where

D
a

Eq. (4.3), one has the result+( 2) ~~d 2(4. 13)and so the "dispersion relation" becomes

letting

2Df(4. 15)As seen from Eq. (4. 15), the fracton dimension d can beobtained knowing the value of the conductivity exponent
p. The fracton dimension d is an intrinsic parameter related to the dynamics of complex systems, and it therefore affects physical properties on a deeper level than any
other exponents.

(4.9)

where the relation d(co )=2codco is used. In analogy tothe usual Debye density of states m" ', Alexander andOrbach (1982) called the related excitations "fractons,
and d the "fracton dimension.
It was also called the"spectral dimension" by Rammal and Toulouse (1983)
because it represented the DOS for the vibrational excitation spectrum.

""

2. Finite-size scaling2)(dc', L) =1

L

B. Characteristics of fractons
1. Localization
Rammal and Toulouse (1983) showed that fractons are
spatially localized at a fracton dimension d(2. They
used the so-called PL function defined by Abrahams
et aI (1979) in t.heir scaling theory of localization:g(L) ~L

Rammal and Toulouse (1983) also derived the vibrational DOS of fractals using a finite-size sca1ing approach.
Consider a fractal structure of size L with fractal dimension Df. The DOS per one particle at the lowest frequency Acu for this system is defined by(4. 10)Aco

L,

where g (L) is the dimensionless conductance of size L,i.e., a quantity of the order of o.
where o. is the conductivity. It is presumed that g(L) follows a power-law
relation on L in the above relation. It is clear that thewave functions are localized when PL is zero or negative.
In the case of percolating networks, o ~(p —p, )~. This
leads to cr ~L "/ for a correlation length g(p) larger
than the size L. The conductance becomes(L) ~L

Assuming the dispersion relation for Am to be
Aco

393

one can eliminate the size L from Eq. (4. 10):(Po(e) ) is the Careen's function. This is defined by(Po(E)) =properties of fractal networks

Dynamical

L

Rev. Mod. Phys. , Vol. 66, No. 2, April 1994

(4. 11)

where

the

—P/v+d —2
scaling~L frelationforfracta1dimension

394

Nakayama,

Yakubo, and Orbach:Df =d —v/P and the definition d =2Df /[2+(pDynamical

P—)/v]

From the above relation, one finds—2)/d. For d -=4/3, the PI function is negaPL =Df(dtive independent of Euclidean dimension d. It is interest+d.
ing to note that for length scales greater than g, d —Thus, for d=3, for example, PL may well be positive, inhave been used.

dicating delocalized vibrations (phonons). As the lengthscale shortens to less than g, d =4/3 leads negative PL
and localization. Hence the crossover can be thought ofas a dimensionality change in so far as localization is concerned: passing through g, d crosses over from d to 4/3,
causing localization (PL changes sign).

2. Dispersion relation
Alexander (1989) gave a simple derivation of thedispersion law from the given form of the DOS such asthat of Eq. (4.9). Consider the vibrations of an isolated
fractal blob of size L. Though high-frequency modes will
not be affected by the change in the boundary conditions,the low-frequency modes will disappear from the spectrum. The crossover occurs at some frequency coL suchproperties of fractal networks

tal structures become relevant. Thus there is a crossover
excitations whenin the nature of the vibrationalA, (co, )-g(p). One has from Eq. (4. 14) or (4. 16) the crossover frequency of the form

(4. 17)
By putting co, =u (p)k at k —I/g(p) in Eq. (4. 17), theconcentration (p) dependence of the phonon velocity becomesvD /d —v~(p —(4. 18)u(p)~(p —p, )p, )'" ~''.

Because the value p —P is always positive, we haveu(p)~0 when p~p, . The results for the DOS are summarized as
d

u(~)

CO

[u(p)]
cod

k,
Df /d

L =A(coL ),where A(co) is the wavelength.
The integrated spectral
weight of the missing low-frequency modes is lumped together in the center-of-mass degrees of freedom of the
disconnected blob (e.g. , the rotational and translationalmodes). This is a number that cannot depend on L andco. Therefore, using Eq. (4.9), one has[A(coL )]co'dco ~ AdcoI =const.

This relation holds for any length scale L, and one has
the dispersion relation for an arbitrary frequency ~:—D /d(4.16)
co ~ A(co)lengthApplying the argument of a frequency-dependent
scale A(co) to waves in a finite homogeneous system(Df =d =d), one obtains the length scale A=2urr/co=i,(where u is the sound velocity), because the lowest frequency of a blob of size A is co(A) =2uvr/A.

3. Crossover from phonons to fractons:
Characteristic frequency

If the wavelengths k of excited modes on percolating
networks p &p, are larger than the characteristic lengthscale g'(p), the system is homogeneous on this scale, andvibrational excitations are weakly localized phonons.
This is because the scattering is determined by the squareof the mass-density fluctuation averaged over regions ofvolume k". Hence, even if the short-range disorder isstrong, the effective strength of the disorder for phonons
with A,g is very weak. If the characteristic length k ofwaves becomes of the order of or shorter than g(p), frac-))

Rev. Mod. Phys. , Vol. 66, No. 2, April 1 994((CO~

1, M&)co

The dispersion relations becomeu (p)k,that1

co &&co, ,
co

))co

where k for cu))m, does not mean wave number due tothe lack of the translational symmetry of the system, butrather it describes the inverse of the characteristic length
A ' [see Eq. (4. 16)]. It should be noted that fractons
refIect two features of fractal structures, namely, the fractality and the lack of the translational symmetry. The
latter leads the localization of fractons.

V. LARGE-SCALE SIMULATIONS AND
PHYSICAL REALIZATIONSIn this section, we erst present results of simulation for
the vibrational DOS for very large percolating networks
with scalar interactions. These data provide rich information about the fracton dynamics. The characteristic
properties of fracton wave functions themselves are also
described. %'e also show the results for the model taking
account of the vector nature of interactions and displacements. In addition, experimental results for the DOS
are discussed usmeasured in real materials
aerogels
ing the concept of fractons.

—

A. Vibrations of a percolating—

network

1. Density of states: Sitepercolation with scalar elasticity
Computer simulations can provide deep insight into
the eigenstates of random systems. Cxrest and Webman(1984) have calculated the DOS of d=3 percolating netroutine forworks, using the standard diagonalization

Nakayarna,

Yakubo, and Orbach:Dynamical

d=3 systems of size L=18. For larger systems,

theyused a recursive technique to calculate the eigenfunctions
and eigenvalues in the low-frequency region. The standard diagonalization routines were sufFiciently accurate,
except at low frequencies. They averaged over threesamples to obtain the DOS. Although there are obvious
di5culties in the treatment of large-scale systems, the situation is changing as array-processing supercomputers
become available. Many numerical methods have beenreported to overcome these difFiculties and have been apof fracton dynamics (Lamplied to the investigation
et al. , 1985; Yakubo and Nakayama,
1987a, 1987b,
1989a, 1989b, 1989c; Nakayama, 1990, 1992; Nakayama
et al. , 1989; Russ et al. , 1989, 1991; Bottger et al. , 1990;
Li et al. , 1990; Montagna et al. , 1990; Stoll and Courtens, 1990; Yakubo, Courtens, and Nakayama, 1990;
Yakubo, Takasugi, and Nakayama, 1990; Lambert andHughes, 1991; Roman et aI. , 1991; Royer et al. , 1991,
1992; Mazzacurati et al. , 1992; Russ, 1992; Stoll et al. ,and Yakubo, 1992a, 1992b). Li,
1992; Nakayama
Soukoulis, and Grest (1990) used the Sturm sequence
method to calculate the integrated DOS and treated d=2
systems of 160 X 640. Royer, Benoit, and Poussigue(1991, 1992) used the spectral moment method, which allowed them to work with a very large percolating network consisting of a square lattice of size L = 1415.
Yakubo and Nakayama (1987a, 1987b, 1989a, 1989b),
Yakubo, Courtens, and Nakayama (1990), and Yakubo,
Takasugi, and Nakayama (1990) succeeded in treating
systems with size number N —10 by applying the forced
oscillator method of Williams and Maris (1985). Russ,
Roman, and Bunde (1989, 1991), Russ (1992), and Bourbonnais, Maynard, and Benit (1989) have also used thismethod to calculate the DOS and the localization
behavior of fractons. This algorithm is based on theprinciple that a complex mechanical system, when driven
by a periodic external force of frequency A, will respondwith a large amplitude in those eigenmodes close to thisfrequency. Yakubo, Nakayama, and Maris (1991) haveformulated a method for judging the accuracy of the calculated eigenmodes
and eigenfrequencies,
and this
method is now used in many different fields. This algorithm can be readily vectorized for use on an array(Yakubo and Nakayama,processing
supercomputer
1987a, 1987b). It should be noted that, for the calculation of the DOS, the algorithm becomes more accuratewith increasing site number.

properties of fractal networks

395

mass (m= 1) on the ith site. The force constant is taken
as K;. =0 if either site i or is unoccupied, and as K, = 1
The displacement u; has only one comotherwise.

jponent.
The calculations for the DOS for d=2 percolating networks with the site number N-10 have been performed
by Yakubo and Nakayama (1987a, 1987b, 1989a). Figure3(a) shows the DOS at the percolation threshold p,(=0.593). This network, formed on a 700X700 squarelattice, has 116 991 atoms. The line through the solid circles has a slope of 1/3. It should be emphasized that thecu'law holds even in the low-frequency region, because
the lower cutoff frequency col is determined from thefinite size of the clusters. One can estimate the cutoff frequency to be coL —10 for the present case, using the relation coL —coD = 2&2. The correlation
coD /N wherelength g(p) diverges at p =p„and the network has afractal structure at longer length scales.
cluster withThe DOS of a d= 2 site-percolating
p=0. 67 is shown in Fig. 3(b). This percolating cluster isformed on a 700X700 square lattice with the cluster size
N =317 672. The results show that the frequency dependence of the DOS is characterized by two regimes. In the
1, the DOS is closely proporfrequency region co, «cotional to co' . The crossover frequency co, corresponds
to the mode of wavelength A, equal to the percolation
correlation length g(p). Therefore the DOS in the frequency regime lower than cu, should be given by the conventional Debye law 2)(co) ~co ', where d is the Euclidean dimension and 2)(co) ~co ' for co))co, . The
simulated result is consistent with this view because thefrequency dependence of the DOS for lower frequencies(co((co, ) clearly obeys the law 2)(co) ~co, as discussed inSec. IV.B.3. Vibrational excitations in this frequency regime behave as phonons.
The region in the vicinity of co, is the crossover region
between phonons and fractons [Fig. 3(b)]. It should be
stressed that the DOS is smoothly connected in this region, exhibiting neither a notable steepness nor a humpin the vicinity of co, . It is remarkable that the DOS does
not follow the u' dependence above co=1. The interpretation on this is given in Sec. V.A. 4 in connection
with the discussion of the missing modes.

«

b. Three-dimensionalcase

The absence of the hump in the crossover region has
also been demonstrated in the case of d=3 percolatinga. Two-dimensional case
Consider a site-percolating network consisting of N
particles with unit mass and linear springs connecting
nearest-neighbor atoms. The equation of motion of theatoms is given by(5. 1)
where u; is the scalar displacement
Rev. Mod. Phys. , Vol. 66, No. 2, April 1994

of the atom with unit

4The force constants K;,. have a different sign from thedefinition of the dynamical matrix elements @', normally used in
the field of lattice dynamics. See Chap. IV of the book by Bornand Huang (1954).
5It is known that excitations in disordered systems with the
Euclidean dimension d ~ 2 should be localized (Abrahams
et al. , 1979). In this sense, phonons mentioned here are localized, but weakly, whereas fractons are strongly localized in thesense of the Ioffe-Regel criterion (see Sec. V.B.2).

396

Nakayarna,

Yakubo, and Orbach: Dynamical properties of fractal networks
1.0

1.0

P = 0.40
N= 1224

Vl

4J

e

I

01

lh

o 0
~

/3

~
Vl

th

C

cc

(J

0.01

0.001

0.01

0 01

C)

C5

I

I

0.1

1.0

10

0.001

0.001

Frequency

1.0

I

0.1

1.0

10

FIG. 4. Density of states per site for a d=3 site-percolating
network at p=0.4 formed on a 70X70X70 simple cubic lattice.
The network size is N =122488. After Yakubo and Nakayama

I(1989a).

N= 317672

tA

I

0.01

Frequency

p=0. 67tA~

0(Loring and Mukamel, 1986; Korzhenevskii and Luzhkov, 1991). In particular, Loring and Mukamel (1986)
suggested a smooth transitionof the DOS at thephonon-fracton crossover, in contrast to the prediction of
the effective-medium theory (Tua et al. , 1983; Derrida
et al. , 1984; Entin-Wohlman
et al. , 1984; Tua and
Putterman, 1986) or the scaling theory (Aharony et al. ,

01

O
th

C

~ 0.01

1985a, 1985b, 1987b).
I

0.001

0.01

0.1

'l.

0

10

Frequency
FIG. 3. Density of states for d=2 site-percolating networks at
two di6'erent concentrations: (a) The density of states per site at
the percolation threshold p, =0.593. The network is formed on
a 700X700 square lattice and contains 116,991 atoms. (b) The
density of states per site at p=0. 67 formed on a 700X700
square lattice. The network size is 317 672. Solid circles indicate the numerical results. The straight lines are only meant as
a guide to the eye. After Yakubo and Nakayama (1989a).

networks (Yakubo and Nakayama, 1989a}. The DOS ofa percolating network at p=0. 4 (p, =0.312) formed on a
70X70X70 simple cubic lattice is shown in Fig. 4. The
network size is X =122448. The DOS in the frequency
region 0. 1 & co & 1 is proportional to cu', as was found in
d=2. The DOS in the low-frequency regime (co((0.1)
obeys the Debye law 2)(m) cc co, where at this concentration the phonon-fracton crossover frequency m, is close
to 0. 1. A sharp peak at co= 1 in Fig. 4 is attributed to vibrational modes of a single site connected by a single
bond to a relatively rigid part of the network.
It is clear that no steepness or hump of the DOS exists
in the crossover region in the vicinity of co, . This featureis also found in the results by Grest and Webman (1984}
for d=3 percolating networks. They found that thephonon-fracton crossover clearly exists for their systems.
The behavior of the DOS at the phonon-fracton crossover has been determined from mean-field treatments
Rev. Mod. Phys. , Vol. 66, No. 2, April 1994

2. Density of states: Bond
percolation with scalar elasticity

The other percolating network, bond percolation (BP),shows some interesting differences from site percolation(SP) networks. The difference in geometrical structure
between SP and BP is short range, as mentioned in Sec.
II.C. As shown in Fig. 2, the fractal nature of the network extends to very short ranges for BP. The DOS and
the integrated DOS have been computed for large-scale(d=2, 3, and 4) BP networks (Nakayama, 1992).
In Figs. 5 and 6, the DOS and the integrated DOS per
atom are shown by the solid squares for a d=2 BP network at p, =0.5 formed on a 1100X1100 square lattice(%=657426) with periodic boundary conditions. The
fracton dimension d is obtained as d=1. 33+0.01 from
Fig. 5, whereas the data in Fig. 6 indicate the value
325+0.002. The DOS and the integrated DOS fordd=3 BP networks are also computed, with results exhibited in Figs. 5 and 6 by the solid triangles (middle).
These data show the averaged DOS and DOS integrated
over three samples
at the percolationthreshold(=0.
100X100X100
249).
The
formednetworks,
onp,
cubic lattices, have 155 385, 114303, and 143026 atoms.
The fracton dimension d is obtained as d=1. 31+0.02
from a least-squares fit, using the data of Fig. 5. Itshouldthe value
be mentionedthatd takes
317+0.003 when using the data of Fig. 6.
d

=1.

=1.

Nakayama,

Yakubo, and Orbach: Dynamical properties of fractal networks

It should be noted that, from these values of d, theconductivity exponents p can be determined using Eq.
(4. 15) as @=1.288 for d=2, @=2.02 for d=3, and
p=2. 45 for d=4. The value p for d=2 is obtained using
the exact values for f3 and v in Eq. (4. 15). The values of pfor d=3 and d=4 are calculated by substituting thevaluesobtained
Monte
Carlo
calculations
by(Grassberger, 1986) into Eq. (4. 15), namely, p=0. 43 andv=0. 88 for d=3, and P=0.65 and v=0. 68 for d=4.

IO

M

10

10
~ ~ ~ si 8

~~

d=

~

~

~

397

~

3. Density of states: Vector elasticity
Cl

10
I

I

10

I

10

10
10
Frequency

FICx. 5. DOSs per atom for d=2, d=3, and d=4 bondpercolating networks at p =p, . The angular frequency co is obtained in units of mass m= 1 and force constant %= 1. The networks are formed on 1100X 1100 (d = 2), 100 X 100 X 100(d= 3), and 30 X 30 X 30 X 30 (d= 4) lattices with periodic
boundary conditions, respectively. After Nakayama (1992).

The DOS and the integrated DOS of d=4 BP networks at p, =0.160, formed on 30X30X30X30 quartic
lattices, are shown, respectively, in Figs. 5 and 6 by thesolid circles, obtained by averaging over 15 samples. The
network sizes are N=8410-64648. The DOS in thefrequency region 0. 12(co&0.9 clearly shows a powerlaw, as was found in the d=3 case. The fracton dimension d is estimated to be d =1.31+0.03 from the leastsquares fitting using the data of Fig. 5.

We have discussed the dynamical properties of perin Sec.
colating networks with scalar displacements
V.A. 1 and V.A. 2. In most vibrational systems under theconditions set forth by Feng (1985a, 1985b), the vector
nature of atomic displacements becomes crucial. Percolating networks with rotationally invariant vector elastic forces have different critical exponents for elastic
moduli from those of the scalar forces (Benguigui, 1984;
Feng and Sen, 1984; Feng et al. , 1984; Kantor and Webman, 1984; Bergman, 1985; Deptuck et al. , 1985; Feng,1985 a, 1985b; Feng and Sahimi, 1985; Roux, 1986;
Sahimi, 1986; Arbabi and Sahimi, 1988; Sahimi and Arbabi, 1991). The same corresponding relation as that forthe scalar displacement (the mapping of the difFusion
problem onto the vibrational problem) is more difficult to
express because of the significant additional complicationof the vector nature of the displacements. The Hamiltonian taking vector displacements into account is givenby (Keating, 1966)
1
1
II = —o. Q IC;J ( u; —u ) r," ]2
g m u;. 2 + —
I'

l

/J

1
+—
p g IC~K;I, (b, 8,k )~ .

(5.2)

ijk

10
~

Here u; is the vector displacement of the ith atom withunit mass (m=1); r,.~, the unit vector between nearest
neighbors (ij ); and b, 8; k, the small change in angle between bonds (ij ) and (ik ) due to the displacements ofatoms. 6 The parameter E;. takes the value unity if bothsites i and are occupied by atoms; otherwise, K; =0 and
a and p are the bond-stretching and the bond-bending
force constants, respectively. The rigidity threshold of
this system is identical to the percolating threshold p, .
Kantor and Webman (1984), Feng (1985a, 1985b), andWebman and Grest (1985) have applied the nodes
blobs model (described in Sec. II.A) to predict the dy-

F0000

0

~

d=4

1

10

~~
~

0~

+

~&

k

el=3

j

10'd=2

CO

O

C3

0)
CD

links-

-2

10

0)

10
I

10

I

I

10
10
Frequency

10

FICx. 6. Integrated DOSs per atom at p =p, for d=2, d=3,
and d=4 bond-percolating networks. The symbols correspondto those in Fig. 5. After Nakayama (1992).
Rev. Mod. Phys. , Vol. 66, No. 2, April 1994

6It should be emphasized that, for square or cubic lattices,
both the equilibrium angles 0;,.k =m/2 and ~ should be involved
From the model taking into account onlyin this Hamiltonian.
0;,.k =m. /2, the angular force along linear links becomes irrelevant, so that the rigidity threshold becomes larger than the
percolation threshold.

398

Nakayama,

Yakubo, and Orbach:Dynamical

namic properties of percolating networks with bondbending force constants. That model describes well thefeatures of percolating networks in which the backbone
consists of a network of quasi-one-dimensional
strings(links), tying together a set of more strongly bonded regions (blobs) T.he typical separation of the nodes forming
the macroscopically homogeneous network is equal tothe correlation length g(p) (see Fig. 1).
There exist two kinds of characteristic length, g(p) andfor percolation networks with Uector elasticity. The
is the correlation length,
length g(p) scaling as ~p —p,the crossover length scale from a homogeneous to a fractal structure. The mechanical length I, determines the
crossover length scale below which bond-stretchingmotion is energetically favorable and aboUe which thebond bending becomes dominant. This crossover lengthl, depends only on the force constants a and P, namely,
1, ~ &P/a, as shown below. One can connect the characteristic lengths g(p) and l, with two characteristic frequencies, co& and col, respectively.

I„

properties of fractal networks

where the level spacing Am is taken to be equal to thelowest finite frequency ~L of a fractal structure withfinite size L, . This is expressed by
IC (L)

1/2

M(L)where M(L) and E(L) are the mass and the effective
spring constant of the system at length scale I, . The
effective spring constant is obtained for Li, from«g «Eq. (5.5a),~C

a. Theoretical prediction based on
the nodes-links-blobs

model

Kantor and Webman (1984) claimed that the efFective
spring constant K of a blob of size g(p) is given by
—1

-+ L

rc '

(5.3)

ikV»'

where the blobs are assumed to be perfectly rigid and I, ,denotes the number of links (red bonds). The criticalbehavior of the links (red bonds) was described by Coniarguglio (1982a, 1982b) using a renormalization-groupment. The mean number of red bonds varies with p as[see Eq. (2. 13)jLi

If g(p)

p. )

(5'

(5.4)

Pp)

« l„where l, is proportional to p/a, the firstv

term of the effective spring constant Eq. (5.3) (stretchingmotions) dominates. One has

K~

fdic
(5.5a)

L, 1

This implies that the elastic energy of the system is primarily associated with the stretching force constant. For
the case of l, «g(p), the bond-bending spring constantbecomes dominant, and the e6'ective force constant isE(5.5b)

L ikS»'Let us derive the formula for the DOS of "stretching"fractons according to the theories of Kantor and Webman (1984) and Feng (1985a, 1985b) for a system of size
L «g'. The DOS at the lowest finite frequency coL for
this system takes the formXl(coL )=1

Rev. Mod. Phys. , Vol. 66, No. 2, April 1994

rC

I.

(L)

(5.6)

where Eq. (5.4) is used, replacing g(p) by L. Using the relation M (L) o- L f, the lowest frequency col is expressedin terms of the length scale
as

I

coL

~1.

—(Df + 1/v)/2

By using this dispersion relation and replacing coL by anarbitrary frequency m, we obtain the DOS for stretching
fractons,
2vDf /(, vDf +1)—1(5.7)
2) co o- coNote that the exponent is determined only through thestatic exponents Df and v. This is due to the assumptionthat one-dimensional links dominate the elastic properties, and that the blobs are assumed to be "rigid" (or superconducting) [see Eq. (5.6)].
The nodes-links-blobs model predicts a direct relationship between the dynamic exponent p and the static exponents. The conductance G of this model is given by
G cc

1
1

Uswhen the blobs are assumed to be superconducting.
o. and G,
ing the relation between the conductivity~ o.L(see Sec. IV. B.1), the conductivity can be ex-

6pressed as~L 2 d

1/V

This is written as a function of p, for L large (p close top, ),)v(d —2)+ 1
CAs a result, one has

j

7The conductivity a. ;, between site i and corresponds to theelastic force constant E;~ in Eq. (5.2), as seen from the mapping
relation between the resistive and elastic networks (de rennes,
1976a).
The value of p given by Eq. (5.8) constitutes a lower boundfor the conductivity exponent, except at d=6 (where it is exact).
This is because blobs are assumed to be superconductors in thismodel, and the actual conductivity of the network is necessarily
smaller than that predicted by the relation (5.8).

Nakayama,

Yakubo, and Orbach:p=v(d —2)+1 .
By using(5 8)

Eq. (5.8) and the hyperscaling

where

2af

relation2+(f —P)/v

p/v, one can obtain the relation between Df
Df =d —and p. This yields, using Eq. (5.7) for stretching fractons,

X(~) ~~The dispersion relation for bending fractons is given by—db /D

2Df

Comparing Eqs. (5.8) and (5. 12), one has the relation between p andin the nodes-links-blobs model,A(co) ~ to(5.9)

2+(p —P)/v

Note that the exponent d, takes the same form as d given
in Eq. (3.15). Thus the nodes-links-blobs model for vector elasticity predicts that stretching fractons belong to
the same universality class as scalar fractons. It should
also be noted that the stretching elasticity is, in general,
different from scalar because the stretching force constant becomes relevant only along the bond connection,
whereas scalar displacements respond to any deformation. Nevertheless, under the condition L &)theyboth belong to the same universality class.
Consider now the opposite case, l,«g(p). The
effective spring constant K is given by Eq. (5.5), andbending motions become relevant. The equation corresponding to Eq. (5.6) becomesl„«L

K(L) ~L(5. 10)Using Eq. (5. 10), we see that b, co becomes—f Df +( 1/v)+ 2] /2
o(ACO

L

As a result, one has
CO~CO[2vDf /(vDf +2v+ 1)]—1

The elasticity exponent
defined as(5. 11)f for the Young's modulus Y is

Y-(p —p, )ff

can be derived in terms of thenodes-links-blobs model as follows: Because one has therelation between Y and K as K ~ YL
(this is analogous with the relation between the conductance and the
conductivity), using Eq. (5. 10) for the case l,
one has the relation

The critical exponent«L «g,y ~L(5. 13)

d —1

where

d, =

399

properties of fractal networks

Dynamicalff =p+2v .

)(5. 14)f)

Because v 0, one hasp (Kantor and Webman, 1984;
Webman and Kantor, 1984). This implies that db &d,(=d). The bending-fracton dimension db for 2d per=3.96
colating networks, using the known values(Sahimi, 1986), v=4/3, P=5/36, and Df =91/48, is estimated to be
78. This indicates that the DOS weakly diverges at very low frequencies. The value of the exhas been evaluated numerically (Feng and Sen,
ponent1984; Feng and Sahimi, 1985; Arbabi and Sahimi, 1988)
and experimentally(Benguigui, 1984; Deptuck et al. ,1985; Benguigui et al. , 1987; Forsman et al. , 1987; Sofoet al. , 1987). The upper/lower
boundsfor3.67 &&4.17 for d=2 and 3.625
3.795 for d=3
percolating networks (Havlin and Bunde, 1991), provide
bounds for d&.
The vibrational correlation between blobs connectedby zigzag chains longer than I, becomes irrelevant for the
stretching modes, as seen from Fig. 7. This implies thatstretching fractons with very low eigenfrequencies do notexist, but bending fractons can (see footnote 6). This is
the reason bending fractons become dominant in the regime below ~1C .
Liu and Liu (1985) have found for the Sierpinski gasket
that bending fractons do not dominate over stretching
fractons, in contrast with the case for percolating networks. They suggest that the disagreement is purely a result of geometry, namely, the Sierpinski gasket is stabilized by central forces alone, but percolating networks
are not. In this connection, it is interesting to note thework by Garcia-Molina, Guinea, and Louis (1988) andsucceeding comments by Tyc (1988), Roux, Hansen, and

f

-0.

f

f

f

—d —1/v

This implies thatf =vd+1 .

(S. 12)We should note that this relation also gives a lower
bound for the dynamic exponent
as explained in footnote 8. The rigorous bound is expressed throughd+v1 f&& vd + vd;„(Havlin and Bunde, 1991).
Inserting Eq. (5. 12) into Eq. (5. 11), we see that the
DOS for bending fractons becomes—1db2)(co) ~ cof,Rev. Mod. Phys. , Vol. 66, No. 2, April 1994

FIG. 7. Blobs connected by zigzag links.

f,

400

Nakayama,

Yakubo, and Orbach:Dynamicat(1988), and Day and Thorpe (1988), who consider
triangular networks connected by central forces.
Alexander (1984) claimed theoretically that an elastic
network with rotationally invariant elastic force constants is not the only approach for describing the elasticiHe
ty of tenuous objects and amorphous materials.
showed that there are scalar contributions to the elastic
energy in stressed systems, noting that real materials always have internal stresses.
in the following
This section can be summarizedmanner.
(i) Percolating networks with p )P, andl, . As
shown in Fig. 8(a), fractons are excited in the frequency
range co, «co, whereas the vibrational excitations in the
low-frequency regime (co, ))co) are phonons. There are
two classes of localized modes, depending on the frequency regimes: co, (&co«~& and co& (&~. We call the
Canyong)

CC

excitations "bending" fractons and the latter"stretching" fractons. Stretching fractons belong to thesame universality class as fractons with scalar displacements under the nodes-links-blobs model (Feng, 1985b).
This result holds only under the assumption of perfectly
rigid blobs.
. As illustrated in Fig. 8(b),(ii) The case of g(P)phonons directly crossover to stretching fractons at m„and bending fractons are not observed.
former(l,b. Simulated results for vector displacements

The first attempt to calculate the DQS for the vector
displacement model was made by Webman and Grest(1985). They focused on the limit A(m)))l, when thebending fractons dominated, and treated a system with. They found that the DQS was weakly divergent
The DOS of excitations forat low frequencies.
A(co)l, showed no crossover from bending to stretching fractons because co& is a high frequency. Lam andBao (1985) used a recursion method to calculate the vibrational DOS of a site-diluted central-force elastic percolating network on a triangular lattice. They found the—1
db
DOS in the fracton regime to be proportional to ~ 'with db =0.625. This estimate should be regarded with

X-10

))

C

'

bending
fraction

~

I

'

t l

'. ~ i,ij'm

gj

j ji& j

'
I

jiil'

h

0

co

{a) ( ) I

FIG. 8. Schematics({b) &lillustrating the correspondence between
length scale and frequency. The upper line is for the lengthscale where a is the lattice constant, and the lower one shows
the corresponding frequency region.

Rev. Mod. Phys. , Vol. 66, No. 2, April 1994

properties of tractal networks

caution because of the very narrow frequency interval in
the fracton regime from which they estimated the valueof db. Day, Tremblay, and Tremblay (1985) calculated
the DOS of percolating networks formed on a triangularlattice with central forces and 60 bond-bending forces.
They found values for the rigidity percolation thresholdp, and the fracton dimension db in the ranges
0.4~@, ~0.405 and 1.25 ~db ~ 1.3, respectively. The result suggests that the 60' bond-bending model does not
fall into the same universality class as a full bond-bending
model. In order to calculate the DOS for triangular elastic BP networks with central forces and full bondbending forces, Bottger, Freyberg, and Wegener (1990)
performed a homeomorphic coherent-potential
approximation (HCPA), a recursion technique, and a replica
trick calculation. They obtained db =1.0 from the recursion method and the HCPA, whereas the replica method
did not show fracton behavior. Liu (1984) calculated thefracton dimension of an elastic Sierpinski gasket using/v=d —1 (Bergman and Kantor, 1984) andthe relation
found d& =2Df /(Df +1). Arbabi and Sahimi (1988) performed numerical simulations for the elastic properties of
4=3 percolating networks in which both central andbond-bending forces were taken into account. Rahmani
et al. (1993) considered the model incorporating thenearest- and next-nearest-neighborinteractions.
In this subsection, we present simulation results for the
DOS of large-scale percolating networks with vector displacements by Yakubo, Takasugi, and Nakayama (1990;
see also Nakayama and Yakubo, 1990). We discuss firstthe crossover behavior of the DOS from bending tostretching fractons. For this purpose, Yakubo, Takasugi,
and Nakayama considered the situation in which thebond-bending force constant P is larger than the stretching one u, so that the characteristic frequency coI ismuch smaller than the Debye cutoA frequency coD. The
network was prepared at p =p„so that the conditionl, (g always held. The calculated DOS is shown in Fig.
9, where the percolating network formed on a 500X500
square lattice has 53 673 occupied sites, and the set of theforce constants [a, P] in Eq. (5.2) was chosen as[0.0133,0. 133]. This network had a cutoff' frequency~D=2.0784. The steplike decrease of the states in thehigh-frequency
regime in Fig. 9 indicates that thestretching motions are not excited above some frequency
coo, whose value is determined by the force constants a.
For a=0.0133, this frequency ~o is estimated to be
0.2309 from the relation coo=2+a. This value coincides
with the observed value in Fig. 9.
In order to clarify the individual contributions fromthe bending or stretching motions, the ratio of the potential energies [Eq. (5.2)] was calculated as a function of
frequency ro. The solid curve in Fig. 9 shows the ratio
between the potential energy attributed to the stretchingmotion (E„) and the total potential energy (E„,) obtained by substituting the displacements of calculated
eigenmodes into the potential-energy
expression of Eq.

fCYakubo, and Orbach:Nakayama,1lhDynamical

properties of fractal networks

401

10

pl

1PQ

w

4

~
LLI

—0.5 &
UJ

~ ~

10-1

P=P,

IC

N

2
+ lp—

C3

10
IQ

10

10

1Q

,

=53673~: ~=10o: ~=-0.12

-3
lQ .

0.0

10

v)

I

l

10

10

1o'

0 1Q
~ %IRIS

th

&

10—
IO

—

C

C

1P3
10

~

N =

~

53673

~: u=1. 0

P=0.01~=0.12

P =Q.12

l

I

0-1

10

1

10

1Q

Frequency

FIG. 10. Density of states(a) and the integrated density ofstates (b) of percolating networks with vector displacements.
Networks have the same structure as that of Fig. 9. Solid circles indicate the result for the net with stretching force constanta=1.0 and bending force constant P=0.01. Open circles arefor u=P=0. 12. The straight lines on the left-hand side of (a)and (b) are drawn by least-squares fitting, indicating a law pro—1ddb
and co with db=0. 79, respectively. The
portional to ~straight lines on the right-hand side indicate a density of states
After Yakubo,
and co ', respectively.
proportional to co'
Takasugi, and Nakayama (1990).
db
jb' —1(co ') with dI, =0.79.
10(b)] are drawn according to co
value
This value agrees well with the predicted—0.78), in contrast with the case of stretching frac(d&tons.
In the case of p/a=0. 01 (solid circles in Fig. 10), themechanical length scale I, becomes close to the lattice
constant, resulting in the crossover frequency ~1 'sCbecoming too large to distinguish the crossover frequency
region. For the case of p/a=1. 0 (open circles in Fig.
10), the crossover frequency can be estimated from theevaluation of E„/E„, as in the case of Fig. 9. The crossover frequency co& from bending to stretching fractons

becomes close to co=0. 1. Note that the simulation does
not exhibit any noticeable change in frequency dependence of the DOS around crossover, as shown by theopen circles in the vicinity of co=0. 1. The DOS for thecase p/a = 1.0 does not exhibit a distinct crossover tostretching fractons at cuI . Similar behavior is found forC

Rev. Mod. Phys. , Vol. 66, No. 2, April 1994aI

Frequency

C3U(5.2). The crossover frequency co& is estimated from thecondition E„(co) /E„, (co) = —,', leading to co& —0.005. The
DOS in the vicinity of coI is independent of frequency.
The crossover region from bending to stretching fractons
extends over at least two orders of magnitude in frequency. This is because the ratio E„(co)/E«, (co) increases
logarithmically, as shown in Fig. 9. This observation isin contrast with the sharp crossover from phonons to
fractons for scalar displacernents described in Sec. V.A. 1.
The straight line through the solid circles in Fig. 9 isdrawn according to the law 2)(co) ~ co' [see Eq. (5.9)]. It
is not clear from the data of Fig. 9 in the frequency region between col and coo that the co' law holds. This is
crucial for decreasing frequencies. A physical interpretation of this discrepancy has been given in Sec. V.A. 3.a.
In order to clarify the contribution of bending fractons, the DOS for the case p/a=0. 01 and 1.0 has been
calculated. These sets of force constants allow exclusive
examination of the DOS for the bending-fracton regime
because the stretching-fracton regime is shifted into thehigh-frequency region. Figures 10(a) and 10(b) show theresults for the DOS and the integrated DOS, respectively,
for percolating networks at p =p, for the same networkas that of Fig. 9. The sets of force constants [a, p] in Eq.
(5.2) were taken as [1.0,0.01] (solid circles) and [0.12,0. 12](open circles), respectively [see Fig. 10(a)]. The cutofF frequencies are the same as the previous cutoff frequency,
coD =0.2784, by virtue of the above choice of force constants. The DOS, given by solid circles in Fig. 10(a),weakly diverges as co~0, in accord with the theory by
Feng (1985b). The value of the bending-fracton dimension db obtained by a least-squares fitting from Fig. 10 islines on the left-hand side
db = 0.79. The straight
through the solid and open circles in Fig. 10(a) [Fig.

P= 0.01
P= 0.1210Frequency

FIG. 9. Calculated density of states, shown by the solid circles.
The network is formed on a SOOX SOO square lattice at the percolation threshold p, with 53 673 sites. Force constants a and P
are taken as 0.0133 and 0. 1333, respectively. The straight line
through solid circles is only a guide to the eye for the law
2)(co) ~ co' . The solid curve indicates the ratio of potential enAfter Yakubo,
ergy E„/E„, as a function of frequency.
Takasugi, Nakayama (1990).

~

0

402

Nakayama,

Yakubo, and Qrbach: Oynamical properties of fractal networks

the case P/a=10. 0 in Fig. 9. For comparison, the lines
on the right-hand side are proportional to co'(co).
Note that the agreement with the data is unsatisfactory.
Furthermore, it should be emphasized that the magnitudes of the DOS in the bending-fracton
regimes aredifferent for the two sets of force constants (see thedifference between solid and open circles in Fig. 10). We
shall show that this implies that the missing modes tend
to accumulate in the high-frequency region (co) coo) (seethe next subsection).
In this subsection, simulated results for the DOS with
vector displacements have been presented. In the case(strong bending force), stretching fractons are excited in the high-frequency regime. It has been shownthat the crossover region from bending to stretching fractons is rather broad. The opposite choice of force constants, such as the case of open circles shown in Fig. 10,
makes it difficult to clarify the bending-to-stretching
crossover in the DOS. It has been shown that the—0.79 fordimension takes a value dbbending-fracton
d=2 percolating networks. This is close to the predictedvalue from scaling theory. However, the calculated DOS
for stretching fractons is not in accord with the valued, =4/3. The results given in this section will be usefulin discussing the characteristics of the DOS for real
disordered materials in which the vector nature of displacements is crucial.

P))astrict similarity in the fractal regime, one expects2)f, (co, p) =2)f,(co, p, ), where the DOS per particle is normalized by Jo" 2)(co)dc0=1. Because d is always larger
than d, X)~h is smaller than 2)r, when the latter is extrapolated to phonon frequencies.
As the integration of2)f,(co,p, ) is normalized to unity, some modes must be
missing for l)(co, p )p, ) in view of the existence of thephonon regime. Their spectral weight must be recovered
somewhere, and it was argued that the most reasonable
place for accumulation is near co„ leading to a hump in
the DOS and to a corresponding hump in the lowtemperature specific heat (see Fig. 11). The main point isthat a hump is seen neither in simulations of thephonon-fracton crossover nor in actual experiments on
silica aerogels (Courtens et a/. , 1987a, 1987b, 1988;
Courtens, Pelous, Vacher, and Woignier, 1987; Courtens
and Vacher, 1988; Conrad et a/. , 1990; Buchenau, Morkenbusch, et a/. , 1992), the latter to be presented in Sec.

V.C.2.
b. The hump at high frequencies:
geometricalinterpretation

To explain the absence of a hump in the crossover region, we write the scaling form for the DOS of a percolating network for p & p, :2)(co,p) = A (p)co"

4. Missing modes in the density of states'F (co/co, ) .

(5. 15)For an infinite percolating network, the phonon-fractoncrossover frequency scales as co, =Q(p —p, ) f . The
crossover frequency ~, is defined as the intercept of theasymptotic phonon and fracton lines in a doublelogarithmic presentation of the DOS vs co (Alexander
et a/. , 1983). The scaling function takes the forms
F (x) =1 for x 1 and F(x) =x "for x l. Equation(5. 15) yields a prediction for the p dependence of 2)(co, p)in the phonon regime,
vaf /da. Missing modes at low frequencies: scaling arguments
The simulation results for the DOS for scalar displacements confirmed that the crossover between the phononand fracton regimes is smooth, with no visible accumulation of modes or a "hump" in the DOS around the crossover frequency co, (Yakubo and Nakayama,
1987a,
1989a, 1989b; Russ et al. , 1989; Li et a/. , 1990; Royer
et a/. , 1992). This is in accord neither with earlier predictions based on scaling considerations(Alexander
et a/. , 1983; Aharony et a/. , 1985a, 1985b, 1987b), norwith arguments based on the e6'ective-medium approximation (Derrida, 1984; Derrida et a/. , 1984; Sahimi,1984), nor with the results given by a recursion technique(Lam et a/. , 1985). The scaling considerations attributed
the origin of this hump to the fact that the crossoverfrom fractons to phonons is accompanied by "missing
modes" in the normalized DOS. In this subsection, thewhereabouts of these missing modes are discussed according to the work of Yakubo, Courtens, and Nakayama (1990).
The early scaling arguments about missing modes
should first be recalled. The DOS of a percolating network above threshold (p p, ) was characterized by two
regimes: the fracton DOS, 2)f, (co, p) cc co" ', for high frequencies
the
and
( co
co, ),
phononDOS,', for low frequencies (co & co, ). Assuming
X~h(co, p) ~ co))«X)(~) ~ A (p)(p —p, )Simulation results are illustrated in Fig. 12, which exhibits the validity of Eq. (5. 15) in the phonon-fractoncrossover region. Because the ordinate in Fig. 12 is the))Rev. Mod. Phys. , Vol. 66, No. 2, April 1994

FIG. 11. Crossover of the DOS (solid line). The dotted{dashed) lines represent the continuation of the phononton) asymptotic behavior into the crossover regime.

(frac-

Nakayama,

Yakubo, and Orbach:Dynamical

100
~

0 0

oo

—

~o
P
P

1

0-1

p = 0.593
p = 0.670
10

10

I

I

I

]0 2

10'

10

10

FIG. 12. Calculated density of states, normalized to one particle, divided by co', and plotted as a function of the frequency~. The solid circles correspond to the critical percolation density p, . The open ones are for p=0. 67. The horizontal lines aremeant as guides to the eye. After Yakubo, Courtens, andNakayama(1990).

weight is rather uniformly distributed over the lowfrequency region.
The above considerations can now be extended to percolation with a correlation length g(p). The discussion isfacilitated by adopting the nodes-links-blobs picture as illustrated in Fig. 1. The typical separation of the nodes
forming the macroscopicallyhomogeneous
network
thecorrelation
equalsWith
lengthg(p).
23~h= 3 (p)co/co, , and 2)t, = 3 (p)co', one calculates
the number of "missing modes" associated with the phonon regime, M h, asI

M h ——

02) h)des(2)t, —

1=—
A (p)co4.

quantity Xl(co)/co'~, the fracton regime corresponds to ahorizontal line of height A (p). Data at p, =0.593 areplotted as solid circles, where the network was preparedon a 700X 700 lattice with %=116,991. The data for the
DOS for a network with p=0. 67 (N= 317,672) are plotted as open circles. One can clearly discern the two regimes in Fig. 12, with a crossover frequency co, =0.1. It
is clear that this simulation does not exhibit any noticeable hump near co, . Further, the magnitude of the DOS
in the fracton regime is diferent in the two simulations,invalidating the equality 2)t, (co,p) =2)t, (co,p, ) assumed by
Aharony et al. (1985a, 1985b, 1987b). This justifies a
nontrivial dependence of A (p) on p in Eq. (5. 15). Finally, one notices that the two curves of Fig. 12 could not
possibly be made to scale towards the upper end of thefracton range. This is the region where modes "missing"from the low-frequency regime have accumulated, exhibiting the violation of the scaling hypothesis of Eq. (5. 15).
To understand these observations, it is helpful to consider the simple models illustrated in Fig. 13. From theSierpinski gasket of Fig. 13(a), one can construct largescale homogeneous systems in different ways. A first exarnple, illustrated in Fig. 13(b), is the carpet treated bySouthern and Douchant (1985). The modes of the simple
gasket, Fig. 13(a), have been investigated in considerable
detail by Rammal (1984a). They can be classified into
"hierarchical" modes, where the density at d= 2 peaks at
co= v'5, and into "molecular,
or strongly localized,modes with a highest density near the upper cutoff at
co=v'6. These modes are only slightly modified by thehigher coordination of a few sites (z=6) in Fig. 13(b).
The higher z simply produces a few modes at frequencies
above the Sierpinski gasket "band" in the region
v'6&co&3. One notes that co=3 is the upper cutoff ofthe d=2 triangular lattice. An alternative way to construct a large-scale homogeneous system is illustrated in"Rev. Mod. Phys. , Vol. 66, No. 2, April 1994

403

Fig. 13(c). That model corresponds more closely to theintuitive picture of fractal networks that reach theircorrelation length by "growing into each other. " In thatcase, the whole region v'6 & co & 3 becomes rather densely
populated with modes, at the expense of the DOS in thefracton regime. The corresponding "missing" spectral—
err ~ r ~ooooo~csi~~oQ~
0

~ ~

0

properties of fractal networks

C= —A (p)Q"(p —p, )(5. 16)= 13 from Fig. 12, one
With A (p ) = A (p, ) = 0.4 andvDf———
estimates M h 3(p p, ) f. The number of occupied
sites on the infinite network in the correlation volume for
d=2, g2, is gwhere P„=Pe(p —p, )~, with @=5/36
and Po —1.53. Hence the actual number of missingmodeswithinthecorrelationarea
is
P M&h —3 OPp.
Hereisdefined=obyg=:-o~p —independent of p —p,p, because the exponent
2v+/3+ vDf i—
s identicallyzero. Using:-O=0.95, we find the number of missing modes to be of
the order of unity.
Thus there is one missing mode per
area g' . One should also note that, for any non-negligible
g, the number M h relative to the total number of modes
with co(co, is very small compared to one, the corresponding area in Fig. 12 being overemphasized
by thelogarithmic presentation.
Numerically more important is the number of missing
modes Mf, produced by the depression of the fracton
density from A (p, ) to A (p). Ignoring the hump near
the high-frequency cutoff, this number is

0

P„,

~,

Mf = J

0[X)t,(ni, p, )—2)f, (ni, p ) ]de = 1

A (p)

3 p,

(5. 17)

There is a certain degree of arbitrariness in the absolutedefinition of dipl, as illustrated, for example, in the work of Kapitulnik et al. (1983), where two rather di6'erent values arefound (one in their Fig. 2, and another further in the text). The
value of:- we derived was taken from the mean-square size ofthe fracton. The value of Pp is taken from a series analysis using Pade approximants, by Sykes, Gaunt, and Glen (1976a).

404

Nakayama,

Yakubo, and Orbach:Dynamical~~ VVV
AAF V

AFV
AHA~~
(c)

For the second equality, use was made of Eq. (5. 15) andof the fact that the integral of 2)~, (co,p, ) over the full frequency range is normalized to unity. The simulated
values of M&, for several values of p are shown in Fig. 14,
a critical behavior, M&, =Ma(p —demonstratingp, )', , gives Mo =4. 1.
The solid line, drawn with m = —This behavior can be explained as follows. Within thearea g, a number of sites have higher coordination thanin the network at p, . The number of these sites is muchlarger than the small number of nodes that eventuallyform the homogeneous system and whose relative density
is 1/(g P„)=M~ .hBased on the naive picture of Fig.
13(c), one could expect that the number of these sites is
proportional to the length of the perimeter, i.e. ,In the case of the percolation cluster, the perimeter at
is a fractal of dimension DI —1, and its length is g ~The total number of occupied sites within this perimeter
being g ~, the relative number— of modes rejected to high
D~
D~
~ I
frequency is then M&, —/g ~ ~ (p —gp, ) . In fact,taking a square of side g, the ratio of occupied perimetersites to occupied area is Mr, =4//=4. 2(P —P, )', and one
notices that v=4/3 for the d=2 case. Both the exponent
and the amplitude agree well with the simulated values(Fig. 14). This supports the validity of this interpretation. Although there is, strictly speaking, no well-defined"perimeter" at g for which the connectivity of all sites increases, the concept appears to remain well defined from(1Q

properties of fractal networks

FIG. 13. Lattice models based on the Sierpinski gasket: (a) a gasket up to the fourth level ofhierarchy; (b) a unit cell made of such a gasket
and an empty triangle; (c) a denser unit cell,possibly more representative of real fractal objects, obtained by the junction of two gaskets,
such as in (a).

an average point of view. The effect of the higher connectivity is to depress the number of modes throughout
the fracton regime.

B. Localized properties of fractons

1. Mode patterns of fractons
The first realization of mode patterns of large-scale
fractons was obtained by applying the numerical method
mentioned in Sec. V.A (Yakubo and Nakayama, 1987a,
1989b). The network at p, =0.593 was formed on a
700X700 square lattice with the number of occupied
sites %=169576. The magnified color picture of themode pattern of a fracton (snaphot) on a d=2 network isshown in Fig. 1 in the paper by Yakubo and Nakayama(1989b), where the eigenmode belongs to an angular frequency m=0. 01. To clarify the details more directly, thecross section of amplitudes of the mode pattern is shownin Fig. 15. Figure 15 exhibits this cross section of the vibrational amplitudes for the fracton mode along the lines
A and 8 drawn in Fig. 1 in the paper by Yakubo andNakayama (1989b). The solid circles and the open circles
represent the occupied sites and the vacant sites in the
percolating network, respectively. One sees that the fracton core (the blue part with the largest amplitude)
possesses very clear boundaries for the edges of the excitation, almost of steplike character, with a long tail in the
direction of the weak segments. This is in contrast withthe case of homogeneously extended modes (phonons) inwhich the change of their amplitudes
is correlatedsmoothly over a long distance. It should be noted that
displacements of atoms in "dead ends" (weakly connected portions in the percolating network) move in phase,and the vibrational amplitudes fall off sharply at theiredges. In addition, it is interesting to note that the tail(the portion spreading from the center of the figure in theupper-right direction) extends over a very large distancewith many phases changes. '

10'

2

P-P,

FIG. 14. Relative number of missing modes, M&„produced by
the depression in the fracton DOS, presented vs p —p, . The error bars are standard deviations. The straight line of slopev 3 is shown to agree with the asymptotic behavior for p ~p,After Yakubo, Courtens, and Nakayama (1990).
Rev. Mod. Phys. , Vol. 66, No. 2, April 1994

This is a natural consequence of the orthogonality conditionof eigenmodes, since vibrational modes belonging to eigenfrequencies co %0 must be orthogonal to the mode of co =0 (uniform displacement).

Nakayama,

Yakubo, and Orbach:Dynamical

~
Line

T

TTTTTT

properties of fractal networks

f TQ

T

T

TTfT

T

A
Ill

I1Il1 Il I%IIII
I

0.

CD

Il

-0-0- CXXXD -0- - 00 00 - 0 CCNXXXXXD

Il&

ii

II IIIIII' I&&

0 - - ~ CCXXD - - - EO 0~

405

FICx. 15. Cross sections of a single fracton shown in Fig. 1 in thepaper by Yakubo and Nakayama(1989b). The upper figure corresponds to line A in Fig. 1 in thepaper and the lower one to line
B. The signs of the amplitudes(plus and minus) are switched
for convenience. After Yakubo
and Nakayarna (1989b).

IllllIllll

W - —- - CXD- -00 -"1111kllLine B

2. Ensemble-averaged

fractons

Localization of waves in disordered systems has received much attention since the work of Anderson(1958). The progress made in understanding electron localization (Mott, 1967; Thouless, 1974; Abrahams et al. ,1979) has had implications for other excitations in disordered media such as phonons, photons, and spin waves(John et aI. , 1983; Akkermans and Maynard, 1985). In
particular, it was predicted by John, Sompolinsky, andStephen (1983) that the vibrational excitations on d ~2
disordered systems would always be localized, and the localization length A would behave as A ~exp(1/co ) for
d=2 and as A o-m foI d=1.
Rammal and Toulouse (1983) applies the scaling
theory of localization (Abrahams et a/. , 1979) to fracton
excitations on a percolating network. The key parameterin the scaling theory, as described in Sec. IV.B.1, is the
exponent Pl, which they expressed in terms of the fractal
and fracton dimensions Df and d as

Px=

Df

2).
(d —

', for percolationin any Euclidean dimenBecause d = —sion d, it is clear that fractons are always localized. In
this context, Entin-Wohlman,
Alexander, and Orbach(1985; see also Alexander, Entin-Wohlman, and Orbach,1985a, 1985b, 1985c, 1986a, 1986b, 1987) supposed thatthe ensemble average of the fracton wave function on
percolating networks was localized with the formd&Pf, ) ~exp

Rev. Mod. Phys. , Vo!. 66, No. 2, April 1994

(5. 18)where A(co) is the frequency-dependent
fracton lengthscale (dispersion or localization), and r a radial distancefrom the center. The exponent d& denotes the strengthof localization. Note that this is an ensemble-averagedenvelope function. Many studies of the value of d& havebeen performed:
theoretical (Aharony et al. , 1987b;
Harris and Aharony, 1987; Levy and Souillard, 1987;
Bunde et al. , 1990; Aharony and Harris, 1992; Bunde
and Roman, 1992), experimental (Tsujimi et al. , 1988),
and numerical (de Vries et al. , 1989; Nakayama et al. ,1989; Li et al. , 1990; Lambert and Hughes, 1991; Romanet al. , 1991; Terao et al. , 1992). Levy and Souillard(1987) suggested that d& =Df ld. For d=2, because
Df =91/48-1. 896, this gives d& =1.42. Localized stateswith d&) 1 are called superlocalized modes. Harris andAharony (1987) found that averaged fracton excitations
decay exponentially (d&=1).
Van der Putten et al. (1992) have experimentally estimated the superlocalization exponent d& from measurements of the d, conductivity of carbon-black-polymer
composites. They obtained the value of the exponent
d&=1. 94+0.06 as well as the value of the conductivity
exponent p = 2. 0+0.2.

a. Shape of core region
The localized nature of fractons, focusing on the value
d& for the core region of fractons, hasbeen numerically investigated by Nakayama, Yakubo,
and Orbach (1989; see also Yakubo and Nakayama,1990). They performed numerical simulations on d=2
percolation networks to determine d& for the cores of 2d
fractons. The core has a large amplitude around theof the exponent

Nakayama,

406

Yakubo, and Orbach:properties of fractal networks

Dynamical

center of a localized fraction (r 5 A), as described in Sec.
V.B.l. They prepared nine 2d site-percolating networks
at p, formed on 700X700 square lattices in order to takean ensemble average. The maximum network size was
X = 171 306 and the minimum size was X = 76 665.
Smoothly varying ensemble-averaged mode patterns wereobtained. The ensemble-averaged
shape of the fracton
core was calculated by averaging over 129 fractons at
co =0.01. They obtained d& =2.3.
In addition, d& and A(co) were calculated for fourco=0.005, 0.006, 0.007, anddifferent eigenfrequencies,
0.008, excited on five percolating networks. As seen from
Fig. 16, the localization length A(co) depended on frequency. The straight line drawn using least-squares', showing good agreefitting indicates that A(co) ~co—d /Dfment with the theoretical dispersion law A(co) ~ cowith d /Df =0.705. For all those frequencies, they foundd&=2. 3+0.1, independent of co. It is now understoodthat this large value of d&=2. 3 applies only to the coreregion of fractons, and that two exponents are required
to characterize the localized nature of fractons, namely,
for the core and for the tail (Roman et a/. , 1991).
b. Asymptotic behavior

The averaged profile of tails (asymptotic behavior) of
fractons has been numerically investigated by de Vries,
de Raedt, and Lagendijk (1989), Li, Soukoulis, and Grest(1990), Lambert and Hughes (1991), Roman, Russ, andBunde (1991), and Terao, Yakubo, and Nakayama (1992).
Roman, Russ, and Bunde (1991) have solved the vibrational equations for large percolation clusters (300 sheets)with d=2, and have averaged over ten individual fractonmodes for each frequency. The amplitudes ~P(r, co)~ fortypical fracton modes are exhibited in Fig. 17. The data

0

~

»

&

&

~

»

&

i

[

r

I

~"t «~a rrrr rr~-rr
&

—

~

r

~

—ivmcr~p~Mf l~~l

I

I

~f

I

863

4:
2

0«t

5

&

I

t

&

&

&

I

» « »
I

r

r~ ~"- cos(ir/d

&

~«

I

i

&

i

t

I

&

30

s

&

r

I

i

~l

i

r

r

i1

15

)

—D /d
2/d
FIG. 17. Plot of —ln[~P(r, co)~co ~ ] vs rescos(aid ),—1.896 and d =2.87, for frequencies co = 0.20
with Df =-0. 10 (circles), and 0.07 (crosses).
After Roman, Russ, and Bunde (1991).
(squares),0. 15 (triangles),strongly support the value d&=1 and the expected frequency dependence of A. Note that a large crossover regime exists when r ~ A, with an effective exponent d& & 1
which coincides numerically with the result of Nakayafor
ma, Yakubo, and Orbach (1989). Asymptotically,r )&A, the curves approach a straight line in a semilogarithmic plot, and hence d& = 1.
Bunde and Roman (1992) have given an analytic explanation for the asymptotic behavior of fractons. For
scalar vibrations, the envelope function of the fracton,density P(r, t) the~cd(r, co)~, is related to the probability
probability of finding the random walker after time t at asite a distance r from its starting point:P(r, t)= J0dcol)(co)~(()(r, co)~exp(

co

t),

—19)
(5.

where 2)(co) is the DOS normalized to unity. For a large
class of networks, including percolating networks at
P ( r, t ) decays,
over
upon
averagingtypicalconfigurations, as (Havlin and Bunde, 1989)

p„

30—

(5.20)
C

O

+
2ON
qf

D

I

I

I

5

6

7

Frequency

l

8

9 10

(x10 )

FICx. 16. Values of localization length A(co) plotted as a function of frequency on a log-log scale. The straight line
drawn using least-squares fitting shows that
[A(co) ~co
fractons in the ensemble follow the correct fracton dispersion
law. After Yakubo and Nakayama (1989a).

"]Rev. Mod. Phys. , Vol. 66, No. 2, April 1994

where d is the diffusion exponent defined in Eq. (3. 12).
Using the asymptotic result Eq. (5.20), one obtains d& =1
upon taking the inverse Laplace transform of Eq. (5. 19)withthemethodof steepest
descent,
using
2/d
and a a constant of order
A(co) '=a cos(vrld )counity (Roman, Russ, and Bunde, 1991).
We warn the reader that the ensemble average of matrix elements will be very different, in general, from thematrix element using ensemble averages for the fracton
functions (Nakayama, Yakubo, and Orbach, 1989). For
example, the Raman-scattering intensity is proportional
to the square of the elastic strain induced by fracton excitations. For this case, the ensemble average of the matrix
element for indiuidual fractons should be taken into account [see Eq. (6.5)].

Yakubo, and Orbach:Nakayama,

Dynamical

c. Multi fracfal behavior

407

i,

A multifractal analysis (see, for example, Paladin andVulpiani, 1987), which originated from the study of turbulence (Mandelbrot, 1978), is useful for the analysis of
the distribution of various physical quantities (measures)on fractal supports such as the voltage drop distribution
on a random resistor network (Rammal, Tannous, Breton, and Tremblay, 1985; Rammal, Tannous, and Tremblay, 1985; de Arcangelis, Redner, and Coniglio, 1985,
1986; Coniglio, 1987), the growth probability of adiffusion-limited
aggregate (DLA; see Meakin, 1986;
Blumenfeld and Aharony, 1990; Mandelbrot and Evertsz,
1990; Schwarzer et al. , 1990), and the spatial profile of a
localized excitation (Castellani and Peliti, 1986; Bunde,Havlin, and Roman, 1990; Evangelou, 1990; Evangelou,
1991; Schreiber and Grussbach, 1991, 1992; Bunde andRoman, 1992; Roman, 1992a). Such analyses give information about the distribution of physical quantities (measures) on a fractal support.
Consider a measure f, on the ith site of a fractal structure of size L. The multifractal nature arises when theqth order moments of f, , averaged by the distributionfunction n (f, L), scale aswith a nonlinear function r(q). Typical examples of multifractal distributions are growth probabilities Ip;] for
DLA and voltage drops [ V, I on percolating resistor networks. In many cases, the multifractal properties are related to a very broad distribution function n (f, L), sothat different moments of are dominated by different
parts of the distribution function.
As mentioned in Sec. V.B.1, the profile of a fracton
eigenfunction P&, is extremely complicated, and the distribution function n ( Pshould be very
) of amplitudes
broad. It is natural to consider the possibility that suchlarge Auctuations may show multifractal properties. Petri and Pietronero (1992; see also Petri, 1991) have performed a multifractal analysis for fracton wave functions
on d=2 percolating networks. They calculated fracton
eigenfunctions using a direct diagonalization techniqueand obtained the (Euclidean) length-scale dependence of

f "

J

~

properties of fractal networks

the qth moments of the measures ~Pr,
which are thesquared amplitudes at the ith sites. Their results suggestthat the function r(q) is nonlinear with respect to q (seeFig. 18), showing multifractal behavior.
%'e note here that an exponentially decaying averaged
wave function does not recover multifractal behavior fora Euclidean length scale. The distribution of ~PI;~ does
not exhibit multifractality, at least for r &) A, where A isthe localization length. Therefore the observed multifractality must arise from the core region r A, as in
the case of Anderson localization, for which localized
wave functions within r + A do have multifractal properties (Wegner, 1980). Since Petri and Pietronero (1992)
chose relatively high frequency modes, the fracton cores
handled by them are very small (of the order of a lattice

constant).
Another kind of multifractal property for fracton wave
functions was proposed by Aharony and Harris (1992),
Bunde and Roman (1992), and Bunde et al. , (1992) that
does not con Aict with an exponentiallydecaying
ensemble-averaged
wave function. They introduced rdependent
moments( Pr, (r, cu) «), where Pr, (r, co)represents the amplitude of a fracton with frequency co ata distance r from the center of the fracton mode. The angular bracket () denotes the average over all fracton
states with frequency cu and all possible realizations. An
exponentially decaying wave function leads to the moment ( ~Pr, (r, co)i ~) decaying exponentially with r This.
is not multifractal in the usual sense. However, if the
moments take the form~~

(5.21)
where y (q) is a nonlinear function with respect to q, thewave function ~Pr, can be regarded as multifractal undera new degnE'tion in which exp[ (r/A)~] is consid—ered as
the scale for fracton modes, instead of L.
Assuming that the moments of P&, at a fixed chemical
distance l from the center of a fracton are expressed asql/A], which holds for deep impurity
Pq(l, co)-exp[ —states with energies far from the band edge (Harris andAharony, 1987; Aharony and Harris, 1992), Bunde et al.
~

12

10—

In multifractal

analyses, measures are usually normalized

asg, f; = l. For example, choosing the mass m; of the ith site ina fractal —network as the measure f;, one should takeD(m; ) =L ~. The moments(mP) are calculated asi.

Therefore the exponent r(q)(m, ~) = g,m, ~=L .L
becomes ~(q) =DI(q —1), namely, w(q) is a linear function withrespect to q. In this case the distribution is called unifractal. In
general, the distribution function n (f, L) can be expressed by asimple scaling form as n ( f, L) = "F( «/L) for unifractals.
L
.

~

D

0—

qD

f

Rev. Mod. Phys. , Vol. 66, No. 2, April 1994

f

6

8

10

FIT+. 18. Simulated results of w(q) for a fracton at the band
center (co=0.5107). After Petri and Pietronero (1992).

408

Nakayama,

Yakubo, and Qrbach: Dynamical properties of fractal networks(1992) have suggested that the moments of the fracton
wave function obey the form Eq. (5.21). They have also
predicted a critical value q, below which, q & q„ the moments ( ~Pf, (r, co) q) exhibit multifractal behavior for) r *(q) .Here, r *(q) ~ q~d,'", where„describesd'".
~the average chemical distance as ( l ) r
For q q„
( ~Pf, (r, co) I) has the simple unifractal exponential
behavior e ~' . For multifractal fracton wave funcr

min

)

~

rnin
tions, the exponent y (q) becomes proportional to q
in the theory of Bunde et al. (1992). They have estimated a value for q, from their numerical simulations using
50 fractons with frequencies n = O. 1. They foundq, =213. Further details and numerical simulations concerning these predictions have been reported in Bunde
et al. (1992), Bunde and Roman (1992), Roman (1992b),
and Aharony and Harris (1992).

C. Observation of fractons in real materials:
Neutron-scatteringexperiments

3. Fractality of silica aerogels
and other disordered systems

namely,Kistler (1932) found a way to dry gels without collapsing them, and produced extremely light materials withporosities as high as 98%, which are ca11ed aerogels.
They have a very low thermal conductivity, solid-likeelasticity, and very large internal surfaces. As a consequence, aerogels exhibit unusual physical properties,
making them suitable not only for a number of practicalapplications, such as detectors of Cerenkov radiation,
supports for catalysts, or thermal insulators (see, for example, Fricke, 1986, 1988; Brinker and Scherer, 1990),
but also for basic research studies. For example, a number of experiments have shown that these porous structures can have profound effects on critical phenomena(Chan et al. , 1988; Wong et al. , 1990; Wong and Chan,
1990; Frisken, Ferri, and Cannell, 1991; Mulders et al. ,1991. See also the conference proceedings edited by
V'

Fricke, 1992).
The initial step in the preparation of silica aerogels is
the hydrolysis of an alkoxysilane Si(OR)„, where R is
CH3 or C2H~ (Kistler, 1932; Prassas et a/. , 1984). The
hydrolysis produces silicon hydroxide Si(OH)~ groups
that polycondense into siloxane bonds -Si-O-Si-, andsmall particles start to grow in the solution. These particles bind to each other by cluster-cluster aggregation,
forming more siloxane bonds and eventually producing adisordered network Ailing the reaction volume, at which
point the solution gels. The reactions are normally notcomplete at this gel point, and the cluster networks continue to grow in the alcogel phase. After suitable aging,if the solvent is extracted above the critical point, theopen porous structure of the network is preserved anddecimeter-size monolithic blocks with a range of densities
from 50 to 500 kg/m can be obtained.
Small-angle scattering techniques using neutrons and xrays are very well suited to systematically investigate theRev. Mod. Phys. , Vol. 66, No. 2, April 1994

structure of silica aerogels. It has been found (Schaefer
and Keefer, 1984, 1986; Courtens and Vacher, 1987;
Vacher, Woignier, Pelous, and Courtens, 1988; Vacher,
Woignier, Phalippou, Pelous, and Courtens, 1988; Woignier et al. , 1990; Posselt, Pedersen, and Mortensen, 1992)
that aerogels take fractal structures, as in the cases ofcolloidal gels (Dietler et a/. , 1986, 1987). Direct electron
microscope observations on silica aerogels were done by
Brinker and Scherer (1990), Brinker et al. (1982),
Rousset et al. (1990), Vacher et al. (1991), and Buckley
and Greenblatt (1992). Bourret (1988) and Duval et al.
(1992) reported high-resolution
electron microscopy(HREM) observations that are compatible with a fractal
geometry. Beck et al. (1989) and Ferri, Frisken, andCannell(1991) used light-scattering
techniquesforcharacterizing
geometrical features. Phalippou et al.
(1991) applied thermoporometry for an in situ study ofsilica aerogels.
The scattering diQ'erential cross section measures theFourier components of the spatial fluctuations in the
scattering length density. For aerogels, the differential
cross section is composed of the product of three factors,= Af (q)S(q)4(q)+8 .
Here 3 is a coefficient proportional to particle concentration,(q) is the elemental particle for-m factor, the structure factor S(q) describes the correlation between parti
cles in a cluster, and 4(q) takes account of the cluster
cluster correlations. B gives the incoherent background.
The Fourier transform of the particle density-density
correlation function G (r) givesfS(q}=1+—~G(r) —1 ~e'q'dr .
V v

I(5.22)

Fractal (self-similar) structures, extending up to a correlation length g, can be modeled by the correlation function G (r) (Teixeira, 1986; Courtens and Vacher, 1989),

G(r) —1 ~rD—3exp( —
r/g).

(5.23)

This is the most convenient, but not unique, choice.
Equation (5.23) is a consequence of the mass M(r) withina sphere centered on a particle at r=0, scaling as
f. Hence the density in the sphere scales as
M(r) ~r Df—
Dff 3
~
r
. Two limiting regimes are of interest. Atp(r)small q, qg(&1, S(q) is almost independent of q. Whenqg))1, one obtains by substituting Eq. (5.23) into Eq.
(5.22) the following simple result for the scattering function,

S(q) ™q(5.24}In the ideal case, the value of Df can be deduced fromthe slope of the observed corrected intensity versus
momentum transfer in a double-logarithmic
plot. This is
illustrated in Fig. 19, which shows the results of elasticscattering experiments on silica aerogels (Vacher, Woignier, Pelous, and Courtens, 1988).

Nakayama,

Yakubo, and Orbach: Dynamical properties of fractal networks

The various curves are labeled by the macroscopic
density p of the corresponding sample, N095, meaning"neutrally reacted" with p=95 kg/m . The solid lines
represent the best fits. They are extrapolated into theparticle regime (q) 0. 15 A ') to emphasize that the fits
do not apply in that region, particularly for the denser
samples. Remarkably, Df is independent of sample density to within experimental accuracy: Df =2. 40+0.03
for samples N095 to N360. Furthermore, g scales withe
as g ~ p '6 — . The departure of S (q) from the q
the presence of particles
dependence at large q indicates0with gyration radii of a few A.
The lower three curves in Fig. 19 are the results ofS(q) for samples prepared under basic catalysis. They
are very different from the upper three curves. Comparing N200 to B220, one notes that the extension of thepower-law region is very different for these two curves.
A most striking effect is found in the value of Df. All
curves from base-catalyzed samples with an extended
fractal range have Df = 1.8+0. 1.
To summarize, silica aerogels exhibit three different
length-scale regions: At short distance, elemental particles of radial size R are found. The particles are aggregated into clusters with size g at intermediate distance,
and a gel is formed by connection of the clusters at large
distance. At intermediate length scales, the clusters possess fractal structure, and at large length scales the gel is
4

10

I

I

I

I

l

I

I

I

l

I

I

I

I
l

10

160

175
1 — 200
220

260
284

310
356

cn

-6

I

II

a homogeneous porous glass. The fractal structure of silica aerogels and their dynamics have been reviewed by
Courtens, Vacher, and Stoll (1989), Vacher, Courtens,
and Pelous (1990), Kjems (1991, 1993), and Courtens andVacher (1992).'

2. Observed density of states
A direct way to obtain the fracton dimension d for realmaterials is to measure the density of states (DOS). A
inelastic neutronvery direct method is incoherent
scattering experiments, which measure the amplitudeweighted DOS. The scattered intensity is given by

where n (co) is the Bose-Einstein distribution function.
The wave vectors k and k' correspond to the incident andscattered neutrons, respectively, and q=k' —k. D, (co)and 8' are the DOS and the Debye-Wailer factor
characteristic of the ith site, respectively. The summation extends over the different sites for the atoms, each ofwhich contributes (proportionally to the amplitude of vibration at frequency co) to the incoherent-scattering crosssection. Incoherent neutron scattering from protons
chemically bonded to the particle surfaces can be used todetermine the DOS in porous media (Richter and Passell,1980). A careful analysis has been applied to extract the
intrinsic DOS by extrapolation
to zero-momentum

transfer.
of the incoherent inelastic scattering
Measurements
have been performed in aerogels (Courtens and Vacher,
1988; Coddens et a/. , 1989; Conrad, Fricke, andReichenauer, 1989; Conrad, Reichenauer, and Fricke,
1989; Page, Buyers, et al. , 1989; Page, Schaefer, et al. ,1989; Pelous et al. , 1989; Reichenauer, Fricke, andBuchenau, 1989; Vacher and Courtens, 1989a, 1989b;
Vacher, Woignier, Pelous, et al. , 1989; Vacher, Woignier, Phalippou, et al. , 1989; Conrad et a/. , 1990; Courtens, Vacher, and Pelous, 1990; Schafer, Brinker, et al. ,1990; Schaefer, Richter, et ah. , 1990; Vacher, Courtens,
Coddens, et al. , 1990; Courtens and Vacher, 1992).
Vacher and Courtens (1989a, 1989b), Buchenau, Morkenbusch, et al. (1992), and Kjems (1993) have reviewed
these results.
Investigations on the phonon-fracton crossover in silica aerogels have been made using inelastic neutronscattering experiments. These have been done on back-

118

p

409

I

I

0. 001SCATTERING

I

Il

I

I

I

0. 01
VECTOR

tl

I

0. 1
q

(A

)

FICx. 19. Scattered intensities for 11 samples. From top to bottom: ten untreated, neutraHy reacted samples of increasing density, and one oxidized sample. The curves are labeled with p in
kg/m . After Vacher, Woignier, Pelous, and Courtens (1988).
Rev. Mod. Phys. , Vol. 66, No. 2, April 1994

There are several reports on disordered materials which arepolymers
said to exhibit fractal structures and properties:(Fischer et al. , 1990; Zemlyanov et al. , 1992), vitreous silica(Dianoux et al. , 1986; Dianoux, 1989), fumed silica (Page,
Buyers, et al. , 1989; Page, Schaefer, et al. , 1989), smokeparticle aggregates of silica particles (Richter et al. , 1987), andsuperionic borate glasses (Fontana et al. , 1987, 1990).

410

Nakayama,

Yakubo, and Orbach:Dynamical

andspectrometers
Fricke,(Conrad,scattering
Reichenauer, 1989; Conrad, Reichenauer, and Fricke,
1989; Pelous et aI. , 1989; Conrad et al. , 1990; Vacher,
Courtens, and Pelous, 1990) and using the spin-echo
technique (Courtens, Vacher, and Pelous, 1990; Schaefer,
Brinker, et al. , 1990; Schaefer, Richter, et al. , 1990).
The backscattering technique has the advantage that the
low-frequency Debye range is seen as a constant-intensity
level extending from the elastic line to the crossover frequency. ' Thus any excess modes at the phonon-fractoncrossover would show up as a peak in the scattering intensity at that frequency, if present. However, such apeak has not been observed in the two backscattering experiments (Conrad et al. , 1990; Vacher, Courtens, andPelous, 1990). Rather, a gradual decrease is observed as
one passes through the crossover regime.
The neutron-scattering spin-echo technique has several
advantages over these other techniques. The larger spectral range makes it a suitable tool for the determinationof the fracton dimension.
The crossover frequencies determined by both backscattering and spin-echo measurements are generally ingood agreement with those determined by 8rillouinscattering (Courtens et al. , 1987b; Courtens, Pelous,
Vacher, and Woignier, 1987). As far as investigations at
higher frequencies are concerned, there are several early
results (Reichenauer, Fricke, and Buchenau, 1989; Vacher, Woignier, Pelous, et al. , 1989; Vacher, Woignier,et al. , 1989). These results exhibited a
Phalippou,change of slope in the log-log plot of the DOS at 200
6Hz, giving a stronger increase with frequency at higher
frequencies.
Investigators interpreted their data as acrossover from fractons to vibrational modes within theparticles. Courtens and Vacher (1989) compared thedata at high frequencies to a formula which has been derived for the DOS of small particles (Baltes and Hilf,
1973). In the regime of particle contribution, theeffective slope of the DOS is around 1.5. This seems tooriginate from the contribution of both surface (proportional to co) and bulk (proportional to co ) particle modes.
The energy resolution was insufhcient to observe thecrossover to the long-wavelength phonon regime.
Measurements(Coddens et al. , 1989; Vacher, Woignier, Phalippou, et al. , 1989) at higher resolution on asample prepared in a different manner have confirmed
the extended fracton region. The DOS in silica aerogels
has been studied in a wider frequency range by combining data from neutron time-of-Aight and spin-echo experiments (Vacher, Courtens, and Pelous, 1990). These results, exhibited in Fig. 20, indicate that different types ofmodes contribute to the scattering in the fracton range,

In general, the Debye range of dense solids cannot be obtained by the backscattering technique because of the extremely
small value of 2)(m)/co =(1/2m )(1/vI +2/UT). In silica aerogels, however, the sound velocities are lower by a factor of
about 40 (Conrad et al. , 1990).
Rev. Mod. Phys. , Vol. 66, No. 2, ApriI 1994

properties of fractal networks
I

I

I

[

I

I

I

100

c

l

0.01
0.00

10
CL

I

'e

I co2

(D

10

DOS
=

&Co&

LL

210 kg/m

10

I 10

e i

~+ y ~

+

Cl

10

0. 1

/o

BEND

10
FREQUENCY

100

1000

10000

(GHz)

FIG. 20. Density of states of neutrally prepared silica aerogel.
The open circles are time-of-Right measurements.
The dottedcurve indicates the DOS that fits the neutron spin-echo data.
The dashed lines indicate the asymptotic phonon as well as theindependent bend and stretch contributions. Inset: small-angle
neutron-scattering data from the same fit described in Fig. 1 of
Vacher, Woignier, Pelous, and Courtens (1988). A straight line
shows the Porod region at high q. After Vacher, Courtens,
Coddens, et al. (1990).
suggesting that the modes have predominantly
bendingcharacter at low frequencies and stretching character athigher frequencies. In the fracton domain, a power law
is observed over more than one order of magnitude.
It
has a slope d, —1 —1.2.
We should mention neutron-scattering experiments onother disordered materials, which were analyzed in termsof the fracton theory. Freltoft, Kjems, and Richter(1987) measured the low-frequency density of states for
fractal silica aggregates by inelastic neutron scattering.
They obtained their fracton dimensionalities.
Page,
inelasticBuyers, et al. (1989) performed
neutronscattering measurements on fumed silica and compared
the results with analogous results for amorphous quartz.
They saw no evidence for a hump' in the DOS near thephonon-fracton crossover. It was also found that neither
the temperature and wave-vector dependence of the intensity nor the absolute intensity was in accord with simple phonon models. Fontana et al. (1990) reported aof low-frequency
vibrational
study
dynamics
andelectron-vibration
coupling in AgI-doped silver borate~4Earlier inelastic neutron-scatteringexperimental results thatexhibit steepness or a hump in the DOS (Buchenau, Nuecker,
and Dianoux, 1984; Buchenau et al. , 1986; Rosenberg, 1985)
had been analyzed
according to the fracton viewpoint.
Buchenau, Nuecker, and Dianoux (1984), Buchenau et al.
(1986), and Dianoux et al. (1986) have suggested, however, thatthe observed hump in the vitreous silica can be attributed tosome intrinsic modes peculiar to amorphous materials.

Nakayama,

Yakubo, and Orbach:Dynamical

glasses. By using both time-of-Bight neutron-scatteringand Raman-scattering spectroscopies, they were able todetermine the vibrational DOS and the frequency dependence of the electron-vibration coupling. The fracton dimension of this system was determined to be 1.4. Zemneutronet al. (1992) employedinelastic
lyanov
scattering measurements to study low-frequency vibrational excitations in polymethyl metacrylate (PMMA).
The DOS obtained in the 2.5 —10 meV range follows apower law in energy, with a spectrum corresponding to a
fracton dimension d=1. 8+0.05. Dianoux, Page, andRosenberg (1987) and Arai and Jdrgensen (1988) performed inelastic neutron-scattering experiments on epoxy
resins. Dianoux, Page, and Rosenberg (1987) found thatthe DOS above E=1.2 meV was proportional to E"
with d=1.5, whereas the DOS below 1.2 meV was proportional to E . They also found a rapid rise in the DOS
near the phonon-fracton crossover energy. The resultsby Arai and Jelrgensen (1988) exhibit a fracton dimension
d=1. 9+0.1, with a crossover energy of 1.8 meV, butthey found no rapid rise in the DOS around E= 1.8 meV.
There are experiments that investigate the dynamical
properties of fractal materials, but that do not involve
scattering techniques.
Helman, Coniglio, and Tsallis(1984) have shown that a proper description of the temperature dependence of the spin-lattice relaxation rate of
low-spin hemoproteins and ferrodoxin measured by Stapleton et al. (1980) and Allen et al. (1982) requires thatboth the fractal structure of the protein backbone andthe cross connections between segments of the folded
chain be taken into account. This work led to several investigations regarding fractons in protein dynamics (Stapleton et al. , 1985; Herrmann,
1986; MacDonald andJan, 1986; Yup Kim, 1988). It is obvious that the exponent d is obtained by spin-lattice relaxation experiments; but the analysis of data is not straightforward, because one cannot use the ensemble-averaged fracton wave
functions to calculate matrix elements.
Theoreticalworks (Alexander, Entin-Wohlman, and Orbach, 1985a,
1986; Entin-Wohlman,
Alexander, and Orbach, 1985;
Shrivastava, 1986, 1989b) have been done assuming theensemble-averaged superlocalized fracton wave functions
given by Eq. (5. 18). The arguments taking into accountthe proper averaged procedure for the matrix elements as
done by Alexander, Courtens, and Vacher (1993) for inelastic light scattering are required. Kopelman, Parus,
and Prasad (1986) and Fischer, von Borczyshowski, andSchwentner (1990) have determined the fracton dimension d by considering exciton recombination in disordered systems. Kopelman, Parus, and Prasad (1986)
measured the exciton recombination characteristics ofnaphthalene-doped
microporous materials. This technique yields the fracton dimension of the embedded
naphthalene
structure, or, equivalently,
the effective
random-walkdimension of the porous network. The
values of d obtained are between 1 and 2. Fischer, vonBorczyshowski, and Schwentner (1990) have studied the
trap-depth distribution of dibenzofuran singlet excitonsRev. Mod. Phys. , Vol. 66, No. 2, April 1994

properties of fractal networks

411

and the temperature-dependent
energy migration by
time-resolved spectroscopy via synchrotron radiation andtwo photon laser excitation. They obtained d= 1.14 fromtheir experimental results. Krumhansl (1986) stressedthat for polymers consisting of long chains, the translational symmetry is effectively valid; namely, the conventional phonon picture is valid for chains. It should be
kept in mind, however, that all configurations of linear
polymers have d= 1 (Alexander, 1986).
Sintered metal powders possess a random structure
consisting of small metal particles of the size a -500 A.
The study of the vibrational modes of these materials
the anomalous
might be important for understanding
Kapitza resistance at millikelvin temperatures (see the review by Nakayama, 1989). Malieppard et al. (1985) andPage and McCulloch (1986) measured the ultrasoundpropagation in sintered metal powders. They found thata band edge exists, at A, -10a, below which sound does
not propagate. They suggested that this edge is associated with a transition from phonon to fracton vibrationalmodes of the sinter. Wu et al. (1987) have investigatedfracton modes using thin metallic planes, which are randomly degraded square lattices of bonds. Their systemmaps onto the percolation model with scalar displacements. Hayashi et al. (1990) have studied low-frequency
vibrational modes in sintered copper and silver powders
using Au Mossbauer spectroscopy. The energies of thevibrational modes coincide with the values estimatedfrom ultrasonic data (Malieppard et al. , 1985; Page andMcCulloch, 1986) using a fracton model, providing support for the localized nature of the modes.
Finally, we should mention experiments on fractal superconducting networks. Alexander (1983) has pointed
out that the linearized Ginzburg-Landau
equation close
to the superconducting transition temperature T, can bemapped onto the equation for random resistor network(or elastic network with scalar displacements).
Modernlithography techniques have enabled the fabrication ofsuperconducting
networks
percolatingor Sierpinski
gaskets. These are obvious candidates for a quantitative
experimental test of the fracton concept. These weredone by Gordon and Goldman (1987, 1988a, 1988b),
Gordon, Goldman, and Whitehead (1987), Gordon et al.
(1986), Yu, Goldman, and Bojko (1990), Yu, Goldman,
Bojko, et al. (1990), Senning et al. (1991), and Meyer
et al. (1991; Meyer, Martinoli, et a/. , 1992; Meyer,
Nussbaum, et a/. , 1992).
O

VI. SCALING BEHAVIOR OF THE DYNAMICALSTRUCTURE FACTOR

Scattering experiments yield S (q, co), providing rich information on the dynamic properties of fractal structures. A variety of scattering experiments have been performed so far for physical realizations of fractal materials, such as sol-gel glasses, silica-aerogels, and borate
glasses. In this section, we review theoretical and experi-412

Yakubo, and Grbach: Dynamical properties of fractal networks

Nakayama,

for this quantity.
mental developments
Scaling arguments on the dynamical structure factor S(q, co) will bepresented in Sec. VI.A. 2, in order to interpret experimental data or simulated results on S (q, a~).
Before discussing the results for S(q, co), we should
clarify the meaning of energy width of fractons. Obviously, one could find exact eigenstates of the random
structure which would have no energy width: they would
be precisely defined in energy. It is only when one projects them onto plane-wave states that a lifetime is generated, equally in frequency or wave-vector space becauseof the linear phonon-dispersion relation. When we calculate an energy width for the fractons, it should be understood to be that width that a plane wave would experi-

ence.

—
i g (q u; )eS(q, co) =g 5(co —coi )(5pg(q)5pi( —q) ) .

In general, the intensity I(q, co) of inelastic neutron or
light scattering with a frequency shift co( =co —coo) is proportional to the Fourier transform of the density-densityr', t)
correlation
function,definedG (r —by= (p(r, t)p(r', 0) ), where p(r, t) is the density and the angular brackets denote an equilibrium ensemble average.
A general form is obtained by introducing the density
fiuctuation 5p(r, t) defined by

5p(r, t) =p(r, t) —p(r),(6. 1)

where p(r) is the static density given byp(r) = g 5(R, —r),and R; is the equilibrium position of the ith atom. One
usually neglects the term contributing
only to elastic
scattering (co=0). S(q, co) is then expressed in terms of
the Fourier transform of the density fluctuation 5p(r, t),

2' Jdte''(5p(0)5p (t)) .

(6.2)

Because the density fiuctuation 5p(r, t) induced by lattice
vibrations
withdisplacementsu;(t) is writtenr) 5(R; —r)], the Fourier
5p(r, t)= g; [5(R,. +u, (t) —transform 5p (t) becomes

—

(6.5)

For convenience, the reduced dynamical structure factor

S(q, co) will be used hereafter. The usual mode quantization and thermal factor [n (co)+ 1]lco, where n (co) is theBose-Einstein distributionfunction, are factored out
from S(q, co) by the relationin Eq. (6.S) into a frequency in-S(q ~) =&(~)(5pdq)5pd

1. Expression for the intensity of inelastic scattering(6.4)

Substitution of Eq. (6.3) into Eq. (6.2) yields

Changing the summation
tegral, one has

A. Theoretical treatments of the
dynamical structure factor S {q,co)S(q, co)=5pi„(q) =(6.6)

where the angular brackets denote the average of 5pi(q)over all modes k with frequencies close to co. Finally, one
has the expression for the intensity of inelastic scattering,1(q, ~) ~n (co)+1&(~)(5pi(q)5pi( —q))

In principle, S(q, co) can be calculated analyticallyfrom Eq. (6.5) or Eq. (6.6) if one knows 5p&(r) [or thefracton wave function Pi(r)] for a specific realization.
However, this is not straightforward, because of the extremely complicated character of fractons (see, for example, Fig. 1S). Although some general remarks can bemade about S(q, co) using scaling arguments (Aharony
et al. , 1988) and an analysis can be carried out within theeffective-medium
approximation(EMA; Polatsek andEntin-Wohlman, 1988; Entin-Wohlman, Orbach, and Polatsek, 1989; Polatsek et al. , 1989), its explicit form is notknow~ except for deterministic fractals. Sivan et al.
(1988) and Entin-Wohlman et al. (1989) have analytically calculated S(q, ai) for the d=2 Sierpinski gasket interms of Green's function, where S(q, ai) was defined
through the displacement-displacement
correlation function. They have shown that fracton excitations on the
Sierpinski gasket obey a single-length-scalepostulateto S(q, co) being peaked at(SLSP), in additionindicating the appropriate dispersion.
q „=co+,2. Scaling arguments on 8{q,co)Decomposing u;(t) into normal modes u; = g& u, e
one obtains

5pq(t)= +5pi(q, t)+O(uwhere

5pdq t) =e'5pdq»

and 5p&(q) is defined by
Rev. Mod. Phys. , Vol. 66, No. 2, April 1994

),

(6.3)

Alexander (1989) and Alexander, Courtens, and Vacher (1993) have produced arguments
thesupporting
asymptotic behavior of the dynamical structure factor
S(q, co) based on the SLSP (single-length-scale postulate;see Sec. IV.B). The following argument is a summary of
the theory proposed by Alexander et al. (1993).
If the SLSP is valid, the correlation function S(q, ai)
should have the following scaling form, depending only
on the single length scale A(co),S(q, co) =q~H(qA(co)),(6.7)

Nakayama,

Yakubo, and Orbach:where the dynamical structure factor is a function of= Iql because of the spherical symmetry for the averq
The asymptotic
aged structure of random networks.
behavior of the scaling function M(x) for x
1 and forx
1 is of power-law form,

x', x «1;'

II(x) ~ x ', x))1.
Here a and a' are new scaling indices. Accordingly,
has

S(q, co) ~'q»—ad /D(6.8a)(6.8b)In the case of q A« 1, Eq. (6.4) can be expanded as'g (q R", )(q u, ),e
5p~(q) = —

where

g Iu;

u(co)=.

I

and N& is the number of sites contained in the region ofvibration, vz (i.e., (Xz) ~A f }. Thus the new exponent
o. characterizes an effective length relevant to an aUeragestrain, analogous to the relationship between the chemical and Euclidean lengths.
— /2 The magnitude of u(co) is
proportional to [A(co)] Dff because of the normalization condition g; u; = 1 (footnote 15). From Eqs.
(6. 10) and (6. 11) and from the use of the dispersion relaIf, qA(co) (&1;a'd /D'cof, qA(co)))1 .

q»+'coone

413

ton modes is given by«))properties of fractal networks

Oynamical

co,—d /DfA~(1993) obtaintion

S(q, co) ~q 4co

I

Alexander,(2o —4)](,d /D

f,
)

—1

Courtens,

andVacherqA &&1 .

Thus the exponents y and a in Eq. (6.8a) are determinedthrough o. as

(6.9)

=2'

Df

a =4+

Df —

g

(6. 12)

R; =R; —R& and R& is the center of the k-mode

fracton. The summand in Eq. (6.9) can be written as
q I R; u; ] q in terms of the dyadic product. Choosing
the center of the fracton as the origin, i.e., R&=0, andg; R,. =0 from the condition g; u; =0 (see footnote 10
in Sec. V. B.l)fip~(q) = —g q R,'
I

I:

'

u, —
u~]

].q

where u& is the amplitude at the center of the A, -mode
fracton, and the summation is restricted to a vibrating region v& which is chosen as the smallest region for which
the boundary condition plays no significant role for thevibration X. If an average strain tensor, e&, is defined as

2' .

(6. 13)
iq. R,.

'
in Eq. (6.4)
In the case qA))1, the phase factor e
is uniform (coherent) only over small regions of size
' ( (&A) (see Fig. 17). Provided that the vibratingl~ ~ qregion v& is divided into "blobs" of size I «A, the number of blobs in the region vz is proportional to (qA) f
and each blob has —(qa) f particles. Then, as for thederivation of Eq. (6.9), one can expand Eq. (6.4) for small
q as

6pz(q) =$ee[i (qa)

f(q. U~&) — (q r;)(q. u; )],$l(6. 14)one obtains
where~pz(q)= —q. Q (R~@R~)e~ - q .
The R; in v& are all at most of order A, so that the magnitude of 5pz(q) can be estimated as
D +2
6p~~q2A f e~ .

Using the definition of S (q, co) from Eq. (6.6), one has((e )')

S(q, co) n(co)q [A(co)]Alexander,

Courtens,

R& is the centerof mass in the "blob" /3 and-r;=R, —R~. The factor U~~—(qa) D g, ~u,. is the aver-(6. 10)and Vacher (1993) assume thataged motion of the blob /3 in the eigenmode A, . The
second summation in Eq. (6. 14) is taken over the regionof the blob /3. Inserting Eq. (6. 14) into Eq. (6.6), one hasphase factors exp[ —iq (R& —R&, )]. Because there is nocontributionof the scattering from different
coherent
blobs in the limit (q A )& 1), only terms with P =/3'
remain. Alexander, Courtens, and Vacher (1993) suggestthat the first term dominates the second term in Eq.
(6. 14) for q A &) 1.

((ez) ) has a scaling form, which to leading order is[((e )')„]'"(6. 11}[A(co) ]where the root-mean-squared

amplitudeRev. Mod. Phys. , Vol. 66, No. 2, April 1994

u (co) of the frac-~5For S(q, cu) defined by Eq. (6.5), the normalization
becomes g,. Iu~l~= [n (co}+11/co.

condition

414

Nakayarna,

Yakubo, and Orbach:Dynamical

When the number
large, i.e. , (qa) f
sumed to beq((U~q))of particles in each blob becomesf, S(q,—Df /2u (co) ~ [A(cv) ]Using the relation
Courtens, and Vacher (1993) have obtainedpressed by
2 — +xS(q, co) o-cv 'q Df

structure(6. 15)Alexander,rv), ex-systems were obtained by thesame numerical technique employed in a series of works(Yakubo and Nakayama, 1987a, 1987b, 1989a, 1989b,
1989c; Yakubo, Courtens, and Nakayama, 1990; Yakubo,
Takasugi, and Nakayama, 1990). By using this algowith
rithm, they excited several modes simultaneously
frequencies close to a fixed frequency m, thereby decreasof the excited modes.
ing slightly the monochromaticity
Their algorithm then automatically performed the frequency average ( . . ) in Eq. (6.6) for S(q, co). Using
these mode-mixed displacement patterns j v; I, which arenormalized by g; (v; ) =1, we see that S(q, co) is givenbythis expression with Eq. (6.8b), one has

Comparing2—
Df +x =y —a'(6. 19)(6. 16a)anda'= Df (d —1) .

(6. 16b)dEquations (6. 12) and (6. 16a) for y yield(6. 17)

To summarize, the scaling argument based on the
SLSP predicts, by introducing the averaged strain exponent o. , the dynamical structure factor behaving asqf,(2' —4)(,d /D ) —1

qA(co)((1, (6. 18)

fto" ', qA(co)))1 .

B. Numerical simulations of S(q, co)Computer experiments are crucial for gaining insight
into the properties of the dynamical structure factor
S (q, co), as well as for calculating the DOS or the dispersion relation for fractal structures. In this subsection,
numerical results for the dynamical structure factor of vibrating percolating networks are presented and compared with the scaling arguments of Alexander, Courtens, and Vacher (1993).
Montagna et al. (1990), Pilla et al. (1992), and Mazzacurati et al. (1992) have calculated the dynamical structhe
ture factor S (q, co) by numericallydiagonalizing
dynamical matrix. They have obtained results for sitepercolating (SP) networks formed on 65X65 square lattices and 29X29X29 cubic lattices. Stoll, Kolb, andCourtens (1992) have calculated S (q, co) for bondpercolating (BP) networks by a direct diagonalizationd= 3
andfor d= 2 (68 X 68) lattices
technique(21 X 21 X 21) lattices.
Nakayama and Yakubo (1992a, 1992b) have performed
numerical simulations for S(q, co) for very large-scale SP
networks (SOOX 500). They considered d=2 SP networks
at the percolation threshold (p, =0.593) with a periodic
Rev. Mod. Phys. , Vol. 66, No. 2, April 1994

where the angular brackets denote the sample average.
Scalar displacements
have been employed,
whenceq. v; =qv; . The Fourier transform of the correlation
function ( v;"v ) has been calculated by assuming acorrelation
functionsymmetric
sphericallyG "(r, —This leads torj ) = ( v, v,).

x =2(o. —1) .

4
q cofactors

S(q, rv) for percolation))1, the magnitude of ((U~&) ) is as-&(U~)'). =(qa)'&(tt )')The dynamical

condition.

boundaryThus one has the resultS(q, co) ~2)(co)[A(co)]properties of fractal networks

g (v;)exp[ —iq (R, —R~)] ~v

g RG (R)JO(qR),

where R = ~r, —rj and Jo(x) is the 0th Bessel function.
The ensemble average has been taken over five percolating networks formed on 500XSQO square lattices. The
maximum site number of our systems is N = 110 793; theminimum, N =93 382.
The results for S(q, co) are shown in Figs. 21 and 22.
Figure 21 shows the calculated results of the q depenof S(q, co) for 50 different
frequencies
dence(0.005 ~ cv ~ 0.5). The abscissa indicates the reduced
~

10

D
O

a$
LL
Q)

P

Ty(()„
-1

il

10

-2

O

10

E
10
C3

10
10

10

10

Wave number
FICx. 21. Wave-number(q) dependence of S (q, co) for 50
diferent frequencies (0.005 ~ co ~ 0.5). The abscissa indicatesthe reduced wave number q/qo. Solid circles are plotted byaveraging data over a narrow wave-number range close to the
reduced wave number q/qo. The ensemble average has beentaken over five percolating networks formed on 500 X 500
square lattices. Error bars indicate the statistical errors of thedata. After Nakayama and Yakubo (1992b).

Nakayama,

Yakubo, and Orbach:Dynamical

properties of fractal networks

415

by Alexander, Courtens, and Vacher (1993).
Stoll, Kolb, and Courtens (1992) have10'O

computed

S(q, co) for BP networks. They treated 68 X 68 square lattices and 21X21X21 cubic lattices and employed the

C3

05
LL

10

standard

il

C3

6

L

II

4h

il

Q)

.o 10

~i

E
CQ

Il

Ci
10

10

10

10

Frequency

FIG. 22. Frequency (co) dependence of S(q, co) for 125 different
wave numbers (2m. /250~ q + m). The abscissa indicates the reduced frequency co/coo. Solid circles are plotted by averaging
data over a narrow frequency range close to the reduced frequency m/coo. The ensemble average has been taken over five
percolating networks formed on 500X500 square lattices. Error bars indicate the statistical errors of the data. After Nakayama and Yakubo (1992b).

wave number q/qo(co), where qo(co) is the wave number
at which S(q, co) has the maximum value„(co) foreach fixed frequency. The values of S(q, co) are rescaled„(co). Solid circles are plotted by averaging overbydata within a narrow range close to the reduced wave
number q/qo (one solid circle is obtained by averaging
about 100 data points). The results exhibit a S(q, co) for
different co which can be scaled by a single characteristic
wave number qo. In particular, the wave-number dependence obeys the power law: S(q, co) ~q" — below qo,+— aboveand S(q, co) ~q
qo. Figure 22 shows the results of the co dependence of S(q, co) for 125 different
wave numbers (2ir/250 ~ q ~ ~). The abscissa represents
the reduced frequency co/coo(q), where coo(q) is the frequency at which S(q, co) has the maximum value„(q)
for each fixed wave number. The values of S(q, co) arealso rescaled by„(q). Solid circles indicate the average values over the data within a narrow frequency range
close to the reduced frequency cu/coo. Their result
demonstrates universal behavior scaled by the single frequency coo. This result also shows that the asymptotic
behavior of S(q, co) can be expressed as S(q, co) ~co'i —0''
inthe
co && coo,andfrequency
regime
as

S,

S,

'

'

'

S,

S,

'

— for co&&coo.
Nakayama and Yakubo (1992b) obtained the values ofthe exponents a, a', and y from four asymptotic forms (qdependence for both q ((qo and q))qo, and the cu
dependence for both co«co, and co»co, ) «S(q, ~).
They found a =3.2+0. 1, a'=2. 4+0. 1, and y =0. 8+0. 1,
which explain consistently four asymptotic relations of
Eq. (6. 18).
the simulated
Insertingresult y = 0. 8+0. 1 [ora =3.2+0. I] into Eq. (6. 12) tor into Eq. (6. 13)], one findsthat o. takes the value of 1.1. This value of o. is larger
than unity and in agreement with the prediction (o. & 1)

S(q, co) ~coRev. Mod. Phys. , Vol. 66, No. 2, April 1994

diagonalization

technique.

As mentioned

inSec. II, the scale range for fractal geometry of a BP network is much wider than that of a SP network. Therefore BP networks are more suitable for determining thenature of fracton eigenfunctions belonging to high eigenfrequencies. Their results were obtained by averaging 40
and 20 realizations of d=2 and d=3 BP networks, respectively. Their values for the exponent a defined in Eq.
(6.8) are a=3.32 for d=2, and a=3.65 for d=3. The
value a=3.32 (d=2) is in accord with the result of
Nakayama and Yakubo (1992b). They obtained the exponent cr defined by Eq. (6. 11): o =1.05 and 1.11 for
d=2 and d=3, respectively, close to the value obtainedby Nakayaina and Yakubo (1992b).
In this subsection, we have described experiments,
scaling arguments, and numerical simulations for the
dynamical structure factor S(q, co). The scaling argument postulates that, for the strongly disordered fractals,
one is always in the Ioffe-Regel strong scattering limit, sothat three distinct length scales (a wavelength, a scattering length, and a localization length) collapse to one(SLSP). For percolating networks, this length scale—d/Dshould have the frequency dependence A(co) ~co
Namely, all waves with the wavelength X(g satisfy the
Ioffe-Regel condition A,(Ioffe and Regel, 1960), indicating that fractons are strongly localized with the localization length A(co) (see also Aharony et al. , 1987a). For
weakly localized phonons, the characteristic lengths have
difFerent frequency dependencies(John, Sompolinsky,
and Stephen, 1983). The numerical simulations by Stoll,
Kolb, and Courtens (1992) and Nakayama and Yakubo(1992a, 1992b) confirm this postulate and show that theasymptotic behaviors of S(q, co) can be characterized by
the exponent o. introduced by Alexander, Courtens, andVacher (1993). In summary: (i) The SLSP holds for fractons in percolating networks; (ii) the asymptotic form of
the dynamical structure factor S(q, co) follows a powerlaw in both q and co; and (iii) the exponents cr characterizing the average strain take the values 1.05 and 1.11 for
d=2 and d=3, respectively.

-l,

C. Inelastic light scattering for fractal materials

1. Raman-scatteringexperiments

Boukenter et al. (1986, 1987) made the first attempt tomeasure the Raman-scattering
intensity for silica aerogels. They analyzed the results, assuming the ensembleaveraged form for the fracton wave functions, Eq. (5. 18),
introduced by Alexander, Entin-Wohlman, and Orbach(1985a, 1985b, 1985c, 1986a). Keys and Ohtsuki (1987)
questioned the validity of the analysis, but also used theaveraged wave function. Boukenter et al. (1986, 1987)

416

Nakayama,

Yakubo, and Qrbach:Dynamical

used two kinds of silica aerogel samples in their Ramanscattering experiments. Sample A was prepared by acidand
condensationof siliconcatalyzedhydrolysis
tetraethoxide in ethanol, and sample B by base-catalyzedhydrolysis and condensation of silicon methoxide in alcohol. The condensation leads to gels dried by a hypercritical procedure. They extracted values for the fracton
dimension from their analysis of d= 1.21 for type A and
d=1.31 for type B. Since these pioneering studies, anumber of studies on the dynamical properties of fractal
materials have been performed using inelastic lightscattering. The dynamical properties of silica aerogels
have been vigorously investigated by Courtens, Vacher,and their collaborators (Courtens, Pelous, Vacher, andWoignier, 1987; Woignier et a/. , 1987; Courtens et ai. ,1987a, 1987b, 1988; Courtens and Vacher, 1987, 1988,
1989, 1992; Tsujimi et al. , 1988; Vacher, Woignier,Pelous, and Courtens, 1988; Vacher, Woignier, Phalippou, Pelous, and Courtens, 1988; Vacher and Courtens,
1989a, 1989b; Vacher, Courtens, Pelous, et ai. , 1989;
Vacher, Woignier, Pelous, et a/. , 1989; Vacher, Woignier, Phalippou, et al. , 1989; Xhonneux et ah. , 1989;
Courtens, Lartigue, et al. , 1990; Courtens, Vacher, andPelous, 1989; Vacher, Courtens, Coddens, et al. , 1990;
Vacher, Courtens, and Pelous, 1990). Mariotto et al.
(1988a, 1988b) have reported low-frequency Raman spectra of gel-derived silica glasses annealed at diIterent temperatures. The very low-frequency spectrum is interpreted in terms of localized "extra modes" coexisting withphonons. Besides these experiments on silica aerogels,inelastic light-scattering measurements have been reported for polymers such as PMMA (Malinovskii et al. ,1988); epoxy or diglycidyl ether of bisphenol A (DGEBA;
see Boukenter, Duval, and Rosenberg, 1988); glasses suchas silver borate glasses (Rocca and Fontana, 1989; Fontana et al. , 1990) or lithium borate glasses (Borjesson,1989); and amorphous arsenic (Lottici, 1988).
Information on the DOS can also be obtained bymeans of Raman scattering. Saikan et al. (1990) havemade Raman-scatteringstudies for the DOS of epoxy
resins. Fontana, Rocca, and Fontana (1987) have determined the crossover frequency co, between the phononand fracton regime from the low-frequency inelastic lightscattering for superionic borate glasses of the type
Duval et al. (1987) have per(AgI), (Ag20 n8203),
formed very low-energy Raman scattering on Na-colloids
in NaC1. They analyzed the data from the viewpoint of
fractal structures.
Raman-scattering
Low-frequency
spectra in yttria-stabilized zirconia (YSZ) were measured
by Yugami, Matsuo, and Ishigame (1992). They foundthat the frequency dependence of the Raman intensity
can be separated into two regions, namely, the region
below the characteristic frequency ~„obeying the m law
and the region above not following Debye theory. They
analyzed the data in terms of fracton excitations above
In these cases, the observed intensity involves the
product of DOS and the square of the polarization amplitude produced by the strain of the fracton excitation.

„.

Rev. Mod. Phys. , Vol. 66, No. 2, April 1994

properties of fractal networks

0. 8—
x

0, 2—
Q

0

u/27t

FIG. 23. Brillouin-scattering(GHz )

spectra (points) and fits (lines) as0, 0=39', q/q, =0.53; 0, 0=90',q/q, =0.92; 4, 0=180, q/q, =1.25. The intensities have been
explained

in the text:

normalized to their peak values. The arrow indicates the position of the average crossover co, /2~ for p=201 kg/m'. After
Courtens et aI. (1988).

This means that the observed power law involves morethan a single exponent.
All these experimental results have been analyzed byfracton wave function ofusing the ensemble-averaged
Eq. (5. 18) (Montagna, Pilla, and Viliani, 1989; Shriv'astava, 1989a). ' As first pointed out by Nakayama, Yakubo,
and Orbach (1989), and as mentioned in the previous section, the use of an ensemble-averaged form for the fracton wave functions to obtain the matrix elements of thestrain tensor is not correct in general. Theoretical arguments avoiding this error were made by Alexander(1989), Mazzacurati et al. (1992), Pilla et al. (1992), andAlexander, Courtens, and Vacher (1993).

2. Analysis of inelastic light-scatteringresults for silica aerogels
We have shown in Sec. V.A. 3 that there exist two
characteristic frequencies co& and coI for networks incorporating vector forces between particles. Under realistic
conditions of the network (g) I, ), bending fractons
characterized by d& are important below ~ & A@I, whereas
stretching fractons are important for co) co& (see Fig. 8).
Courtens et al. (1987a, 1988; see also Courtens, Pelous,theVacher, and Woignier,1987) have measured
Brillouin-scattering spectrum for silica aerogels. Figure
23 presents the results for a silica aerogel with density
C

CC

Benoit, Poussigue, and Assaf (1992) have calculated the Raman intensity of the Sierpinski gasket without using the averaged fracton wave function. The results show that, in the fracton regime, the Raman intensity behaves with a power law.

Nakayama,

Yakubo, and Orbach:Dynamical

in polarized scattering[VV (verticalp =201 kg/m
vertical) polarization], in which the relevant frequency
region is low (to (co& ). They have analyzed the data, assuming the following form for the scattering intensity
Cproperties of fractal networks

4't 7

0.20—

I(q, w ),I(q, co) =

I

Av q

~2

(~2+1 2

U2qZ)2+41 2 22(6.20)

0. 10—

where A is a numerical constant, and v and I are thegroup velocity of phonons and the decay rate of phonons,
respectively.
The dispersion relation is expressed as cu=vok in the~- —k Df ' in the fracton regime,
phonon regime and as
whereas the decay rates are taken as I ceto /co, in thephonon regime (Rayleigh-scattering regime) and I cc co in
the fracton regime (Ioffe-Regel regime). Courtens et al.
(1988) have used the following relations for U(co) andI (co) in order to connect smoothly the phonon and fracton regimes in Eq. (6.20),

U(co)=UO[1+(co/co, )

]"

80000

40000

/db

/2~

(cm

)

FIG. 24. Plot of the values of co, vs q, derived from various
Brillouin-scattering measurements performed on a series of mutually self-similar aerogels. Different symbols correspond todifferent sample densities, as indicated in units of kg/m . The
points for each individual sample were obtained at various
scattering angles. After Courtens and Vacher (1988).

(6.21)

andFig. 25, showing the scattering spectra for a silica aerogelin depolarized 90' scattering [VHwith p=357 kg/m
(6.22)

Substituting Eqs. (6.21) and (6.22) into Eq. (6. 11), andusing the experimental data, they obtained the following
values for the parameters: m=2, co, /2~=9. 926p ''cm/sec. From these values,
Hz, and vo =4. 79p'
they have obtained the acoustic correlation length de6ned
A. The value of g„ isby g„=uo/co, =8. 3X 10 plarger than that of the static correlation length g estimatexperiments (Courtens anded from neutron-scattering
Vacher, 1987). They find $„=5/. From the value of g'„,the fractal— dimension was extracted using the formula
1/(D
d)
gccp
Df =2.46. It should be stressed that thisvalue of Df is very close to the value obtained fromneutron-scatteringexperiments, Df =2.4 (Vacher, Woignier, Pelous, and Courtens, 1988).
Courtens and Vacher (1988) also obtained the disperfrom the data for various densision relation co, cc g, ,ties of silica aerogels, as shown in Fig. 24. Substituting
relation into Eq. (4. 16), and using
this dispersion
Df 2.46, they obtained db = 1.3. Vacher, Courtens,
inelastic neutronCoddens, et al. (1990) performed
scattering experiments for silica aerogels with a density
210 kg/m and Df =2.4. They estimated the crossover
frequency ~& /2m. =10 GHz, above and below which the(vertical-horizontal)
polarization]. In the VH polarization the contribution from the bending fracton is mostimportant, and that of longitudinal phonons is irrelevant.

NEUTRALLY REACTEDOXIDIZED

'

f:

'

=

CO

l—10
CQ

lcj
UJ
CO

Ct

357
t

i

I

10

C

fracton dimensions were estimated to be db=1. 3 andd, =2.2. The value for d& is very close to that obtainedby Courtens et al. (1988) from light scattering.
Tsujimi et al. (1988) have reported the results of depolarized Raman spectroscopy on fractal silica aerogels at
frequencies from 0.3 to 50 cm '. They found in the fracton region a power-law dependence of the Raman susceptibility on frequency. A typical spectrum is shown inRev. Mod. Phys. , Vol. 66, No. 2, April 1994

FREQUENCY (cm ')FIG. 25. Raman susceptibilities

I(co)/n(co) for four samples
by their densities in kg/m . The corresponding
acoustic correlation lengths g„are 750, 480, 300, and 170 A, inorder of increasing density. The straight lines are fits with theindicated slopes, whereas the tin curves are meant as guides tothe eye. The different symbols correspond to the four different
mirror spacings L. After Tsujimi et ah. (1988).
designated418

Nakayama,

Yakubo, and Qrbach:Dynamical

They have shown, by comparing polarized (VV) and
depolarized (VH) scattering intensities, that the effect of
the far wing of the elastic line is negligible for the spectra
beyond 6% of an order away from the central line. To
cover the spectral region of interest with sufficient overlap, they have selected four mirror spacings (1.=0.015,
0.029, 0.075, and 0. 165 cm). The spectra were divided by
the Bose-Einstein factor to obtain the Raman susceptibilities, and both Stokes and anti-Stokes channels wereaveraged over logarithmically spaced increments. They
matched the results obtained with different spacings byusing constant multiplicative factors, leading to the presentation in Fig. 25.
Qn the four curves of Fig. 25, one recognizes a linear
region in the logarithmic plot of the susceptibilities, as
indicated by the straight lines. For the lightest sample,
this behavior extends over at least 1.5 orders of magnitude in co. The large extension to lower frequencies can
02 cm ' derived
be related to the low value of co,from the Brillouin data on this sample (Courtens et al. ,1988). For the heaviest sample, the same data give
3 cm ', and this corresponds qualitatively to theco,onset seen in that region in Fig. 25. The rounding-out of
the curves measured on the three heaviest samples at low
frequencies scales with their respective phonon-fractoncrossover frequencies as determined from the Brillouinexperiment (Courtens et al. , 1988). This establishes theorigin of that feature, and also that the straight-line
behavior corresponds to the fracton regime. The Ramansusceptibility is given by

-0.

-0.

I(q, co) In (caulk~ T) ~ cox =0. 35-0.39. Alexander, Courtens, and Vacher(1993) have given an equation for the exponent x incordipole-induced-dipole(DID)
porating the long-range
withmechanism and introducing an averaged strain exponento. . The exponent x is expressed as

x = 2[( db /D~ )(Dg —d —a ) + 1] .
the observed values x =0.36, d& = 1.3, and=2.
4, we can estimate the exponent o. for strain to beD&properties of fractal networksN1[n (co)+1] 2)(co)where the Raman intensity

I(co) isand p, (t) is .the induced dipole of the ith atom. When the
induced dipole is proportional to the local strain, the Raman intensity I(co) becomes equivalent to the dynamical
structure factor S(q, co) as q~0. The results for C(co) of
Montagna et al. (1990) and Mazzacurati et al. (1992)
suggest that the DID Raman coefficients depend linearlyon co for both d=2 and d= 3 percolation networks at p„
which is not in accord with the simulation results by
Stoll, Kalb, and Courtens (1992). The source of disagreement may lie with the use of SP lattices by Montagna
et al. (1990) and Mazzacurati et al. (1992). As shown by
Stoll et al. (1992) (see Fig. 2), the SP lattice does notreach the scaling regime until roughly ten lattice sites
have been covered. Only a few lattice sites are requiredfor BP. This means that the geometry of a spatial regionof characteristic dimension less than approximately ten
lattice sites for SP would not exhibit fractal structure.
Hence any vibrational excitations localized over spatial
ranges less than this amount would not exhibit the scaling properties expected for fractons. Therefore the lackof scaling noted by Montagna et al. (1990) and Mazzacurati et al. (1992) may be attributable to the small latticesize used by them for their simulations.
Stoll, Kolb, and Courtens (1992) have calculated theRaman coupling coefficient C(co) for the DID scattering
process. The results are shown in Fig. 26. The symbol F
means the full-DID calculation in which the polarizationof an atom is induced by polarizations of all remaining
atoms, while NN means the polarization is affected byonly the nearest-neighbor polarizations. Their results indicate that the coefficients Cz(co) follow the relations
for d=2 and d=3 BP netand ~co
CF(co) ~co 'works, respectively. These results are in accord with the

Using
o.

=1.

Montagna et al. (1990), Pilla et al. (1992), and Mazzacurati et al. (1992) have calculated numerically the polaractivity coefficient C(co) for theized Raman-scattering
DID effective polarizability model for SP on 65 X65 lattices for d=2 and on 29 X 29 X 29 lattices for d = 3 (footnote 17). The Raman coefficient C (co) is defined by

5

0.

@

0.01

1

DlD

x 2d-BP-F+ 2d- BP-NN
II

Rev. Mod. Phys. , Vol. 66, No. 2, April 1994

3d-BP-NN~ 3d-SP -F~ 3d - SP - NN

~7The net electric field incident on a given atom is composed

scatteredfields. When the incident field, modified by the effect of the index of refraction is much stronger than the scattered field, one
can neglect the latter. This corresponds to Raman scattering by
the direct process or neutron scattering. When the scatteredmechanism becomes
field is strong, the dipole-induced-dipole
relevant.

0.001

o 3d-BP-F

u 0.001of the incident field plus the suIn of all previously

0. 1

5 + a a ~I

0. 1
Reduced frequency

I

I

I

0.000
I

I

I

I

I

0.01

a

FIG. 26. Relative DID coupling coefficients for the six cases indicated. The straight lines were drawn according to a theory ofAlexander, Courtens, and Vacher (1993). After Stoll, Kolb, andCourtens (1992).

Nakayama,scaling predictions

Yakubo, and Orbach: Dynamical properties of fractal networksof Alexander, Courtens, and Vacher

Vll. MAGNONS AND FRACTONS INPERCOLATING MAGNETS

Diluted Heisenberg magnets represent realizations offractal structure. We discuss the nature of spin-wave andmagnetic fracton excitations in this section and use isotropic percolating Heisenberg magnets. %'e focus, inparticular, on the dynamical property of percolating antiferromagnets, because they can be readily prepared andstudied.

The Hamiltonian for a diluted Heisenberg magnet on apercolating network is given by

g J,"S, .SJ .

(7. 1)

Here the symbol S; denotes the spin vector at the site i,
and J; the exchange coupling between nearest-neighbor
spins, respectively. The coupling constant J; is taken as
ifboth sitesi and are occupied, and J; =OotherJ,wise.
We first discuss the case & 0 in this subsection, namely, the ferromagnetic system. The linearized equations of
motion for spin deviations S;+ from perfect ferromagnetic order are expressed in units of 2S/4= 1 (S is the magnitude of single spin) by

j

"=J,

J

(7.2)

:

S;+:—
S; +iS~. The same equation(—S, —iSf ).

'holds for S,

For p &p„ there is no long-range magnetic order ofspin configurations, because no infinitely connected cluster exists for p. Shender (1976b) has shown that theCurie temperature T, is proportional to (p —p, )" . At@=1, one has conventional spin waves with dispersion relation

(p,

—Jk

It has been shown, using a variational approach (Murray,1966), that spin waves persist above the percolationthreshold. For the case of p )p, and A, ))g, where A, isthe spin-wave wavelength, Edwards and Jones (1971) andthat weakly
Tahir-Kheli (1972a) have demonstrated
damped spin waves exist on diluted percolating ferromagnets, for which the dispersion relation is given by

co=D (p)kwhereo (p) =

2kBgo&„(p)D (p),(7.3)

D(p) is an effective spin-stiffness coefficient. The
dependence of D (p) for ferromagnetic spin

concentrationRev. Mod. Phys. , Vol. 66, No. 2, April 1994

(7.4)

where go is the conductance of a single bond. From this,[see also Eq. (3.4)j

D(p) becomes'

D(p)~

o (p)

&„(p)

~(p —
p, )" ~.

(7.5)

ro=k fF(kg),(7.6)

where zf is a dynamic exponent (Halperin and Hohenberg, 1967, 1969; Kumar, 1984; Christou and StinchThe scaling function E(x) satisfies
combe, 1986a).
2—
zbecause dispersion relation (7.3)
for xE(x) 0-x
must hold in the hydrodynamic limit; and E ( x ) = const
for x
1, because the dispersion relation cannot depend
magnetic
on g in this limit. Thus, for long-wavelength
2
2
fk
From
has
.
ro~g
excitations (kg(&1), oneEq.
(7.3), D (p) ~ g'~ "' ', and one finds

))1,

«

'

P)/v .
zf =2+(p —(7.7)

gut zf =2Df /df (Alexander and Orbach, 1982; Rammal
and Toulouse, 1983), where df is the fracton dimensionof ferromagnetic fractons, so that
2vDf

where

co

limit has been determined byrelated the stifffness
coefficient D(p) with the conductivity o. (p) of the network, using the corresponding relationship between theequation of spin motion (7.2) and Kirchhoff's equation of
the resistor network. He found(1973b). He has

To make the connection with fracton excitations, we
introduce dynamic scaling for the dispersion relation

A. Fractons on percolating ferromagnets

H=

waves in the hydrodynamic

Kirkpatrick(1993).

419—p+2v '
I

(7.8)

This equation for df is the same as that for the fracton
dimension d of the vibrational problem. This is a naturalconsequence of the equivalence between the equation of
motion (7.2) for spin waves and Eq. (5. 1) for vibrations
with scalar interactions. The only difference is the powerof the eigenfrequency co in the corresponding secularequations. The above scaling argument suggests that ferromagnetic fractons belong to the same universality classas that for vibrational fractons with scalar interactions.
Magnetic excitations on percolating ferromagnets werediscussed in some detail by Shender (1976a, 1976b) prior
to the formulation of the scaling theory of vibrational

It might be expected from Eq. (7.5) that the ordered groundspin
state for p &p, would be stable against long-wavelength
fluctuations. However, the derivation of Eq. (7.5) is based on
the linearized equations of motion valid when the magnetization is near full polarization, that is, (S; ) is large and one is at
low temperatures.

420

Nakayama,

Yakubo, and Orbach:fractons (Alexander and Orbach, 1982; Rammal
Toulouse, 1983).
The DOS of ferromagnetic fractons is given by
df /2 —12)f(co) ~ coDynamical

and

(7.9)

and the dispersion relation becomesA(co) ~ co—d /2D(7. 10)where A has the meaning of a localization length, a wavelength, and a scattering length, as in the case of vibrational fractons. From Eq. (7. 10), one sees that the crossover frequency co, from magnons to fractons is given by(7. 11)
which is different from the case of vibrations [see Eq.
(4. 17)] by a factor of 2 arising from the difference in orderof the time derivative in the equation of motion.
Numerical calculations for the ferromagnetic systems
have been performed by Lewis and Stinchcombe (1984),
Evangelou
and(1986a), Evangelou,
Papanicolaou,
Economou (1991), Argyrakis, Evangelou, and Magoutis(1992). Lewis and Stinchcombe (1984) calculated the
DOS of magnetic excitations for d= 2 diluted Heisenberg
ferromagnets.
They treated percolating networks at p,
on 64X64 square lattices with periodic boundary conditions. The computation was carried out for 15 di6'erent
random clusters. They found that the DOS is propor/2 —1
tional to co df
with df = 1.34+0.06. Evangelou(1986a) also calculated the DOS of spin waves in d=2
site-percolating
formed on
Heisenberg
ferromagnets
square lattices by the Gaussian elimination technique(Evangelou, 1986b). The results obtained for systems—10 sites follow the power law
with
~
co
indicating2)f (co)df = 1.36+0.04.
Evangelou also obtained the dispersion relation from theconcentration dependence of the crossover frequency ~,from magnons to fractons, checking the relation Eq.
(7. 11). The DOS of d= 3 ferromagnetic fractons has been
calculated by Evangelou, Papanicolaou, and Economou
method (Jaynes,(1991) using the maximum-entropy
1957a, 1957b; Mead and Papanicolaou, 1984). They estimateda value for the fracton dimensionofdf = 1.52+0.05. Argyrakis, Evangelou, and Magoutis(1992) have calculated the DOS of spin waves in the SP
network using the Lanczos method.
Their results
demonstrate that the DOS is proportional to the powerdf /2 —1
law cowith df =1.32 and df =1.30 for d=2 andd=3 networks, respectively, and that it is smooth at themagnon-fracton crossover.
Effective-medium approximations (EMA; see Yu, 1984and Wang and Gong, 1989) suggest that df =1 for percolating ferromagnets in any Euclidean dimensions d.
The DOS of percolating ferromagnets above p, have been
calculated, and results show a sharp crossover from themagnon spectrum [2)(co) ~co"~ '] for co&&co, to thefracton spectrum [2)(co) o-co ' ] for co))co„. The cross-—,Rev. Mod. Phys. , Vol. 66, No. 2, April 1994

properties of fractal networks

over frequency co, depends on the concentration p
through co, ~(p —p, ) . Provided that the AlexanderOrbach conjecture, df =4/3, holds, the DOS of ferromagnetic fractons should obey 2)(co) ~co '~ and thecrossover frequency co, should be proportionaltofor d=3. The difference between the ex(p —p, )ponents 1/2 and 1/3 is due to the fact that the spatial
correlations of percolating networks are neglected in the
EMA. The DOS at the crossover frequency obtained by
the EMA exhibits very sharp structure. There is no signof structure in the crossover region from numerical simulation studies (Evangelou, 1986a).
The values of zf and df have also been calculated
analytically by Stinchcombe and Harris (1983) and by
Pimentel and Stinchcombe (1989). Stinchcombe andHarris (1983) treated ferromagnetic spin-wave dynamics
near the percolation threshold by the renormalizationgroup technique and the continuum approach, valid forsmall wave vectors and long correlation lengths g. Bothapproaches yielded the same dynamic exponent zf =2.
This implies df =Bf for ferromagnetic fractons. Pimentel and Stinchcombe (1989) studied spin-wave fracton dynamics using Nagatani's model (Nagatani, 1985) as ad=2 deterministic fractal model for the BP network.
They obtained the values of zf =ln22/ln3=2. 81 anddf=2ln8/ln22=1. 34. These results are in good agreement with the numerical results and the AlexanderOrbach conjecture. The dynamical structure factor of
damped magnons (kg1) in percolating ferromagnets
has been obtained through a diagrammatic perturbation
technique (Christou and Stinchcombe, 1986a, 1986b).
The DOS of ferromagnetic fractons have been obtained
experimentallyof the temperatureby measurementsof the magnetization
dependence
M ( T) of diluted
Heisenberg ferromagnets (Salamon and Yeshurun, 1987;
Yeshurun and Salamon, 1987). Because each spin excitation reduces the magnetization of the ferromagnet by one
Bohr magneton, the well-known Bloch's law for homoBT3~, follow—s fromgeneous systems, M(T)/M(0)=1
a Bose-Einstein integration of the spin-wave DOS perunit volume (Xl-co' ) (Bloch, 1932; Keffer, 1966). Magnetic fractons in percolating Heisenberg ferromagnets
cause a deviation in the temperature dependence of themagnetization from the Bloch T
law because of theiranomalous
DOS (Stinchcombe and Pimentel, 1988).
Salamon and Yeshurun (1987; Yeshurun and Salamon,
1987) have measured the temperature dependence of theof amorphous (Co~Ni, ~)75P, 686A13 almagnetization
loys with 0.34~p ~0.5. They found departures from
Bloch's T law and obtained the DOS of magnetic fractons. Their results agree well with the numerically obtained DOS.

«

B. Fractons on percolating antiferromagnets
1. Scaling arguments
In the case of diluted Heisenberg antiferromagnets, theHamiltonian(7. 1) is taken, but J')0. The linearizedNakayama,

Yakubo, and Orbach:Dynamical

equations of spin motion are given by

i

BS;+

=o; g JJ(S++S,+),J

and the same equations hold for S; . In Eq. (7.12}, o., istaken to be + 1 for the site i belonging to the up-spin sublattice and —1 to the down-spin sublattice. These equations have quite different symmetry from the equations of
motion for ferromagnetic spin waves, Eq. (7.2), or vibrations with scalar displacements, Eq. (5. 1).
To explore the consequences of this difference, consider the corresponding secular equation coy;=Q;.
where y; is the normal mode belonging to the eigenfre. The matrix
quency co, i.e., S;+(t)= gi Aiy;(A, )eelements Q;~ are given by Q; =o;[J~ —5; gk Jk;]. The
matrix Q is not symmetric (Q;~= —Q, for iAj) and"%0
becausethefactorof.
whereasthe dynamio;,Q,
cal matrix
for lattice vibrations or ferromagnetic spinwaves is symmetric and satisfies the condition QJ D, =0.
These differences are the reason that the equations of
motion for antiferromagnets cannot be mapped onto themaster equation.
Shender (1978) has shown that the Neel temperature
T, of a percolating antiferromagnet is proportional toso that one can expect spin-wave excitations(p —p,
on percolating antiferromagnets for p p„as was foundfor ferromagnets. This has been confirmed by a Green'sfunction approach (Jones and Edwards, 1971) and thecoherent-potential
approximation(Buyers, Pepper, andElliott, 1972; Tahir-Kheli, 1972a, 1972b; Elliott andPepper, 1973; Holcomb, 1974, 1976). These theories predict a linear dispersion relation for low-frequency spinwaves,

g. y,

g.

0

)",

)

co=C(p}k .

(7. 13)Using the phenomenologicalexpression for the hydrodynamic long-wavelength
spin waves, Harris and Kirkpatrick (1977) have shown that the stiffness constantC(p) in Eq. (7.13) is given by

C = y +2 2 /yi .

(7. 14)

Here y is the gyromagnetic ratio; gz, the transverse susceptibility, ' and A is defined as a measure of the energy
needed to create a spatial variation in the staggered magnetization. The quantity 3 is proportional to the conductivity of a related resistor network (Brenig et al. ,1971; Kirkpatrick, 1973a; Harris and Kirkpatrick, 1977).
Thus one can set A ~ (p —p, )&. Breed et al. (1970, 1973)
have shown experimentally that the transverse susceptibility y~ diverges as p —+p, . The p dependence of y~ wasfirst elucidated by Harris and Kirkpatrick (1977), whofound nuinerically that pi~(p —p, ) with ~=0.5 for
d=3. The stiffness constant C(p) in Eq. (7. 14) therefore
varies with concentration p as')(p+ r)/2
Rev. Mod. Phys. , Vol. 66, Np. 2, April 1994

(7. 15)properties of fractal networks

421

The dynamical scaling argument for spin waves on percolating antiferromagnets can be extended with the aid of
the hydrodynamic
descriptions, Eqs. (7. 14) and (7. 16).
The dispersion relation is given, as in the ferromagnetic
case [Eq. (7.6)], by

f0=k 'G(kg'),(7. 16)

where z, is a dynamical exponent for antiferromagnets.
Since the linear dispersion relation, Eq. (7. 13), holds inthe hydrodynamic limit (kg'«1), the scaling function
G(x) should satisfy G(x) ~x ' for x && l. At the opposite extreme, G(x) should be constant for xl.
From these behaviors of the scaling function in two
asymptotic regimes, one finds co ~ k ' with»

1+ @+72v
The dynamical" mensiond(7. 17)

exponent z is related to the fracton di-d, as z, =Df /d„and one obtains
2vDf(7. 18)p+7 +2v

It should be noted that this relation becomes the samewith the expression for df [Eq. (7.8)] or d [Eq. (4. 15)] if
one replaces v by —P. Because ~ and P are positive, andso ~ & —p, one has the inequality
dg

(df .

(7. 19)As seen from the above scaling arguments, magnetic
fractons should exist in diluted Heisenberg antiferromagnets as well as for the case of ferromagnets (Shender,
1978; Christou and Stinchcombe, 1986b; Orbach and Yu,
1987; Orbach et al'. , 1988; Polatsek, Entin-&ohlman, andOrbach, 1988; Orbach, 1989c). The DOS of antiferromagnetic fractons is
d —1(7.20)
2)~(to) ~ cv
and its dispersion relation becomesA(to) ~ co—1/z

(7.21)

where A is the characteristic length of fractons.
The relationship between ~ and other known exponents
has been investigated by Harris and Kirkpatrick (1977).
They considered a system in which all the spins are
directed initially along the z axis, dividing it into small
cells v with volume g". The disorder of the system produces the unbalanced spins S(v~ ) =(S ) ~ QN~,
where X is the number of spins in the cell v . Magnetic fields Hox on v are applied, where x is a unit vector
to z. Provided that the spins on theperpendicularboundary of each cell are fixed to be parallel to z, spinsare tilted by the magnetic field and are expressed byg„sS(r) =S,'
with

g2

1—

z+O„x

(7.22)

Nakayama,

Yakubo, and Orbach:Dynamical

the fracton dimension d, is bounded bycfg„=ak ff sinkx
a=1

where x is the Cartesian coordinate of the e direction
and k =2m/g is chosen so that O„vanishes on the boundary of U . The symbol ak is a numerical constant. Thechange of the total energy E(u ) in u arises from theexchange energy of order g akk and the Zeeman energyof the unbalanced spins. As a result, one has

E(u
where

E(u

)

=c, g"akk

c2S—(u )akHo,c, and cz are numerical constants.
) with respect to a&, one has

ak

-S(u

)g

Minimizing

per

site

mz

-S(u )g

This is the result for the constrained spins on the boundary of a cell. We expect larger magnetization for unconstrained spins. Therefore the transverse susceptibilityy~(=mL/Ho) should be
) k

2vDf

(7.26)

P+ (4 —d)v
2p —Yakubo, Terao, and Nakayama (1993) have found that
theAlexander-Orbach
conjectured = 4/3
is
taken
into
accountininequality(7.25), i.e. ,p=[v(3d —4) —P]/2, the upper bound for d, becomes
unity independent of the Euclidean dimension d. Thusif

(7.27)
for all d ~ 2, a most surprising result.

2. Simulated results of the density of states

f

S(u

d, ~

(7.23)

Ho,

magnetization
so that the transverse[ g-" S(u~ )sin8„dr] is given bymproperties of fractal networks/Ho.

The efFective field
J;.S for the ith spin plays therole of Ho in this case, and this field is proportional toDone/((~g ~ ) in Eq. (7. 14). Using S(u ) ~X
obtains
2+Df —2d +p/v
(7.24)Xl—g~g,

and thusr ~ p —P+(2 —d)v .

(7.25)

There are several arguments on inequality ('7.25) suggesting that this relation should be an equality (Harrisand Kirkpatrick, 1977; Ziman, 1979; Kumar and Harris,
1985). Ziman (1979) has derived the same relation byconsidering the propagation of spin waves on a nodeslinks blobs pictur-e of the percolating network.
The
mean-field calculation by Kumar and Harris (1985) also
generates an equality for Eq. (7.25). Ziman (1985) has
calculated numerically the value of the exponent ~ ford= 3 percolating antiferromagnets by employing thefinite-size scaling technique for the transverse susceptibilP]. For a finiteity per occupied site, gi [~(p —p, )size system, this quantity depends on system size L asgi o-L'+~'/ at p =p, . Ziman (1985) calculated the size
dependence of y~ for percolating systems formed on d= 3
lattices of 10X 10X 10 to 80X 80X 80 available sites. The
results suggest y~~L'
to an excorresponding
ponent ~=0. 79+0.1. The lower bound value of ~ ford=3 percolating systems becomes 0.72, using known
values of p, , P, and v. The value r=0. 79+0.1 is larger
than the value of the lower bound.
Using Eq. (7. 18) and the inequality (7.24), we show that',

Rev. Mod. Phys. , Vol. 66, No. 2, April 1994

Several numerical calculations for the universality
class of antiferromagnetic fractons have been reported sofar. Hu and Huber (1986) have carried out numericalstudies of the DOS of spin waves excited on percolating
Heisenberg antiferromagnets
using eigenvalue-counting
techniques (Dean, 1960; Grassl and Huber, 1984). Theyobtained the averaged DOS over 65 configurations of
d=2 percolating systems formed on 50X50 square lattices at p, . Their results indicate 2), (co) oindicating d, =0.94. This value of d, is very close to the upperbound of inequality (7.27).
Large-scale and more accurate calculations have been
performed by Yakubo, Terao, and Nakayama (1993; seealso Nakayama, 1993, 1994). Their results show the clear
existence of antiferromagnetic fractons for d= 2, 3, and 4
percolating networks. They employed the equation-ofmotion method (Alben and Thorpe, 1975; Thorpe andAlben, 1976), introduced in Sec. VII.A, to calculate thefracton DOS. In the case of d=2 systems, BP networks(11 realizations) are formed on 1000X1000 square lattices at p, with periodic boundary conditions for both x
and y directions. The largest BP network has 605544
sites. The DOS and the integrated DOS for d=2 percolating antiferromagnets are shown by solid squares in
Figs. 27 and 28, respectively. It is remarkable that an almost constant DOS for co &&1 is found in Fig. 27 for all
three dimensions.
The least-squares fitting for solidin Fig. 27 leads to 2), 0sosquaresd ' —1d, =0.97+0.03. It should be emphasized that the mlaw holds even in the very low-frequency region, as in thecase of Fig. 3. We see from Fig. 27 that the DOS does—1
not follow the power-law dependence co 'above co-1.
This is because the system is not fractal on a length scale
shorter than the wavelength corresponding to co-1. The
value d, =0.97 agrees well with the upper bound given
by inequality (7.27).
The DOS and the integrated DOS for d= 3 percolating
antiferromagnets
at p, (=0.2488) are shown in Figs. 27
and 28 by solid triangles, respectively. The BP networksof 13 realizations {the largest network has 177 886 spins)are formed on 96X96X96 cubic lattices. The value of d,obtained by least-squares fitting takes d, =0.97+0.03.

co,

co;

Nakayarna,

10

go~~

~

Polatsek, Entin-Wohlman, and Orbach (1989) have calculated the DOS of percolating antiferromagnets
interms of the effective-medium
approximation(EMA).
', for antiferromagThey find a fracton dimension d, = —netic fractons in any Euclidean dimensions d, in disagreement with the results of numerical simulations.
This
discrepancy is not surprising for the same reason as thatfound for ferromagnetic fractons. Pimentel and Stinchcombe (1989) have analytically calculated the values of z,
and d, using Nagatani's model (Nagatani, 1985) for BP
networks.
They obtainedz, = ln22/ln9 = 1.41 andZ, =21n8/ln22=1. 35, exceeding the upper bound expressed in Eq. (7.27).

~

~~

10

d—
3

O~

10Q

J~

d=2

C
m,

L

~~a~~~

423

Yakubo, and Orbach: Dynamical properties of fractal networks~

h ~ ~rr
r

e 10

~

~

0

CO

10
10
l

10

I

C. Dynamical structure factor of antiferromagnets

I

10
10
Frequency

10

FIG. 27. Density of states (DOS) per spin for d=2 (squares),d=3 (triangles), and d=4 (circles) BP antiferromagnets at
p =p, . The results have been obtained by averaging over 11,
13, and 12 realizations of BP networks formed on 1000X1000,
96X96X96, and 28X28X28X28 hypercubic lattices for d=2,
d=3, and d=4, respectively. After Yakubo, Terao, and Nakayama (1993).
Solid circles in Figs. 27 and 28 indicate the DOS and theintegrated DOS for d=4 BP networks at p, (=0.160), respectively. The BP networks of 12 realizations (the largest network has 26060 spins) are formed on 28X28X28
X28 hypercubic lattices. From the DOS data, we obtain
fractons.
d, =0.98+0.09 for d=4 antiferromagnetic
These values of d, do not depend on the Euclidean dimension d, but they agree with the upper bound of Z, expressed through inequality (7.27); that is,Z, =1 for all d ( ~2) .

102
(0

C
6$

1

10

100

C5

CD

O
Cl

10'-

D

0)

@$2
10

CD

0)

10
I

10
10
Frequency

10

FIG. 28. Integrated DOSs per spin for d=2 (squares), d=3 (triangles), and d=4 (circles) BP antiferromagnets at p =p, . After
Yakubo, Terao, and Nakayama (1993}.
Rev. Mod. Phys. , Vol. 66, No. 2, April 1994

1. Theories and numerical simulations
The dynamical structure factor of antiferromagnets
has been analyzed with several theoretical treatments.
Perturbationor
coherent-potentialtechniqueapproximation (CPA) calculations of S(q, co) for diluted
antiferromagnets have been investigated so far by various
authors (Jones and Edwards, 1971; Buyers, Pepper, andElliott, 1972; Tahir-Kheli, 1972b; Elliott and Pepper,
1973; Holcomb, 1974, 1976). These calculations, however, do not suggest critical properties, such as the fracton
DOS or the anomalous dispersion of spin-wave excitations in diluted antiferromagnets.
In order to describe fracton dynamics, an EMA calculation was performed by Yu and Orbach (1984; see also
Orbach and Yu, 1987), Orbach et al. (1988), and Polatsek, Entin-Wohlman, and Orbach (1988, 1989). They
treated the isotropic Heisenberg antiferromagnet formed
on a bond-percolating network and obtained the dispersion relation, the DOS, and the dynamical structure factor S(q, co) for percolating antiferromagnets
slightly
above p, . The EMA leads to the fracton dimensiond, =2/3 and a cubic dispersion relation. The dynamical
structure factor S(q, co) within the EMA is expressed bya quasi-Lorentzian form which is characterized by aneffective stiffness constant C(co) and a linewidth r '(co).
The linewidth for magnon excitations (co «co, ) follows
the Rayleigh law (r ' ~ co +'), whereas the fracton excitation line width obeys the Ioffe-Regel condition forstrong localization (Ioffe and Regel, 1960). Though thequalitative features of S (q, co) appear consistent with theexperimental results (see Fig. 31), the EMA results are
not quantitatively correct because they ignore the spatial
correlation of spin con6gurations.
The intrinsic properties of S(q, co) for magnetic fractons and, similarly, for vibrational fractons, are given by
the single-length-scale postulate (SLSP). The SLSP leads
S (q, co) to be expressed, by analogy with Eq. (6.2), byS(q, co)=q'H [qA(co)],where A(co) is the unique characteristic(7.28)
length of a spin-

Nakayama,

Yakubo, and Orbach: Dynamical properties of fractal networks

wave fracton. Christou and Stinchcombe (1986a, 1986b)
have presented an analytic expression of S(q, co) for
1 satisfying
the dynamicalscaling hypothesis.
qg
technique and a
They employed a Green's-function
dynamical scaling argument for hydrodynamic magnons.
The dynamical structure factor they obtained takes a
Lorentzian form with respect to frequency. It seems natural that the I.orentzian form of S(q, co) keeps its profile
even in the fracton regime. In this case, the dynamical
structure factor S(q, co) for antiferromagnetic fractons is
written in the form (Terao, Yakubo, and Nakayama,
1994; Yakubo, Terao, and Nakayama, 1994)((

S (q, co) =I (q)

r(q)

[co

—co (q)] +I (q)(7.29)

where co~ is the frequency at which S (q, co) takes its rnaximum value for fixed q. The symbols I (q) and I(q)represent the width of the line shape and q-dependent intensity, respectively.
Terao, Yakubo, and Nakayama(1994; see also Yakubo, Terao, and Nakayama, 1994)
have shown, using Eqs. (7.28) and (7.29), that the profile
of S(q, co) can be characterized by one unknown exponent y, . The SLSP requires that both the peak frequency co and the width I have the same wave-numberzzdependence, i.e. , co~(q)=cooq ' and I'(q)=I oq ', wherez, is the dynamical exponent given by Eq. (7. 17). Thus
the right-hand side of Eq. (7.29) is written in the form of
I(q)G [qA(co)]/co, where G is a function of qA(co). The
scaling function H [qA(co)] in Eq. (7.28) is then given by

H [q A(co) ] = (q) q

I'G(qA)(7.30)

Because the right-hand side of Eq. (7.30) should be afunction of the variable qA, the function I(q) should be
proportional to q 'Therefore, from Eq. (7.29), one
obtains S (q, co) in the form

'.

a

ya

Og

S(q, co) =So(co—a)oq z') 2 +I 22z(7.31)

O2q

where So is a numerical constant.
The asymptotic
behavior of frequency dependence of S (q, co) given by Eq.
(7.31) is cofor co))co (q), whereas the wave-number
2z —dependencies
are q ' y and q ya' for q ((q (co)1/z(=[co/coo] ') and for q &)q~(~), respectively. The validity of Eq. (7.31) can be checked by numerical simulations.
One can evaluate the upper and lower bounds of the
new exponent y, as follows. From Eq. (7.28), the equaltime correlation function C (q) defined by the integrationof S(q, co) over the whole range of co is given by'

C(q) ~q '(7.32)

The correlation function C (q) should not be zero at q= 0.
This leads to a lower bound of the exponent y, : y, z, .
The Fourier transform C(r) of Eq. (7.32) becomes pro-)Rev. Mod. Phys. , Vol. 66, No. 2, Apral 1994

z —d
y —
portional to r ' ' . Because the spatial correlation
function C(r) should decrease with increasing distance r
between two spins, one has y, &z, +d, which generates
an upper bound for y, . Consequently, the value of the
exponent y, is bounded byza &ya &za+d(7.33)The explicit value of the exponent y, can be determinedfrom numerical computations of S(q, co) for magnetic
fractons.
Terao, Yakubo, and Nakayama (1994) have performed
numerical simulations on S(q, co) of magnetic fractons on
percolating
Heisenberg
antiferromag nets. They considered BP at 0= 2 networks at the percolation threshold(p, =0.5) formed on 62X62 square lattices with periodic
boundary conditions. The dynamical structure factor
S(q, co) of Heisenberg antiferromagnets is expressed byspin-wave eigenmodes y; (A. ) as (Terao, Yakubo, andNakayama, 1994)

S(q, co) =—g 5(co — )Ri g ecoil'g;(A,),(7.34)

where V is the volume of the system and R; is the positional vector of the site i. The symbol R& is defined by
Ri =f;(A, )/g;(1, ), which is independent of i, where g,. (A, )is related to y, (A, ) as g; 0;g, (A, )y;(A, ')=5ii. The direct
diagonalization technique has been employed to obtain
fracton eigenmodes y;(A, ), and S{q,co) has been calculated from the Fourier transforms of y;(A, ), whose eigenfrequencies are close to m. The dynamical structure factor
S (q, co) as a function of q ( = q~ ) and co has been obtainedby a directional average over vector q and an ensemble
average over 54 realizations of percolating networks.
Calculated results of q dependence of S(q, co) for several
values of co are shown in Fig. 29. The value of the exponent y, in Eq. (7.28) is evaluated by least-squares
fitting for the q dependence of S (q, co) with fixed values ofqA(co). They find y =3.0+0.3. This value satisfies condition (7.33).
Figure 30 is a plot of the calculatedvalue
H(qA(co)) =q 'S(q, co), with the above value of y„as afunction of qA(co). Solid circles represent the averagevalue over data within narrow range of the scaling variable qA(co). The results show that S(q, co) obeys Eq.
(7.28), with the single scaling function H(x) having a
power-law asymptotic form both for x
1 and forx
From Fig. 30, Terao, Yakubo, and Nakayama(1994) obtained the wave-number dependence and thefrequency dependence of S(q, co) for these cases. Forco »co (q),the frequency
of S (q, co)dependence
behaves as S(q, co) ~co
The wave-number depen— for ((q (co) and for
dencies are q
and q
q
With
respectively.
valuestheofq &) q~ (co),z, = 1.83+0.08 and y, = 3.0+0. 3 taken into account,these results are consistent with Eq. (7.31).
~

«

))1.

' ' '.

Nakayama,

Yakubo, and Orbach:Dynamical

S(q, ro)properties of fractal networks

425

10

10

10

10

O

~g

~a» ~~ (o=0.10 q~&» co=0.15

r

G

O

D

O~O~O

O~O

~p ~o~ U~

0.20
ro-0 25

O~ to 0.30

FIG. 29. Numerically obtained S(q, co) of percolating antiferromagnets at p, . The solid lines are only meant as guides to the

I

10

10

10

qA(o))

FIG. 30. Scaling function of the dynamical structure factor of

d=2 percolating

Heisenberg antiferromagnets
at p, as a function of the scaling variable qA(co). Solid circles are plotted byaveraging over a riarrow range of qA. The ensemble averagehas been taken over 54 BP networks formed on 62X62 squarelattices. Error bars indicate the statistical errors of the data.
The asymptotes have the theoretical slopes 2z, and 0 withz, =1.83. After Terao, Yakubo, and Nakayama (1994).

eye. After Terao, Yakubo, and Nakayama (1994).

2. Experiments
In this subsection we describe experimental work leading to the observation of fractons in antiferromagnets.
reveal the
Inelastic neutron-scatteringexperimentscharacteristic features of fracton excitations most easily,
as mentioned in Sec. VI. ' In order to compare experimental results with theoretical predictions of fracton excitations, one should prepare samples with the following
properties: (i) The coupling between spins should be described by the Heisenberg interaction; (ii) the anisotropyof the system should be negligible; (iii) the spin interaction should be short range; (iv) the diluted magnetic ionsshould be distributed uniformly over the entire system sothat the configuration of magnetic ions has a percolation
structure; and (v) randomness should not introduce spin
frustration. Mn„Zn& „F2 is one material that satisfies
most of these conditions [it does not satisfy property (ii)].
MnF2 is a representative d=3 Heisenberg antiferromagnet with the rutile structure. Below the Neel temperature, T& =67.4 K, the Mn spins align along the c axis because of the weak anisotropic Ising interaction between
the Mn moments. The antiferromagnetic
exchange in-

There is an experiment other than inelastic-scattering measurements that observes antiferromagnetic
fracton excitations.
Ito and Yasuoka (1990) have investigated magnetic excitationsin antiferromagnets
Mri Zn&
NMR techniques.
F& using
They observed the temperature dependence of the ' F resonancefrequency v(T), which follows the relation v(T) =v(0)(1 —aT )for extended spin waves. They found that v(T)/v(0) of thissample deviated slightly from the T law, and attributed this deviation to fractons.
Rev. Mod. Phys. , Vol. 66, No. 2, April 1994

teraction between the Mn moments located at the body
center and the corner of the rutile crystal is muchstronger than this anisotropy energy and the coupling between Mn spins on the same sublattice. Therefore therandomness introduced by diluting MnF2 does not giverise to frustration. Diffuse scattering measurements (Birgeneau et al. , 1980; Uemura and Birgeneau, 1986, 1987)
suggest that this material can be regarded as a d=3 sitepercolating spin network on the body-centered-cubic lattice. Because the threshold concentration of the bcc
site-percolation network is 0.245, one can expect spinwave excitations above 25% Mn concentration.
Coombs et al. (1976) performed inelastic neutronon
M 0. 78Zn0. 22F2
scatteringexperiments
and
Mn0. 32Zn0. 68F2. For high-density Mn samples, they observed a well-defined magnon spectrum with the energy
width broadened with increasing wave vector. Suchweakly damped spin-wave excitations supported the validity of the hydrodynamic theory of Harris and Kirkpatrick (1977). In contrast, for the MnQ 3QZnp 6sFp sample,
a very broad spin-wave response has been observed. Thisis a sign of nonpropagating
spin waves in antiferromagnets. Takahashi and Ikeda (1993) have performed inelastic neutron-scattering experiments on d=3 diluted antiferromagnets RbMn Mg,
F3 with x=0.74 and 0.63.
Uemura and Birgeneau (1986, 1987) performed highresolution
inelastic
neutron-scatteringstudieson
Mn„Zn& „F2 with x=0.75 and 0.50. High-quality andquite large ( —10 cm ) single crystals made it possible toperform detailed measurements with an energy resolution
much better than that of previous scattering experiments.
Their results for Mno 75Zno ~5Fz at T=5 K (the Neel temperature of this sample is T& =46.2 K) indicate that themagnon peak is still sharp (well defined) even at high energies. This feature agrees well with the result of426

Nakayama,~~

Dynamical

300—

~ q=Q

400—

+
V

e
0

9
ifi

Jl

0.05
0. 1
0.35
0.2

CO

l—

200—

300—

LI',

C3

(Mno 5Zn0 5)F2

3 400-[q 01]

O
O

300

properties of fractal networks(Mn 0 5Zn0 5 )F &

500—

A

Yakubo, and Orbach:BGl—resolution

2 00—

LLj

Q

l

l

0.5

1.0

ENERGY

0)

K

3.5

(meV)

LLI

100C)
tX:
LU
I

0.0

0.5

1.0
ENERGY g~

1.5
(meV)

FIG. 31. Energy spectra of spin waves in Mno, Zno &F2 measured at T=5 K along the [q01] (in reciprocal-lattice unit)
direction by use of cold neutrons with Ff =3.5 meV. The solidlines are guides to the eye; the dotted line shows the background level. After Uemura and Birgeneau (1987).
Coombs et al. (1976) for Mno 78Zno 22F2 The results obtained for Mno ~zno ~F2 at T=5 K (the Neel temperature
is Tz =21.0 K) are shown in Fig. 31, which displays theenergy dependence of the reduced intensity. Peaks at
higher energies are extremely broad, whereas responses
at the lower energies continue to have sharp peaks. They
identified the sharp and broad peaks with extended magnons and localized fractons, respectively. Line shapes ofneutron intensities at small q are highly asymmetric.
This is because the spectrum for q (1/g may be described as the sum of a sharp component at low energies(magnons) and a broad component extending to high energies (fractons). Such a two-component shape of neuEntintron intensity
was predicted
by Aharony,
Wohlman, and Orbach (1988). At q = q, (crossover wave
vector), the amplitudes of these two components becomecomparable.
Uemura and Birgeneau (1987) observed a double-peakstructure in S(q, tu) near q„as shown in Fig. 32. They
fitted their data from Mn05Zn05F2 with a line shapecomposed of the sum of a sharp C'raussian (magnon spectrum) and a broad Lorentzian (fracton spectrum). Three
distinct energies
the peak energies of the Gaussian coG
and of the Lorentzian coL and the energy width of theLorentzian I L
are plotted in Fig. 33 as a function ofwave vector. The diamond symbols (coG) in Fig. 33 show
the dispersion relation of the magnons. The value of coG
at q=o indicates the anisotropy gap energy. Solid circles
2 (r.l.u. ,open circles (I I ) above q(co& ) andreciprocal-lattice units) show the fracton dispersion rela-

—
—

-0.

Rev. Mod. Phys. , Vol. 66, No. 2, April 1994

FICx. 32. Energy spectra from Mn05Zno, F2 observed at wave
vector q=0. 125 r. l. u. in the [q01] direction at T=5 K. The
solid line represents the fit to the sum of a sharp Gaussian and abroad Lorentzian; the dotted line shows the background level.
A double-peak feature characteristic of the wave vector around15 r. l.u. is demonstrated.
After Uemura and Birgeneauq,-0.

(1987).

tion. Both of them have the same q dependence, indicating that magnetic fractons satisfy the SLSP. The value of
col at q=O exhibits a crossover energy co, . In the case ofMno 5Zno 5F2, the gap energy is about a half of the crossover energy. In order to obtain precise information
about magnetic fractons, it is important to perform exsuch as
periments on more isotropic antiferromagnets
RbMn„Mg, „F3 (Ikeda et a1., 1994).
Ylll. TRANSPORT ON A
YISRATING FRACTAL NETWORKIn this section, we introduce the process of phononassisted fracton hopping through the effects of vibrational anharmonicity
and calculate the characteristic hopo

PL

+

(g

K

2

LLI

R

LLI

~ ~~ ~

58

kW+~+++
0

I

I

I

I

I

0. 1

0.2

0.3

o4

0.5

WAVE VECTOR

q

(r 0)
l

FIG. 33. Best-fit values of the peak energy coL (coG ) and the energy width I I for the Lorentzian (gaussian) part of the energy
spectra of Mn Zn& „F2 at T=5 K. The data at q ~0. 15 r.l.u.
are Atted to the sum of the two parts, while those at q) 0.2 r.l.u.
are fitted to the Lorentzian alone. After Uemura and Birgeneau(1987).

Nakayama,

Yakubo, and Orbach: Dynamical properties of fractal networks

ping distance and the contribution to the thermal conductivity K. We also calculate the change in the velocity
of sound coming from the same microscopic process andcompare it with the results with the two-level-systemmodel of amorphous materials.
A. Anharmonicity

The thermal conductivity at temperature
1

(T)D
a'

(T),

T is given byrandom stroctures. In ordered structures, anharmonicity(through, e.g. , umklapp processes) reduces thermal transport. In random structures, what thermal transport
there is results from anharmonicity.
Thus randomness
causes anharmonicity to "stand on its head": whereas inordered structures, anharmonicity serves to reduce heat
Aow, in disordered structures anharmonicity is the causeof heat Aow. This is but another example of the complexity of random structures and how their properties areconsequences of sometimes quite counterintuitive ideas.

(8. 1)

1. Phonon-assisted fracton hoppingwhere C is the specific heat and D the diffusion constant associated with the mode a'. However, in the absence of diffusion, D is zero and a. vanishes. This is astrong condition, for it implies that whenever the condition for localization in the Anderson sense occurs (Anderson, 1958), thermal transport is forbidden.
Thusthermal conductivity approaches zero where the meanfree path becomes of the order of, or even worse, lessthan the wavelength for vibrational excitations. Unfortunately, the literature abounds with use of the above expression for sc under conditions where the vibrational
states are strongly localized. Transport on structures
where the geometry allows a crossover from phonon to
fracton vibrational excitations faces this problem directly. Fractons are known to be strongly localized (see Sec.
V.B), so that thermal transport can only be accomplishedby the (extended) phonon normal modes.
It is interesting to calculate sc under these conditions.
Clearly, that part of sc associated with the phonons, ~pb,
mill increase with increasing
temperature
for tworeasons. The first is associated with increasing modedensity occupied as T increases; the second is associatedwith the increase in the Bose factor as T increases. This
increase will continue until one exhausts all of the extended phonon states with increasing temperature. The
thermal conductivity from phonon sources will then saturate in the Dulong-Petit regime with regard to the extended phonon states ( k~T))fico b). However, this is
not the case for aerogels, as a continues to increase above
the value where the phonon contribution saturates. The
question presents itself: why? %'e shall show that the introduction of anharmonicity into the phonon-fracton excitation spectrum allows for fracton contributions tothermal transport. Indeed, the form for this increase inthermal conductivity is reminiscent of the universal
features of amorphous structures. Just why is currently amatter of speculation, to which we shall return.
The introduction of anharmonicity
is essential forthermal transport in the fracton regime (Alexander,
Entin-Wohlman, and Orbach, 1986). As we shall see, its
introduction allows for fracton "hopping" in much thesame sense as the "phonon-assisted electronic hopping"of Mott for localized electronic states (Mott, 1967, 1969).
In fact, our treatment wil) follow Mott's arguments closely. That the introduction of vibrational anharmonicity
allows for thermal transport is an interesting feature of~~~Rev. Mod. Phys. , Vol. 66, No. 2, April 1994

The introduction of vibrational anharmonicity resultsin two important vertices exhibited in Fig. 34. The
second vertex, (b), is relevant to phonon-assisted fracton
hopping (Alexander, Entin-Wohlman, and Orbach, 1986;
Jagannathan, Orbach, and Entin-Wohlman,1989). We
introduce the corresponding Hamiltonian,

C ff

gfbop

where the b

g

(A

~

-b b

a, a,I a II

(b

~

b

+H. c. )(8.2)) operators

annihilate (create) phonons
on whether the index a refers tomodes with frequencies less than or greater than the
crossover frequency co, .

or fractons depending

a. Characteristic hopping distance
Because the fractons are strongly localized, the two
fractons in Fig. 34 are, in general, located at different
spatial positions. To determine how far they are separated spatially, we invoke the most probable hopping disDrance of Mott (1969). The region of volume gcontains(co')b, cofractonsof energy in the interval2)f,[co', co . +b, co ]. The differential probability of finding—1
these fractons in a volume element r Dff dr (assuming auniform random distribution of the fractons) is then~

(8.3)
Take the first fracton in Fig. 34(b) (index a") to lie atthe origin. Then, to obtain the most probable hopping

FICx. 34. Schematicsof thephonon+ phonon~fracton;(b)anharmonic

process:(a)phonon+ fracton~fracton.
The wavy lines denote phonon states; the dashed arrows, fracton states.

428

Nakayama,

Yakubo, and Orbach:Dynamical

distance, we integrate Eq. (8.3) up to that distance R (a')
which would give us a second fracton within the volume
with probability 1. Integrating up to the maximum distance g would give us 2)t, (co )hen, i.e. , the total numberDof fractons inside the volume g f. One finds that
1/Df

R (a')=A(co

(8.4)

)

where the energy uncertainty Ace has been taken to beco, (Alexander et al. , 1983). Note that this results in aA(co ), so that in fact the fracton
hopping distance R
distance
relative to its localizationasignificant
hops
length scale. The diffusion constant associated with the
hopping of the a' fracton then becomes
~

~

D

~

)

= R (a) )(8.5)

rt, (co, T)

where rt, (n& ., T) is the lifetime of the fracton of energy
at temperature T associated with its hopping a disco
tanceR . .

properties of fractal networks

with temperature independent of the precise nature of thedensity of states for temperatures above the plateau tem-

perature.
The vertex (b) in Fig. 34 not only determines the fracton hopping rate, but also the phonon lifetime associatedwith fracton hopping. Because the same vertex is involved for both processes, one can express Khpp in termsof the inelastic lifetime mph(soph, T) for a phonon of frequency soph at temperature T. Now there are almost no
adjustableparameters.
Jagannathan,Orbach,andEntin-Wohlman (1989) findad ~I2 Df coD

f2

k~ co,

f(8.7)

This remarkable formula involves only the constantD—
a =5 d 4d /—f, I, an integral of order unity [Eq. (17)of Jagannathan, Orbach, and Entin-Wohlman (1989)];
and the I function with argument Df /d&.

~b. Contribution

to the thermal conductivity

These relationships allow direct calculation of thethermal transport associated with fracton hopping. At
temperatures greater than the crossover energy, inserting
Eq. (8.5) into Eq. (8. 1) yields the contribution to the(Jaganfrom fracton hopping
thermal conductivity
nathan, Orbach, and Entin-Wohlman, 1989):

iihop(T)=

2 617

c ttd

3

~ 8

Co

3

8p "sk ~c

~T .

(8.6)

This equation has very few undetermined constants (e.g. ,6 arises from an integral over frequencies and has a value
close to 1.4). The other terms appearing include coD,
which is the Debye frequency associated with the phonon
velocity of sound v„and p, the mass density. Thus it
represents a quantitative contribution to the thermal conductivity, which, apart from the coupling constant, canbe determined quite accurately. In reverse, knowing &eh, ,one can determine C,z directly.
Another feature of Eq. (8.6) is the absence of any
dependence upon the fracton density of states above thecrossover frequency m, . This is because the dispersion
relation for fractons (Sec. IV.B.2) leads to a rapid spatial
diminution of the fracton size with increasing fracton energy. As a consequence, the fracton overlap associatedwith vertex (b) in Fig. 34 falls off so rapidly with increasing fracton energy, that the principal contribution to ~h parises from fractons in the immediate vicinity of co, .
Hence only the 1owest energy fractons contribute to thethermal conductivity via phonon-assisted
fracton hopping. We shall argue below that this can explain therather universal form found for thermal transport above
the plateau temperature for amorphous materials. Thatis, the thermal conductivity appears to increase linearlyRev. Mod. Phys. , Vol. 66, No. 2, April 1994

2. Temperature dependence of the sound velocity
Even more remarkably, use of the Kramers-Kronig relation allows for the extraction of the velocity of soundchange caused by the (b) vertex above in terms of thefracton hopping contribution to the thermal conductivity. J'agannathan and Orbach (1990) find5U~

= —0. 1Kh~p( T)

g2
m

(8.8)

v,

The term I~'(T)/T is independent of temperature forphonon-assisted
fracton hopping, so that Eq. (8.8) islinear in the temperature but independent of the frequency of the sound wave. This result differs substantiallyfrom that generated by two-level systems (TLS) in amorphous materials. Such models generate (Jackie, 1972;
Tielburger et al. , 1992)

Ck, T
v

1n(coro),(8.9)~here C is a constant proportional

to the product of theparameter P and the square of the coupling
constant between the phonons and the TI.S; Eo is theground-state energy of the tunneling particle; co is thesound wave frequency; and ~0 is a few times the vibrational frequency of the tunneling particle in a single well.
Whereas this quantity has not been measured in the aerogels at temperatures in the vicinity of the phonon-fractoncrossover, it has been extensively studied in amorphousmaterials. We shall discuss the relevance of Eqs. (8.8)
and (8.9) to experiments in Sec. VIII.D.
In conclusion, all three quantities,
tunneling+hop&+ph(~ph~model.
are closely linked within the phonon-fracton
They are all consequences of the vibrational anharmonic

N akayama,vertex (b) of F'hd etermines th
e ermines

Dynamicalal propert'

Y k bo, and OrbacI,

th at knowl ed ge ofhthe quantities that'nd h reefore it cann bee stringentlysy10

sof fractal nnetworks
~

'

\

I

1:

429

~

:K(W/m K)
.

10':-

-

10

104-

B.. Thermal condu
conductivity of thee aerogels. 10

The specificT
1 c heatcata dt ermaal conductivit
e

(K) .

0.1

s

10

y Calemcz u k et al. (1987) and d eurements w

0

~I

oo

~~

op
0

.iver

~

~

~

~ a ~

~~
~

0.275 g/cm3
0.190 g/cm3
0.145 g/cm3

respectivelye" t o the sam

w

~

~ ~

'6 -h eat aneasuremments by Sl eator t l.

P

~ ~

Qoo

lat e d sam leich did not c
on

100

ooo

10'
0. 1

10
K)

nge

FIG. 35. Tem perature d ependence of the t
est

h b

Th

l

" 'heooserma
sa uration
ure such themperature'
excitations
ns off ener
ter thelated
p ratures (an d higher)
onon states
ther a 11y excite
onony
excitations
current byo contribute tothence k B T
e ho non enere Du ong-Petit re im
conductivitfurates at a constanta ue. Howev

in tern p ero the dBTe anharmo'

itribution Kho eenof states
11 d

K—Kphh+KKhop

. Th

d

erdp ohl (1971). After Ber-

(8. 10)&which cann bee written,
writ
for T

) T„

K=A+
inano
s

(8. 11)

et al. (1991
vious not ationn
in a cons'istent mannner for th'e imitin g phonon mean f'eny et al. , 1987). U/8" romeey f"d th't the at'ocalculations
ions basedbon Eq.

'

ional
th at
es

g constant C
experimental oermal conduct
w d ensity L
g/cm (lowi y, HD) in Fiig. 35 The rstrogels in the tparallel

).

t'

'

same f d'ff'
silica b Z ll

th

~

I

~

I

20

perature

T=0. 13 K. This is

.

e see

p

e

!0-

r rom
p onon-dom'esar- ominat ed thermal

d

bo

i

bo t 3 Kryt ea ppo

a make u
el
me
cross

accessibleCo igh tofTfor the thermyin the aero-n- racton m
1

1vlty

Rev.. Mod.. Ph ys. , Vol. 66, No.
R
2 April 1994o. 2,

- ominatedpicts the fo ll owing form

0

~

0

01

0.2

0.3

I

0.4

0.5

T (K)

FIG. 36. ~(T)et0.7 K. The solid lin( T bet
B nasconineI; 1 (1992
toEq (8 .11). Af er Bern

0.7

430

Nakayama,

Yakubo, and Orbach: Dynamical properties of fractal networKs(8.8). Thus the predictions for thermal transport basedon phonon-assisted fracton hopping appear to be quantitatively verified in the aerogels.

SiO
OD

~DD D

— 10

C. Transport properties of glassy
and amorphousmaterials

Qo

I—

The lessons we have learned from the phonon-fracton
concept are attractive because of the simplicity of themodel and the tractability of the equations. We are able
to execute quantitative calculations for quantities as complex as thermal transport associated with the hopping of
localized excitations. In addition, the actual results forthe quantities calculated often bear a striking similarity
to the more or less universal properties of glasses andamorphous materials. We are not saying that such substances are fractal, nor that they exhibit dynamic properties that mimic fractal structures, e.g. , phonon-fractoncrossover. What we are saying is that there may be properties of glassy and amorphous materials that might obey
the same kinetics as those that we have explored for fractal structures. In this subsection, we shall focus on twoof those properties that have been extensively explored:
the thermal conductivity and the temperature and frequency dependence of the velocity of sound. We shallsuggest that the internal structure of lattice vibrations inglasses and amorphous materials is such that a mobility
edge exists, which we term co„above which energy thevibrational excitations are localized. The energy width ofthe crossover region is unknown to us, but undoubtedly it
is connected with the temperature width of the plateau in
the thermal conductivity, exhibited by nearly every glassor amorphous material. At the high-temperature end of
the plateau, the thermal conductivity ~ is known to risewith increasing temperature.
Examples for five amorphous solids are given in Fig. 37. In addition, similar
behavior is exhibited by the epoxy resins, as shown in
Fig. 38. To our eyes at least, the rise of ~ above the plateau value is certainly at least initially linear with increasing temperature, reminiscent of the linear increasein the thermal conductivity manifest in ~h, „of the previous subsection.

1. Thermal conductivity
Many authors have observed (Dreyfus et al. , 1968;
Blanc et al. , 1971; Lasjaunias and Maynard, 1971; Lasjaunias, 1973; Alexander et a/. , 1983; Karpov et aI. ,1983; Karpov and Parshin, 1983, 1985; Akkermans andMaynard, 1985; Buchenau et al. , 1988, 1991; Buchenau,
Cxalperin, et al. , 1992; Sheng and Zhou, 1991; Buchenau,
Galperin, et al. , 1992) that the plateau in the thermal
conductivity, sc, for amorphous materials can be explained if one assumes the existence of a mobility edge
for phonons in the medium. Evidence for localization
can be obtained from the extraction of the phonon mean
free path as a function of phonon frequency from theRev. Mod. Phys. , VoL 66, No. 2, April 1994

Q

Qo

4

Cl

O

C3

QO QOOG

CC

lLI

0

0
300

I

100

0

200

TEMPERATURE(K)

FIG. 37. Thermal conductivity for temperatures above the plateau temperature for five amorphous solids. The data for vitreous silica are plotted according to the scale on the right. After
Cahill and Pohl (1987).

thermal conductivity. We exhibit in Fig. 39 the analysis
of Zeller and Pohl (1971) as an example. The phononmean free path is seen to plunge precipitously for phonon
frequencies just below the plateau frequency (temperature) to a few atomic spacings, suggesting that a IoffeRegel limit (1960) for phonons is reached at this vibra-

EDA2X

E

0.02—

0 0

0

0

)IO

0

Cl

EDA1X

0

/~3

0

0

0 0$
I+

I:
lUl

0

««

I

g

&

g

50

&

I

100

TEMPERATURE (K)FIG. 38. ~(T) for temperatures above the plateau temperaturefor di6'erent samples of epoxy resin. EDA2X contains twice thestoichiometric quantity of hardener EDA (thylene diamine),
while EDA1X contains the stoichiometric amount. The greaterthan amount of hardener, the shorter the value of the crossoverlength scale (corresponding to a greater crossover frequency).
After de Oliveira, Page, and Rosenberg (1989).

Yakubo, and Orbach: Dynamical properties of fractal networks

Nakayama,

10

Si02 A«m

10

10

10

I

1

I

)

j

I

f

J

i

1
I

I

A

j

10'
10
CL

10

UJ
Ct

LL

z

10

0z

10-5

10'

heat conduction channel, namely,by an additional
phonon-assisted hopping of the strongly localized vibrational states with energies larger than cu, . This isdifferent from the "resonance" interpretation of others(Karpov and Parshin, 1983, 1985), where the increase in
sc above the plateau temperature
is caused by a return toextended character for the phonon states. There is clearly a point here where the competing theories can be tested: are there extended phonon states above co, 7 Somepreliminary answers are now available. Experiments by
Cahill et al. (1991) and Love and Anderson (1990) haveshown that there is no contribution to thermal conductivity from phonons at room temperature.
There is an additional complication. Glancing at Fig.
37, one sees that the increase in ~ does not continue
linearly with temperatureindefinitely, that there is a"roll-over" in the vicinity of 100 K. Jagannathan, Orbach, and Entin-Wohlman (1989) have hypothesized that
this is caused by the condition co„h~&„& 1 brought about
Here, cu~h is the phonon that assistsby anharmonicity.
the vibrational
state hopping,andz&„ is theanharmonic-induced
lifetime for the localized vibrational
state that is doing the hopping. This quench of thephonon-assisted
localized vibrational state hopping isquite analogous to the Simons (1964) calculation of the
breakdown of three-phonon anharmonic scattering.
Our approach to the calculation of the increase of ~above the plateau temperature will follow closely the results of Sec. VIII.A. 1, except that we shall substitute thelocalized vibrational excitations above m, for the fractons. The mathematical machinery will be comparable
for the two calculations, leading to some interesting
conclusions.
quantitative
This approach has recently
been criticized by Bernasconi et al. (1992), who argue,"We conclude that the ~ plateau of amorphous SiOz is
most likely not the result of fractal behavior at shortlength scales, because a similar feature is not observed foraerogels in the regime dominated by particle modes.
While there
is a serious criticism it is possible that the0small 20 3 particles may not possess the same structure asconventional vitreous silica. Further, the existence of aplateau in ~ Ineans that the crossover region cannot be
sharp, but must be extended in energy. Neither we nor
anyone else possesses a satisfactory microscopic theory
for dynamics in this energy range for glasses. Rather, wewish to focus on those aspects that are amenable to calculation, and see if the consequences warrant considera.

CL

10"
10

431

I

0.1

t

i

1

1

1

~

I

10

~

&i

!

100 500

TEMPERATURE(K)

FIG. 39. The average phonon mean free path A(T) extracted
from thermal-conductivity
measurements for various glasses using the kinetic formula x= 3 C„u, A(T), where v is the thermal
conductivity, C„ is the phonon specific heat, and U, is the Debye
sound velocity, respectively. The phonon wavelengths are indicated at the top of the figure in the dominant-phonon
approximation for vitreous silica. The X drawn on the SiO& curve
denotes the length scale at which crossover takes place between
extended and localized vibrational states. After Zeller and Pohl
(1971).

tional energy.
As shown by John, Sompolinsky, andStephen (1983), John (1984), and Aharony et al. (1987),
this implies phonon localization. With an assigned phonon localization frequency, say, co„ then, for temperatures above A'co, /k~, conventional heat transport can
only occur via the already excited phonons. We have argued that such behavior leads to a saturation in sc; forglasses and amorphous materials, it is referred to as thep/ateau in the thermal conductivity.
The general argument we have presented for the occurrence of the plateau in the thermal conductivity was
first made by Dreyfus, Fernandes, and Maynard (1968).
Akkermans and Maynard (1985) suggested that Rayleighscattering of phonons could lead to a mobility edge in thephonon spectrum. However, there are strong experimental indications that the mean-free-path dependence may
be inelastic in character for phonon frequencies approaching co, (Dietsche and Kinder, 1979), and therefore
non-Rayleigh-like.
We interpret the increase in ~ at temperatures higher
than that marking the onset of the plateau to be caused

oSee also the detailed analysis of Graebner, Golding, and Allen (1986).

Rev. Mod. Phys. , Vol. 66, No. 2, April 1994

"

tion.
To be specific, we use Eq. (8.7) for vitreous silica, taking r~h(cosh, T) from Fig. 39 at the value of cosh where the
Ioffe-Regal condition is satisfied. We have marked thispoint by an 0 X on Fig. 39. Reading from the axes, wefind /=20 A. The average sound velocity u, in vitreous
silica in 4.4X10 cm/s. The crossover frequency is thenobtained from co, =2m. u, /g, so that co, =1.4X10' s
The dominantphonon approximation
sets co&, ( T)
=3.83k~ T/A, giving a crossover temperature T, =28 K.
To choose the phonon whose frequency and lifetime weNakayama,

432

Yakubo, and Orbach:Dynamical

shall insert into Eq. (8.7), we select an equivalent temperature (dominant phonon approximation) of 10 K. One
extracts from Fig. 39 co &=5.3X10' s ', and a mean
free path of A(10 K) of approximately 200 A. We use
conventional exponents for the fractal parameters whichappear in Eq. (8.7), but they do not affect the numericalresults very much.
Inserting these values into Eq. (8.7), we find at 50 K for
vitreous silica«h, „(T = 50 K) = 5 X 10 ergs/cm

Ks .

This is clearly an upper bound for &h p because of our assumption that 1/~ h as extracted from Fig. 39 is entirely
inelastic. Nevertheless, comparison with experiment isremarkable, remembering that no adjustable parameters
have been introduced. From Fig. 37, we find«, „(T=50 K)=2X10" ergs/cmKs.

The quantitative agreement between the calculation of«h, ~( T) and the increase of « for temperatures greaterthan the plateau temperature for vitreous silica is suggestive. When combined with increasing evidence for vibrational localization in glasses at low energies (Laird andSchober, 1991), it may be that vibrational localization,
which leads to the plateau, and phonon-assisted hoppingof localized states above the plateau are the fundamental
mechanisms that generate the near-universal behavior of«.( T) found for glasses and amorphous solids.
A test of the correctness of the hopping model can befound by inverting Eq. (8.6), using the experimental valueof «(T) above the plateau for vitreous silica, to find theanharmonic coupling coefficient C,s. We obtain (Jagannathan, Orbach, and Entin-Wohlman, 1989)

C,~ = 10'" ergs/cm(8. 12)of the pressure derivative of theUsing measurements
elastic stifFness moduli (Andreatch and McSkimin, 1976),
we find C,z to be about an order of magnitude larger
than the third-order elastic constant found at long lengthscales. A recent experimental realization of a Cantor-likestructure (Alippi et al. , 1992) finds an enhancement of
the nonlinear vibrational interaction of about the same
magnitude. This is discussed in Sec. VIII.E.

The velocity of sound in glasses and amorphous materials exhibits a temperature dependence that is very
difTerent from crystalline materials. The change in thesound velocity is nonmonotonic at low temperatures, involving resonance scattering of the sound waves ofF' theTLS (Hunklinger and Arnold, 1976). In the vicinity of 1
K and above, relaxational scattering leads to a decreasein the sound velocity with increasing temperature as thelogarithm of the temperature (Dreyfus, Fernandes, andMaynard, 1968; Blanc et QI. , 1971; Lasjaunias and Maynard, 1971; Lasjaunias, 1973). With increasing temperature, the sound velocity exhibits a stronger temperaturedependence. It is in this regime, above 10 K or so, that
we suggest that localized vibration hopping accounts forthe temperature dependence of the velocity of sound.
The temperature scale for this regime is set by the experimental observation of the plateau in the thermal conductivity.
The experimental work of Bellessa (1978) demonstrates
that the frequency and temperature dependence of thesound velocity in amorphous materials can be broken
into two parts,
5v,

As derived in Sec. VIII.A. 2, the vertex (b) of Fig. 34
also leads to a change in the velocity of sound with inWe found from phonon-assisted
creasing temperature.
fracton hopping that 5v, /v, varied linearly with T and
was independent of sound frequency [see Eq. (8.8)]. Making the argument that an analogous process involving localized vibrational states takes place in glasses and amorphous materials, is such a dependence found? And if so,
what would be its predicted magnitude in vitreous silica
meausing C,& as extracted from thermal-conductivity
surements, Eq. (8. 12)'?
Rev. Mod. Phys. , Voi. 66, No. 2, Aprii f 994= AT+8 inc@ .

(8. 13)

The two constants A and 8 are independent of temperature and frequency, respectively. This behavior is inconsistent with the TLS result quoted in Eq. (8.9). Examplesof the first term are exhibited in Fig. 40, and that of thesecond in Fig. 41 for vitreous soda-silica. More recent
measurements of Duquesne and Bellessa (1986) on amorphous Se, Ge, and Se-Ge compounds show identicalbehavior, as exhibited in Fig. 42. These figures suggestthat the temperature dependence of the velocity of soundin this temperature range can be associated with soundwave —assisted localized vibrational hopping, whereas thetemperature-independent
term arises from the TLS(Bellessa, 1978).
The magnitude of the velocity change can be calculated immediately from Eq. (8.8). We find for vitreous silica(», )h, ~ = —(1.0X10

K

')T,

vs

while

2. Sound velocity

properties of fractal networks

experimentally,from

Hunklinger

andArnold(1976), we extract(5U, ),„~= —(0. 3X10

K ')T .

vs

Our calculated value somewhat overestimates the slope.
However, the numerical prefactors that led to Eq. (8. 13)
were only approximate, and so the agreement with experiment can be considered quite reasonable. The more important point is the consistency between all calculated
quantities. We find that the magnitude of the thermal
conductivity and the temperature-dependent
part of thevelocity of sound all are related to the phonon lifetime
which we have extracted from experiment. There are no

Yakubo, and Orbach:Nakayama,

Dynamical

properties of fractal networks

433

0

— 10
C)

X

(3

0

)

C3

0
LLi

-5

C)

C3

O

0(0

(Q
LLj

LLi

-10

I—

LL

O

0
LL

O
CL

0 -&5
LLI

I—
LLi

CC

5

10

I

1

TEMPERATURE

IN

15

-20

1

KELVIN50

FIG. 40. Variation in the velocity of longitudinal sound waves
as a function of temperature for different frequencies in vitreous
soda-silica. The velocity variation is relative to the value at 0.4
K. The longitudinal sound velocity is 5.8X 10 cm/s. After
Bellessa (1978).
C)

—0

8 K

0
LU

Z',

12 K

0
CQ

14 K

LLl

0
0
CC

IN

KELVIN

FIG. 42. Variationin the velocity of ultrasonic waves in a-Se
(164 MHz), a-Se75Ge» (150 MHz), a-Se60Ge4o (450 MHz; shear
waves), and a-Ge (220 MHz; Rayleigh wave) as a function oftemperature. The curves are arbitrarily shifted along the velocity axis. After Duquesne and Bellessa (1986).

K

(3

LL

TEMPERATURE

150

300

f6 K

—-2

18 K

—-30

LLI

I—
IO
I

I

I

I

50

100

1

I

FREQUENCY

IN

200
I

1

MHz

FIG. 41. Variation in the velocity of longitudinal sound as afunction of frequency for different temperatures in vitreous
soda-silica. The velocity variation is relative to the value at 0.4
K. After Bellessa (1978).
Rev. Mod. Phys. , Vol. 66, No. 2, April 1994

undetermined
This is strongly suggestive
parameters.
that our basic picture of vibrational transport at temperatures above the plateau temperature has relevance toglasses and amorphous Inaterials.
Very recent experiments of Tielburger et al. (1992) on
vitreous silica agree with neither the form of Eq. (8. 13)
nor the analysis of Bellessa (1978). They find, in the temperature range between 5 and 15 K, that the coefFicient A
depends upon frequency as the logarithm, in accord withthe results of the TI.S model quoted in Eq. (8.9). It isdifficult to know how to reconcile the results from thetwo groups, except to note that the energy characteristicof the plateau for vitreous silica is 28 K [see the analysis
for ~(T) above]. The temperature range of Tielburger
et al. (1992) is therefore too low to have present
significantly many thermally excited localized vibrational
states. Our model would predict a change in the frequency dependence found by Tielburger et al. (1992) in thevicinity of the energy characteristic of the plateau.
Indeed, the experiments of Bellessa (1978) are all at or
above this characteristic energy, consistent with the assumptions of our model.
Above about 50 K for vitreous silica, Jagannathan andOrbach (1990) estimated that there would be substantial

Nakayama,

Yakubo, and Orbach: Dynamical

deviations from the simple perturbative results that led toEq. (8.8). Finite lifetimes for the localized vibrational
will inhibit the fullstates arising from anharmonicity
strength of the vertices in Fig. 34, leading to a diminution in our calculated value for 5U, . This will allow theincrease in coupling with density inhomogeneities
todominate, leading to a minimum in 5U, /U„as observed
around 60 K, and then to a subsequent linear rise up to
the highest temperatures measured, as observed for vitreous sihca.

D. Magnitude of the anharmonic

coupling constant

An important point of phonon-assisted localized vibrational state hopping is the large value of anharmonicity
required to explain agreement between theory and experiment. The value of C,z extracted from comparison withthe absolute values for ~( T) [and, concomitantly, 5U, ( T) ]observed experimentally are too large by roughly an order of magnitude when compared with the values found
for the anharmonicity at long length scales.
Very recently, Alippi et al. (1992) found experimental
evidence of extremely low thresholds for subharmonicof ultrasonic waves in one-dimensional
generation
artificial piezoelectric plates with Cantor-like structures,andhomogeneous
as compared to the corresponding
periodic plates. A theoretical analysis demonstrated thatthe enhancement of the interaction between the localized
vibrational states (in their case, true fractons) and the extended phonon states was caused by favorable frequency
and spatial matching of the coupled modes in the
Cantor-like structure, "wish no need to invoke anomalous modi6cations of the nonlinear elastic constants.
Whereas the driving voltage for subharmonic generation
for the homogeneous and periodic structures was around
25 V, typical values of the lowest threshold voltages observed for the Cantor-like sample were 3 —5 V.
The frequency and spatial matching arose from the nature of the fracton states. Rather than being completely"

gf k~~gkii%~XX%I

25
30 35 4020x (mm)
FIG. 43. Experimental displacement profiles of the normal
modes (a) co„=385.5 kHz, a fracton mode, and {b) co =~„/2, aphonon mode, of the Cantor-like sample. The anharmonic coupling between these two modes, which is responsible for thesubharmonic generation, is favored by the relatively large spatial overlap between the square of the subharmonic displacement (b) and the displacement of the fundamental mode (a) in
the region where the fracton mode (a) extends. Ater Alippi
et al. (1992).

0

5

30

15

Rev. Mod. Phys. , Vol. 66, No. 2, April 1994

properties of fractal networks

overdamped, they were found to possess two or three oscillations before the envelope fell to zero. This allowed
substantial overlap with the square of the subharmonicmodes. An example is shown in Fig. 43. Our results forthe structure factor, as given in Sec. VI (Fig. 22), exhibit
structure similar to that in Fig. 43 a few wavelike oscillations within a narrow (superlocalized) envelope.
The lessons from fractal structures may carry over toglasses and amorphous materials. Anharmonic couplings
on length scales of the order of the localized vibrationalmodes may be much larger (an order of magnitude) than
measured at large length scales.
those conventionallyThis amplification would be all that is necessary to quantitatively match the strength of phonon-assisted localized
vibrational hopping processes to experimental observations.

—

IX. SUMMARY AND CONCLUSIONSThis review has attempted to present a complete picture of the dynamical properties of fractal networks as ofthis point in time. Its size and scope reAect the great activity in this field since the review by Orbach (1986) and
Vacher, Courtens, and Pelous (1990). It is fair to say thattheory and experiment, especially in relation to the silica
aerogels, agree remarkably. True, but perhaps surprising, scaling based upon a single length scale has yet to
fail. We have reviewed the properties of a percolating
network, defining the nature of excitations on this model
structure and displaying the crossover between extended(phonon, magnon) and strongly localized (fracton) excitations. Scaling theory has been introduced which freed us
from these speci6c examples and enabled an analysis of
excitations on any self-similar (fractal) structure. The advent of modern, large, and fast computers has allowed
numerical simulations of sufficient size to shed light on
and to test the scaling theories through explicit simulations. Two physical systems with fractal geometry over arather sigriificant length scale have yielded extensive excitation spectrum data from light- and neutron-scatteringexperiments: the silica aerogels (vibrational excitations)and site-diluted antiferromagnets (magnetic excitations).
Finally, the transport properties of fractal networks
were examined, addressing the disturbing question of
how a strongly localized fracton could transport heat. A
calculation of the thermal difFusivity was described which
involved phonon-assisted fracton hopping. This is the vibration analog of Mott's variable range phonon-assisted
localized electronic state hopping. The theory that has
been developed couples together the assisting phonon
lifetime, the fracton contribution to the thermal conductivity, and the temperature dependence of the velocity ofsound. Knowledge of any one of these quantities determines the other two with no adjustable parameters. Veryexperiments on the silica
recent thermal-conductivity
aerogels have exhibited results consistent with the theory
and have shown that the results scale as the theory would
require.

Nakayama,

Yakubo, and Qrbach: Dynamical properties of fractal networks

The transport results for fractal networks appear to be
similar to those found universally for (nonfractal) glasses
and amorphous materials at and above the temperature
region of the plateau in the thermal conductivity. This
led to speculation that the plateau was a consequence of acrossover from extended to localized vibrational excitations in these materials. If so, and if the localized character of the vibrational excitations continued well above theplateau energy (temperature), phonon-assisted localized
vibrational state hopping might well be an importantmechanism for thermal transport. Calculations were performed for the thermal conductivity arising from thismechanism for vitreous silica, using phonon lifetimes extracted from lower temperature thermal-conductivity
measurements.
Agreement was found with experimentwith nearly no adjustable parameters, but it was foundthat the anharmonic coupling constant extracted fromthe fit to experiment was about an order of magnitudelarger than that for long (continuum) length scales. This
has now been shown to be the case for an artificially constructed one-dimensional (Cantor) fractal, suggesting thatby analogy random structures may exhibit similar properties (A1ippi et al. , 1992; Craciun et al. , 1992).
These considerations point to continued activity in thisfield for some time to come. Numerical simulations for
d=3 percolating networks can shed further light on the
structure factor and on the reasons why single length
scaling works so well. As interesting is the question of
the effects of anharmonicity.
So far only one-dimensional
Cantor-like structures have been analyzed to show how
phonons and fractons interact. It is very important toknow whether the resonances in frequency and space
which are crucial for the enhancement of anharmonicity
for d=1 are still present for d=3. Preliminary evidence,
through the form of the structure factor, suggests that
this may be the case (the fracton exhibits a few spatial oscillations within its superlocalized envelope).
The recent low-temperaturerneasurernentsof thethermal conductivity and the specific heat for the silica
aerogels suggest further work. Lower measurement temperatures would enable comparative measurements of thephonon-fractoncrossover as exhibited through thesemeasures, enabling systematic information to be extracted. In addition, the relationship between phonon lifetime, thermal transport, and variations in the velocity ofsound can be directly tested in the aerogels. It is important to measure U, at temperatures near and above thecrossover temperature
to determine if the phononassisted fracton hopping relationship predicted between ~and U, is, in fact, observed.
Another area of interest is site-diluted magnets. So far,excitations in 0= 3 antiferromagnets have been measured
through neutron scattering for the highly anisotropic antiferromagnet Mn Zn&
F2. Lower anisotropy would allow a greater energy range for magnon excitations formaterials with p closer to p, . It would also be very interesting to measure the magnetic excitation spectrum of
d=2 site-diluted magnets, for which numerical simulaRev. Mad. Phys. , Vol. 66, No. 2, April 1994

435

tions of S(q, co) have been performed. New results fromsimulations and scaling theory suggest that d, is close tounity for all d greater than 2. This is a strong prediction
for the fracton DOS and dispersion law and cries out forexperimental examination. And, of course, it would be
welcome to have numerical simulations to serve as a testof scaling theories and to match with experiment. For
example, is it possible to obtain two peaks in S(qs„,d, co)
as a function of co, one for magnons, the other for fracton
excitations, as has been observed for Mnp 5Znp gF2(Uemura and Birgeneau, 1986, 1987), but which fail toshow up in effective-medium calculations?
Finally, are the vibrational excitations in glasses andamorphous materials strongly localized above energies ofthe magnitude of the plateau temperatures?
This is a
crucial question to test the applicability of the ideas generated by fractal lattices to these nonfractal but disordered structures.
Surprises will continue to abound. in this field. We
hope that this review will serve as a basis for further exploration of the dynamics of random structures.
ACKNOWLEDGMENTS

We are grateful to Shlomo Alexander and Eric Courtens for many valuable discussions and useful suggestions. We thank S. Feng, S. Havlin, J. K. Kjems, M.
Matsushita, A. Petri, S. Russ, E. F. Shender, Y. Tsujimi,
R. Vacher, and O. Wright for fruitful discussions. The
work was supported
in part by a grant-in-aid
forscientific research in the international joint research program from the Japan Ministry of Education, Science, andCulture (MESC), which has made our collaboration possible. This work was also supported by a grant-in-aid forscientific research in priority areas, computational physics as new frontiers in condensed rnatter research from
the MESC. This work was also supported in part by the
U. S. National Science Foundation through grants DMR
90-23107 and DMR 91-12830; the U. S. OfFice of Naval
Research through contract number Nonr-N00014™88-K0058; and the University of California, Los Angeles andRiverside.

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