IC/81/132

INTERNATIONAL CENTRE FORTHEORETICAL PHYSICS

FRACTAL EFFECTS ON EXCITATIONS IN DILUTED

FERROMAGNETS

if)INTERNATIONAL

ATOMIC ENERGYAGENCY

Deepak. Kumar

UNITED NATIONSEDUCATIONAL,

SCIENTIFICAND CULTURALORGANIZATION 1981 MIRAMARE-TRIESTE

IC/81/132

International Atomic Energy Agency

and

United Nations Educational Scientific and Cultural Organization

INTERNATIONAL CENTRE FOR THEORETICAL PHYSICS

FRACTAL EFFECTS ON EXCITATIONS IN DILUTED

FERROMAGNETS*

Deepat Kiimnr**

International Centre for Theoretical Physics, Trieste, Italy

ABSTRACT

The low energy spin-wave l i k e exc i t a t i ons in d i lu ted ferromagnets

near perco la t ion threshold are s tudied . For t h i s purpose an e x p l i c i t use

• f the f r a c t a l model for the tacMone of the i n f i n i t e percola t ing c lu s t e r

due to Kirkpatr ick i s made. Three physical e f fec t s are i d e n t i f i e d , which

causa the softening of spin-waves a s the percola t ion point i s approached.

The importance of f r a c t a l e f fec t s in the ca lcula t ion of densi ty of s t a t e s

and the low temperature thermodynamics i s pointed out .

MIBAMARE - TRIESTE

Auguat 1981

* To he submitted for publication.

•* Permanent addresa: Department of Pnys-icsr, University of Roorkee, Roorkee,

UP India.

Randomly diluted magnets Kith, short-ranged interactions have an1 2interesting and complex Behaviour near the percolation threshold * . It

is now veil recognized that a number of their dynamic and ttiermostatic

properties depend crucially on the geometry of the Infinitely connected

percolating cluster tlPC), when one is above the percolation threshold. In

randomly diluted systems, IPC has a very disordered and ramified structure ,

and in recent years much attention has been paid to develop simplified models

for it with a view, to understand magnetic critical properties as well as

conductance in Inhomogenous mixtures. A significant step in this direction

has been the so-called 'node-link' model introduced by Shal and Shklovshii

and de CJennes . In this model, vhich has Been exhaustively and critically

discussed in recent articles * ' , the biconnected component of IPC, also

referred to as "backbone is visualised as a series of small clusters set on a

superlattice of nodes connected 6y one-dimensional crooked paths. The typical

distance between the nodes is of the order of percolation correlation length

£ "*» (p - p ) p. The average length, L, of the one-dimensional connecting

parth is naturally greater than £ and is assumed to diverge according to

lp ~ P I vith X, > v • The upper limit on the path connecting nodes Is

clearly a self-avoiding walk, vhich means that L < £ ^ SAW or £ < V ^ „,,,.P 5 T 8 P

While the node-link model has served a very useful purpose in

providing a nice intuitive understanding of hov the reduced connectivity in

the IPC affects the various physical properties, it has many dravtaeks

especially In tvo dimensions and to some extent in three dimensions ' .

Through a detailed study of the backbones of IPC in two dimensions, Kirkpatrick

has made several important observations, vhich are briefly: Ci) As p •* p +,

the one dimensional links do not grow faster than £ . Thus a moreP

appropriate assumption Is that L us £ . Cii) Though many regions of the

backbone do look like one dimensional paths, the Intersections usually occur

much closer together than C . Thus E characterises the longest links inP P ,

the backbone rather than the typical ones, tlii) Finally according to

'node-link' model the fraction of sites, BCp), in the backbone is (p - p ) ,

which Is much smaller than the values determined In computer simulations.

Defining an index Bg such that BCp) • {p - p ) B It is seen that

&> - £ Sf £ $_. This means that the longest links form a negligible part of

the backbone close to p .

- 2 -

1.3

Kirkpatrick * haa taken this to imply that although the 'node-link'

model captures some essential physics at length, scales bigger than £ , it

completely misses the texture present at scales smaller than £, . He

models this texture as a self-similar fractal , in the following vay. Since

the backbone must be homogenous on scales "Bigger than £ , one can regard it

as being made up of statistically identical cells of linear dimension ? .P

Now a finer texture, say some connections, are introduced at a scale of

5 /?. -A similar texture is next introduced at the scale of £ A and the

process is continued by getting finer and finer until the scale equals a

lattice constant. The fractal dimension of the 'Backbone, <L, is identified

by calculating the volume of the material in a cell of side %, which is

<-dB(p) a 5d~BBA>p = 5dB.

This model is very attractive from the geometrical point of view and

is also very successful in giving correct exponents for conductivity and Hall

effect in random resistor networks . In this paper we use these considerations

to study spin-wave excitations in f err oraagnet s and the resulting consequences

for low temperature thermodynamics.

From general grounds, one knovs that at sufficiently long wavelengths,

i.e. for q •= < C 1 , the spinwave dispersion relation Is 1 0' 1 1' 1 2 E = D(p)q2.

The stiffness coefficient Dtp) has been exactly related11 to the conductivity,

of a diluted resistor network through the relation

Dtp) PCp) (1)

where P(p) denotes the fraction of sites in the IPC. Thus Dtp) vanishes

at p Q like (p - p j " , with u = t - 5 , where t is the exponent for the

vanishing of conductivity at p^ and 6 is the usual index associated with

the percolation probability. Within node-link model, Ziman has related u

to the index ?, He argues that since the wave moves along a crooked one-

dimensional path, the wave-vector q along the path, which enters into the

formula for energy is not the same as, q, which Is related to the real distance.

For small q, there should be a linear relationship between the tvo, vhich he

argues to be q « (i&)Q. Substituting this into the formula for excitations

on the chain, ID » 2 SJQ , one obtains

Da tp-pc)2(? - V

-3-

Cs)

If the patha L are taken to he 3elf-avoiding walks, this relation is in

good numerical agreement with the numerical values calculated from relation

V = t - B.

But as argued above, the one dimensional links are much shorter and

surely the texture at length scales shorter than Z affects the spin-wave

stiffness. More over the effect of dead ends is not considered in Eq. (3).

So we feel that the numerical agreement does not imply a complete elucidation

of the problem. More importantly, the fractal nature of the backbone implies

a different density of states for spin-wave excitations and hence a qualitatively

different thermodynamic behaviour.

Our calculation of spin wave energies proceeds by identifying,

somewhat artificially, three effects of the ramification and fractal nature

of the IFC. Assuming with Kirkpatrlcft, that the backbone can be regarded as

being made up of statistically Identical cells of linear size % , we apply

the standard formula for spin-wave stiffness, given below in Eq. (3), to

this cell.

(3)3 I

Since here all J, 's and Sj's are equal and are non-aero only for the

nearest neighbours, this formula simply counts the average number of neigh-

bours in the backbone. Clearly this number, a, lies between 2 and 2d

for a d-dlfflenslonal hypercubic lattice.

The second step of our calculation consists in including the effect of

dead ends. One can obtain this effect from formula (3) only, If one recognises

that In applying the formula to IPC, the numerator should Include only the

sites In the biconneoted component, whereas the denominator includes all the

sites In IPC. ZIman' by considering the excitations In a chain to which

dead ends are attached regularly, showed that the stiffness is reduced by a

factor equal to the number of dead ends attached per site. We have verified

this for more complex situations, in higher dimensions. Physically, we may

argue that there is no phase drop (or voltage drop In case of resister net-

work) along dead ends. So if one adds dead ends to the backbone, one does

not alter E(p), while the number of sites or P(p) increases, thus D{p)

gets reduced by the number of dead ends per site. The third step of the

-1*.

calculation la to relate the Q—vector on the fractal to the actual q., which,

say, is sften by the neutron.

The backbone fractal ' is constructed ty taking a cell of size %

and dividing it into 2 parts. A fraction ll-t) of these cells is

removed Cleaving the boundaries), and the remaining cells, 2df, are further

divided into 2 parts each, and from each bigger cell a fraction (l-f) is

removed randomly as before. The process is continued till the lattice size

ia reached. A typical example Is shown in Fig. 1. The parameter f is

related to dg according to the following equation

Now consider the calculation of the numerator, H, of Eq. (3). The

contribution from the longest links is 2d(£ - 2) + 2d. At the next

heirarchy we have d2 f links of length Jj /2 and these contribute

d.2df[2(£/2 - 2) + 2d]. Thus we have a series

t5)

where n = tn^,. Since 2 " f > 1, the su» becomes

N ~

Note that

l .

2Jf -I

C6)

is simply equal to the number of tonda in one cell

of the backbone. Thus the stiffness of the backbone is

-5-

2JSa2 2d~1f-lDEackhone

t7)

However, as argued above, in order to tsie into account the dead ends or

loose spins, we should divide N fly the total number of spins in the backbone,

which is a £d p£p - p ) . Thus we obtain

1-1.Cp - P CB)

To give some idea of magnitudes here, we note that for d «• 2,

6 = 0.15 and f = 3/1*, and_for d = 3 0B = 0.9, 6 = O.lt and

Thus for d = 3, D

VBCP-P C)°-5. E q. (7) for D ^ ^

> 0.5.

is an

underestimate. We have also obtained an upper bound for the square lattice

shown in Fig. 1.

We begin with the cell in which all the sites are occupied, and

calculate the number of free boundaries created as we remove the subcella

at each stage. At first level, we have 1* squares and by removing a

fraction (l-f) of these we form boundaries of length !*(l-f)E/2. At the

next stage we remove h f(l-f) squares and create new boundaries containing

h fCl-f) f- [U a. + 3a ] spins, where a. is the fraction of cubcells

removed from the interim so that for such cells It new boundaries are created

and a is fraction removed from the periphery, so that only 3 new edges

are created. By actually carrying out this process successively, we

determined empirically that after about fourth stage a, = 1/5 and

a - lt/5. low we can count the total number of spins at the boundaries to

be

3 2f-I§* c-t> y

(9)

The stiffness ^-R^V*. e c a n t e calculated by noting that all tile

interior spins have U neighbours, whereas- the ones at the edge have only 3.

Thus

do)

Thus we find that the tvo estimates in E<js. (9) and (io) do not differ much

for d - 2.

To proceed with the third step, we write the spinwave energy as•"* 2

f = D Q , where Qa is roughly the average phase difference between

neighbouring sites on the IPC along the direction In which the wave is

progressing. To fix Ideas, consider the propagation of a long wave length

spinvave on a self-avoiding walk of L steps, so that the end to end direct

distance is % t LUstvV Let the phase difference between two ends be

8 = q£ . Then the average phar.e difference between successive sites is

6/L for waves q » C"1. This gives c - 2J(9/L)2 = 2JU /L)aq2, which•i -r P 1 P

is what Ziman1" ' obtained. Now one would like to apply a similar argumentto fractal. Consider a single cell of size 5 and fix the phase difference

between two opposite edges to 5 .P

The wave-front on a random fractal

is obviously very complicated and the phase drop across the cell would occur

along a very tortuous path. This path i , is necessarily bigger than 5 •

and in a simple theory should be related to the fractal dimension d_. One. d

such Bimple length which satisfies the above criterion is I B = £ . How

I H t * 6/*a, so that the spin-vave energy e becomes

2JSa3

td-2)2af-l

til)

Cl2)

with

2VdB Cl3)

-7-

In Table 1, ve have listed various exponents for d = 2 and d = 3 ana

compared index JJ calculated according to the aBove formula {.131 vith the

exact relation u » t - S. The agreement ia rather good, For larger values

of q, one may employ the dynamical scaling argument given by Btauffer ,

according to which

xvhere f (x) •+ x as x + 0. Applying this to 2q. Cl2), we find

Z ™ 2d/A , and thus the critical modes at shorter wavelengths have dispersion

(15)

In order to calculate the low temperature properties, one needs to

know the density of states. Hitherto, moat authors * have considered

low temperature properties by using the dispersion relation Cl2) In the

usual way to write the density of modes as

pC<") a,Cd-2)/2

[Dtp)]d/2

Cl6)

This gives rise to contributions to specific heat and magnetisation which

strongly diverge as p •* p eg. for d = 3

T << T (p) ClT)

Here one may argue that since the modes are softening as p •* p , their

occupation probability rises vary strongly with temperature, but we would

like to note that since the excitations are on a fractal, the q-space

available to them is not as large as Implied by Eq. (17).

One can use an illuminating scaling argument due to Stapleton et.al."1

to determine the low energy density of states. Consider a sample of macroscopic

linear size L. Its density of statea a CM) is

16

U)I

-8-

Cl8)

Let the size of the sample he changed hy a factor h: L -> tL, then i tfollova from simple hydrodynaraic arguments that ID. •* u. d l / t i . Now

Thus the scaling relation for density of states is

p. ia) = Ci9)

from which one finds at low a

U^-ai/a (20)

Though One may not rule out a concentration dependent factor in Eo_. (l8) ,

we would like to emphasise that the main effect of dilution on density of

•states occurs through fractal dimensionality of the backbone supporting the

exaltations, rather than through factors contained in Eq. (l7).

Let us discuss a 3-cUmensional isotropic ferronsagnet .

Though the fractal dimensionality indicated is ^ 2 , there is other evidence

'frcsa series 3tudies on T (p) to show that long range order does exist

till very close to j , which is the true multiaritical point. From this we

presume that d <- 2+. Then in the small temperature range 0 < T « T^tp),

the thermodynamics is governed hy spin-wave excitations. In the presence of

a field, H, which may be due to anisotropy and/or application, one has

Va/

C21)

122)

f_?*/

-9-

where

W " Th> i- t

C23)

For small fields Eq.s. (21) and (22) assume the forms

NCr)

13

where X(x) denotes the Riemann Zeta-function. This shows that the use of

Eq. (20) leads to a very different Dehavlour, ooth with regard to temperature

and field.

Though Eqs. (,2k) and (25) represent clear-cut differences Detween

diluted and homogenous systems, from the point of viev of experimental veri-

fication, we note that since T (p) itself is small, the temperature range

of validity of the ahove formulae is rather restrictive. Until now we knov

of dilution studies on only one ferromagnetic compound19 On the other hand,

many antiferronsagnetio compounds have teen investigated near percolation

point , and it would he desirahle to extend such considerations for anti-

ferromagnetlc exc it ations.

To summarise, we have identified certain physical mechanism which

cause the softening of spin-wave stiffness in dilute ferromagnets, as the

percolation concentration is approached. We find that an explicit account

of the fractal nature of the infinite percolating cluster is necessary for

these considerations. The fractal effects alter the nature of the low energy

density of states which reflects significantly on the low temperature

thermodynamic properties.

-10-

Table 1.

d

2

3

S

0

0

.11*

.39

P

1.36

.88

e

0

0

B

.5

• 9

(J

1 .

10

62

d

1

2

E

.6

. 0

u-

0 .

1 .

a-B

96

33

0 .

l .

98

, 1

Ik, D. Stauffer in Amorphaua Magnet I apt I I , edi ted by Ft.A, Levy and

R. Ha3egava (plenum Pre s s , 1977) p .17 .

15. E,F. Snender, J . Fays. C9, L309 U.976"); Eov. Fhys. JEEP, 1*8, 175 Cl978).

16. H.J. Staple ton , J . P . Al len , C.P. Flynn, D.G. Stinson and S.R. Kurtz,

Pliys. ! t e , Le t te r s 1*5, 1I156 (19SO).

IT- One may obtain a speci f ic dependence on D by writ ing u t DQ and

flU) a .UB-2)/2/[DtP)]S/2 = tp - p c r V V B ) / 2 A " 2 ) / 2 , whichi s a weaker dependence on tp - p' ) than in Eq. ( 1 6 ) .

18, E. Brown, J.W. Essam and CM. P l a c e , J . Phys . C8, 321 (1975) .

19 . K.E. Loche e t . a l . , Phys. Rev. B23, 5928 ( l 9 8 l ) .

ACKNOWLEDGMEirTS

The au thor would l i k e t o thank P ro fes so r Abdus Salam, t h e I n t e r n a t i o n a l

Atomic Energy Agency and UNESCO for h o s p i t a l i t y a t t h e I n t e r n a t i o n a l Centre

fo r T h e o r e t i c a l P h y s i c s , T r i e s t e . He a l s o v i s h e s t o thank Prof . U. Kumar for

helpful discussions.

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-11--12-

F I G . 1

KIrkpatrick's self-similar fractal model for the single cell of

the 'backbone of an infinite cluster in two dimensions. The linear size

of the cell i s £ and the backbone i s made By joining s ta t is t ica l lyP

identical cells of this type . f = 3/h.

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IC/81/76 P. BALLOHE, G. GENATORE and M.P. TOSI - Coexistence of vapour-like andINT.REP.* l iquid- l ike phases for the classical plasma model.

IC/81/77 K.C. MATKUR, G.P. GUPTA and U.S. PUNDIR - Reduction of the Glauber amplitudeINT.REP.* for electron impact rotational excitation of quadrupolar molecular ion3.

IC/81/78 N. KUMAR and A.M. JAYANNAVAR - A note on the effective medium theory ofINT.REP.* randomly res is t ive-react ive medium.

»IC/8l/79 'S.H. PANDEY - Some aspects of gravitational waves in an isotropicINT.REP.* background Universe.

IC/81/80 E. GAVA and E. JEMGO - Perturbative evaluation of the thermal Wilsonloop,

IC/81/81 ABDUS SALAM and J . 5TRATHDEE - On the Olive-Montenon conjecture.INT.REP.*

IC/81/B2 M.P. TOSI - Liquid alkali metals and alkali-based alloys as electron-IHT.REP.* ion plasmas.

IC/8I/83 M.A. El-ASHRY - On the theory of parametric excitation of electron plasma.IHT. REP.*

IC/8I/8I4 P. BALLONE, G. SECJATORE and M.P. TOSI - On the'surface properties of aINT.REP.* semi-infinite classical plasma model with permeable "boundary.

IC/81/85 G. CALUCCI* R. JENCO and M.T. VALLON - On the quantum field theory ofcharges and monopoles.

IC/8I/87 V.K. DOBREV, A-.Ch. GAHCHEV and O.I. YORDANOV - Conformal operators fromspinor fields - I : Symmetric tensor case.

IC/81/88 V.K. DOBREV - Stiefel-Skyrme-Higgs models,• their classical s ta t ic solutionsand Yang-Mills-Higgs monopoles.

IC/8I/89 Trieste International Symposium on Core Level Excitations in Atoms,Molecules and Solids, 22-26 June 1901 (Extended abstracts) .

IC/81/90 A. TAWANSI, A.F. BASHA, M.M. MORSI and S.E1-KONSOL - Effects of gamma-INT.REP.* radiation and rare earth additives on the insulation of sodium si l ica te

glasses.

IC/81/91 A. TAWAHSI, A.F, BASHA, M.M. MORSI and 3. El-KONEOL - Investigation ofIHT.REP.* the electr ical conductivity of y~irradiated sodium s i l i ca te glasses

containing multivalence Cu ions.