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Scattering from a model fractal lattice of dimension less than one
Volume 128, number
6,7
PHYSICS
SCATTERING FROM A MODEL FRACTAL OF DIMENSION LESS THAN ONE Harjinder
SINGH
LETTERS
A
11 April 1988
LATTICE
and P.K. CHATTARAJ
Theoretical Chemistry Group, Department of Chemistry, Panjab University, Chandigarh 160014, India Received 18 May 1987; revised manuscript Communicated by A.P. Fordy
received
9 February
1988; accepted
for publication
10 February
1988
Scattering cross sections are obtained for scattering of waves from one-dimensional lattices chosen as simplistic models of Cantor-like sets. The self-similar geometry is described in terms of a fractal dimension. A parametric dependence of the crosssection on the fractal dimension is proposed. The average wavefunction method is used to calculate the differential scattering cross section. The integrated cross section shows a fractional power-law dependence on the wave-vector of the incoming wave. Results are presented for a wide range of energies and two different regimes exhibiting different fractal behaviour are observed.
Characterizing complex non-linear systems in terms of simple parameters is a widely shared goal of theoreticians. A large number of works have appeared over the last decade exploring the relevance of fractal dimensionality for self-similar systems [ l-5 1. In spite of much activity, the physics of fractals is still not completely understood [ 61. One area where a direct correlation between the physics and the fractal behaviour can be explored is heterogeneous reaction dynamics [ 71. Just as the length of a coast-line or the boundary of a cloud is dependent on the scale chosen, the surface area of a solid depends on the scale and method chosen to measure it. The true description of an interface cannot be given in terms of a euclidean dimension. The fractal property has a profound influence on the physico-chemical properties of a surface, e.g., catalysis and other dynamical processes occurring on an interface. The imparticularly portance of scattering processes, gas-surface scattering, in understanding heterogeneous catalysis is well known [ 8 1. While scattering from regular surfaces is more or less well-studied, disordered surfaces present problems that are difficult to handle theoretically. On the other hand, disorder plays a significant role in the chemistry of surfaces [ 91. Fractal dimension can be used as a characteristic of a disordered surface. Fractal surfaces have been studied [ 10,111 and a number of 0375-9601/88/S ( North-Holland
03.50 0 Elsevier Science Publishers Physics Publishing Division )
limitations in this area have been reported [ 121. For a gas-surface interaction, the fractal dimension enters the potential as a description of the geometry of the surface atoms. The dynamics is governed by the interaction potential. Thus the fractal behaviour can be seen as a structure and dynamics property. Such a role of the fractal dimension is well-studied for some biological systems [ 13- 15 1. Stochastic treatments to describe rough surfaces have been shown to involve fractal dimension, for ocean topography as well as solid surfaces [ 16,17 1. Experimental verification of fractal behaviour of porous materials has appeared in a number of works [ 11,181. The optics of waves encountering fractals has been studied before [ 19-211. We have tried to explore scattering from fractal-like lattices using a simple approximation scheme called the average wavefunctions method [22]. Though we consider only a few atoms arranged in a lattice generated by initial steps of an iteration process to produce a fractal eventually, the results are expected to show some correlations for larger systems. The interaction potential for a structureless particle interacting with a solid surface contains information about the geometry of the surface. Thus, it is meaningful to label the potential vdr(r) by the B.V.
355
Volume
PHYSICS
128. number 6.7
fractal dimension d,-.The dimension df is defined by a scaling behaviour, for instance Rx N ‘/dt. where N is the number of atomic sites in a circle of radius R. For a simpler system, e.g. a euclidean one-dimensional lattice, a similar definition based on the selfsimilarity of the system is defined. The potential I”” could simply be an interaction between an approaching atom and a solid surface or it could be the interaction with adsorbate atoms on a surface. The coordinate Ymay consist of internal degrees of freedom other than the translational ones. The quantity of interest is the T-matrix element, (V”‘\yp)
7+‘(k’,k)=(f$,
tr)=@k
cr)
G+(r,r’)V”‘(r’)tyk(r’)
dr’ ,
(2)
where G+ (r, r’ ) is the Green function for forward scattering. If the labelling by df on the T-matrix element in eq. ( 1) is justified, then the analysis of crosssection data for scattering from different lattices provides a means to explore fractal physics. More specifically, if roughness can be characterized by a selfsimilar behaviour, it is possible to explore the effect of roughness on dynamics. The geometry of the target has to enter the analysis for the argument given above to be valid, i.e. approximations that disregard multiple scattering effects will not be consistent with this logic. We use the pairwise interaction approximation, l’(r)=
1 c,(r-r,).
/
356
C i
April I988
In the average wavefunction method ( AWM ) [ 2 I 1. the T-matrix has the following modified form. T(k’,k)=
Ci,;)i
/
jdr&(r)*c,(r-r,)w,(r),
(5) where
I drw,(r)*v,(r-r,)~k/,r). !‘ Iw(r)12u,(r--y,)
yI= *-I
=
4
(6)
dr
and V/k is the AWM approximated
jdrl*(r)*u,(r--,)@k(r).
+
1i /
I
Y,
(7) wavefunction.
dr’ G’(r,r’)q(r’-r,)w,(r’).
(4)
(8)
the term w,(r) is a weighting factor. In the zeroth order of the approximation it is taken to be the zeroth order (distorted) wavefunction. That this is a good choice is based on the fact that o,(r) = Ii/k(r) gives the optimal solution wk (r) = vk (r) [ 22,23 1. Application to model problems of gas-surface scattering and atom-diatom scattering [ 23,241 has that AWM is a useful time-saving shown approximation. We have done preliminary calculations on onedimensional fractal-like lattices with df =log 2/lag 3 and d,-= log 3/lag 5 for the case where there is no background potential. Though simplistic, this is done with the hope that some insight can be achieved into scattering from more realistic and larger fractal lattices. The T-matrix for a spherically symmetric gaussian potential at each atomic site, has the following form for the scattering of plane waves, 7‘(k’, k) x exp( -Ak’/2)
(3)
The interaction potential V(r) is usually a corrugation potential on a background hard wall-like potential. The @ in eqs. ( 1) and (2) should then be replaced by an appropriately distorted wave. In the first order Born approximation, the T-matrix element has the following form,
Vk’.k)=
II
A
(1)
Once the T-matrix is known, other quantities such as the differential cross section can easily be evaluated. The scattered wavefunction I+Y:” is related to the incident wavefunction dk by the Lippmann-Schwinger equation, w:‘”
LETTERS
x 2 l/‘,exp( iAk*r, ) .
erf(iAk/..‘2) (9)
where Ak=k’ -k, erf( 2) is the error function in Z, j; is defined in eq. (6 ) and r, is the atomic position as in eq. ( 3). The wavevectors k (incident) and k’ (scattered) are appropriately scaled with respect to the variance of the gaussian, which is taken to be identical at each site. We have also used h= 1 and m= 1 in the Schrodinger equation to derive eq. (9). The results presented in figs. l-3 show a clear de-
Volume
128, number
PHYSICS
6,7
LETTERS
A
11 April 1988
n =32; n =16; df = log 2/leg
d,=
log 2/1og3
3 ok:
r,’
a
k=.ol ,I
;.* :: .r. .a
. . . .
.
O-
O a
0
8
.
b
n
2ll
8
Fig. I Differential scattering cross section (do/dQ), in units of 4x*a2, plotted against the scattering angle (0). where a is the variance of thegaussian potential, v=exp( -r*/2az) at each atomic site, and the fractal dimension ofthe lattice is log Z/log 3. The number ofatoms is(a) 16(ak=0.001 to 10);and (b) 32(ak=O.l to 10).Theparameterakisthescaledwave-vector(~=l,m=1).
pendence of scattering cross section on the fractal dimensionality of a lattice. Figs. 1 and 2 show the differential cross section plotted as a function of the scattering angle for different generations (hence different scales, in the construction of a fractal) of the lattices. While figs. la and 2a show the results for 16 third generations) and 18 (&log 2/lag 3, (d,= log 3/lag 5, second generation ) atoms respectively, figs. lb and 2b are for 32 (&=log 2/lag 3, fourth generation) and 54 (&log 3/lag 5, third generation) respectively. The increasing number of peaks as one goes from lower to higher energies (ak=O.OOl to 10, in figs. la and 2a and ak=O.l to 10 in figs. 1b and 2b, where a is the variance of the gaussian potential at each atomic site) for incoming waves, is expected because of diffraction effects. The features seen here are expected to be more prominent for larger lattices. In fig. 3, we have shown a logarithmic plot of the integrated cross section,
[J( da/dQ) do] /47t”a2, against the scaled wavenumber of the incoming wave for the two different generations of two different fractal sets considered earlier. Immediately visible is a striking cross-over in the linear slope when ak= 1. While the linear slope confirms the fractal property of the cross-section data, the cross-over for the same value of ak in all the plots is significant. Cross-over from euclidean to noneuclidean dimensions have been seen before in the spin-lattice relaxation of proteins [25]. In the case of optical diffraction on fractals [ 261, the integrated structure factor (proportional to the scattering intensity) is seen to exhibit a similar power law dependence on the wavevector. There is qualitative agreement between our results and some of the first order Born calculations for X-ray scattering off a Menger sponge reported by Schmidt and Decai [ 271. The cross-over from a positive to a negative slope in fig. 3 remains puzzling. It is possible that because of 357
Volume 128. number
5.7
n = 18; df= ak=
PHYSICS
log ~/lo
LETTERS
.A
I I ‘April 1988
n = 54; df = log 3/109
5
268X104 ‘-\
.Ol ,’
\
dc; dn
O a Fig. 2. dn/dQ (see fig. 1). in umts of 4x’a’. plotted against (a) 18 (uk=O.OOl to IO):and (b) 54 (uk=O.l to IO).
8. The fractal dimewon
greater uncertainties in the position variable for ak-c 1, a completely different fractal behaviour arises in this regime. The near constancy of the slope in fig. 3 for the lattices of the same fractal dimension indicates that though we have considered a small number of atoms for the calculations, we can still reach conclusions that are more general in nature. In the regime ak> 1. we see that the lattices with different fractal dimensions give rise to different fractal behaviours in the scattering cross-section output. This is a significant result, though a rigorous justification for this needs to be worked out. Certainly a more realistic calculation would require an ensemble averaging over a reasonably large number of atoms. i.e., a much larger amount of computer storage and time. The results shown here were obtained using a gaussian elimination technique for matrix inversion and 4-point Gauss-Legendre quadrature for numerical integration. 3.58
of the lattice is log 3/lag 5. The number
of atoms is
In summary, we have shown here the results of a set of calculations for scattering from fractal-like lattices using the AWM approximation. We notice a clear dependence of scattering cross-section data on fractal behaviour of the lattice. A more explicit dependence on fractal dimension should appear in eq. (9 ). Further work is in progress. One of us (PKC) wishes to thank Shyamalava Mazumdar of the Tata Institute of Fundamental Research, Bombay, for his assistance in the computations. We also thank Professor B.M. Deb for his unending encouragement to continue this work in difficult circumstances. Thanks are also due to the encouraging and critical comments by the referees. that helped us to understand the problem better. A preliminary report of the theoretical ideas presented here has appeared elsewhere [ 281.
Volume
128. number
PHYSICS
6,7
LETTERS
1 I April 1988
A
I I
log (4
(4
log
i Fig. 3. Logarithm of integrated cross section (integrated over the scattering angle only) plotted against the logarithm of the wave-vector scaled with respect to the variance of the gaussian potential at each atomic site. (a) &= log 2/lag 3: slope at ski I= 1.75 (n = 16). 1.80 (n=32);slopeforak>l=-1.8 (n=l6), -1.72(~=32). (b)dr=log3/log5;slopeatak~l=l.75 (n=l8.54);slopeatak>l=-2.08 (n= 18), -2.1 (~~54). n is the number ofatoms in the lattice.
References [ 1 ] B.B. Mandelbrot, man, San Francisco,
The fractal 1983).
geometry
of nature
[ 141 J.S. Helman, A. Coniglio and C. Tsallis, Phys. Rev. Lett. 53 (1984) 1195. [ 151 S.H. Liu, Phys. Rev. B 32 (1985) 6094. [ 16 JM. Ausloos and D.H. Berman, Proc. R. Sot. A 400 ( 1985) 331.
(Free-
[ 21 L. Pietronero and E. Tosatti, eds., Fractals in physics (NorthHolland,
Amsterdam,
[3] S. Liu, ed., Kinetics Holland, Amsterdam,
[ 17 ] M. Berry and Z.V. Lewis, Proc. R. Sot. A 370 ( 1980) 459. [ 18 ] H.D. Bale and P.W. Schmidt, Phys. Rev. Lett. 53 (1984)
1986). of aggregation 1986).
and gelation
(North-
596.
[4] M. Ghil, R. Benzi and G. Parisi, eds., Turbulence and predictability of geophysical flows and climate dynamics (North-Holland, Amsterdam, 1985). [5] A. LeMehaute,
[ 6 ] L. Kadanoff,
J. Stat. Phys. 37 (1984) Phys. Today 39 ( 1986
[7] F. Pfeiferand
D. Avnir, J. Chem. Phys. 79 (1983)
[S] H. Singh and B.M. Deb, Pramana [9] J.C. Tully and M.J. Cardillo, [ lo] P. Meakin,
665.
) 6. 3558.
J. Phys. 27 (1986)
Science 223 (1984)
(1985)
1231 H. Singh, D.K. Dacol and H. Rabitz, (1986) 1852.
337.
445.
[24]H. Singh, D.K. (1987) 339.
Chem. Phys. Lett. 123 ( 1986) 428.
[ 111 D. Farin, S. Peleg, D. Yavin and D. Avnir, Langmuir 399.
[ 121 D. Avnir, D. Farin and P. Pfeifer, J. Chem. Phys. 79 ( 1983) 3566.
[ 131 J.T. Calvin and H.J. Stapleton,
J. Chem. Phys. 82 (1985)
[19]M.V.Berry,J.Phys.A12(1979)781. [20] M.V. BerryandT.M. Blackwell, J. Phys. A 14 (1981) 3101. [2 I ] E. Jakeman, in: Fractals in physics, eds. L. Pietronero and E. Tosatti (North-Holland, Amsterdam, 1986) p. 55. [22] H. Singh, D.K. Dacol and H. Rabitz, J. Chem. Phys. 84 (1986) 1373.
1
125) 1261 (271 [28]
Dacol
and H. Rabitz:
J. Chem. Chem.
Phys. 84 Phys.
115
K.N. Srivastava, 3. Phys. C 20 (1987) L381. C. Allain and M. Cloitre, Phys. Rev. B 33 (1986) 3566. P.W. Schmidt and Q. Decai. Phys. Rev. A 33 (1986) 560. H. Singh and P.K. Chattaraj, Proc. Ind. A-ad. Sci. (Chem.) 99 (1987) 47.
4699.
359
6,7
PHYSICS
SCATTERING FROM A MODEL FRACTAL OF DIMENSION LESS THAN ONE Harjinder
SINGH
LETTERS
A
11 April 1988
LATTICE
and P.K. CHATTARAJ
Theoretical Chemistry Group, Department of Chemistry, Panjab University, Chandigarh 160014, India Received 18 May 1987; revised manuscript Communicated by A.P. Fordy
received
9 February
1988; accepted
for publication
10 February
1988
Scattering cross sections are obtained for scattering of waves from one-dimensional lattices chosen as simplistic models of Cantor-like sets. The self-similar geometry is described in terms of a fractal dimension. A parametric dependence of the crosssection on the fractal dimension is proposed. The average wavefunction method is used to calculate the differential scattering cross section. The integrated cross section shows a fractional power-law dependence on the wave-vector of the incoming wave. Results are presented for a wide range of energies and two different regimes exhibiting different fractal behaviour are observed.
Characterizing complex non-linear systems in terms of simple parameters is a widely shared goal of theoreticians. A large number of works have appeared over the last decade exploring the relevance of fractal dimensionality for self-similar systems [ l-5 1. In spite of much activity, the physics of fractals is still not completely understood [ 61. One area where a direct correlation between the physics and the fractal behaviour can be explored is heterogeneous reaction dynamics [ 71. Just as the length of a coast-line or the boundary of a cloud is dependent on the scale chosen, the surface area of a solid depends on the scale and method chosen to measure it. The true description of an interface cannot be given in terms of a euclidean dimension. The fractal property has a profound influence on the physico-chemical properties of a surface, e.g., catalysis and other dynamical processes occurring on an interface. The imparticularly portance of scattering processes, gas-surface scattering, in understanding heterogeneous catalysis is well known [ 8 1. While scattering from regular surfaces is more or less well-studied, disordered surfaces present problems that are difficult to handle theoretically. On the other hand, disorder plays a significant role in the chemistry of surfaces [ 91. Fractal dimension can be used as a characteristic of a disordered surface. Fractal surfaces have been studied [ 10,111 and a number of 0375-9601/88/S ( North-Holland
03.50 0 Elsevier Science Publishers Physics Publishing Division )
limitations in this area have been reported [ 121. For a gas-surface interaction, the fractal dimension enters the potential as a description of the geometry of the surface atoms. The dynamics is governed by the interaction potential. Thus the fractal behaviour can be seen as a structure and dynamics property. Such a role of the fractal dimension is well-studied for some biological systems [ 13- 15 1. Stochastic treatments to describe rough surfaces have been shown to involve fractal dimension, for ocean topography as well as solid surfaces [ 16,17 1. Experimental verification of fractal behaviour of porous materials has appeared in a number of works [ 11,181. The optics of waves encountering fractals has been studied before [ 19-211. We have tried to explore scattering from fractal-like lattices using a simple approximation scheme called the average wavefunctions method [22]. Though we consider only a few atoms arranged in a lattice generated by initial steps of an iteration process to produce a fractal eventually, the results are expected to show some correlations for larger systems. The interaction potential for a structureless particle interacting with a solid surface contains information about the geometry of the surface. Thus, it is meaningful to label the potential vdr(r) by the B.V.
355
Volume
PHYSICS
128. number 6.7
fractal dimension d,-.The dimension df is defined by a scaling behaviour, for instance Rx N ‘/dt. where N is the number of atomic sites in a circle of radius R. For a simpler system, e.g. a euclidean one-dimensional lattice, a similar definition based on the selfsimilarity of the system is defined. The potential I”” could simply be an interaction between an approaching atom and a solid surface or it could be the interaction with adsorbate atoms on a surface. The coordinate Ymay consist of internal degrees of freedom other than the translational ones. The quantity of interest is the T-matrix element, (V”‘\yp)
7+‘(k’,k)=(f$,
tr)=@k
cr)
G+(r,r’)V”‘(r’)tyk(r’)
dr’ ,
(2)
where G+ (r, r’ ) is the Green function for forward scattering. If the labelling by df on the T-matrix element in eq. ( 1) is justified, then the analysis of crosssection data for scattering from different lattices provides a means to explore fractal physics. More specifically, if roughness can be characterized by a selfsimilar behaviour, it is possible to explore the effect of roughness on dynamics. The geometry of the target has to enter the analysis for the argument given above to be valid, i.e. approximations that disregard multiple scattering effects will not be consistent with this logic. We use the pairwise interaction approximation, l’(r)=
1 c,(r-r,).
/
356
C i
April I988
In the average wavefunction method ( AWM ) [ 2 I 1. the T-matrix has the following modified form. T(k’,k)=
Ci,;)i
/
jdr&(r)*c,(r-r,)w,(r),
(5) where
I drw,(r)*v,(r-r,)~k/,r). !‘ Iw(r)12u,(r--y,)
yI= *-I
=
4
(6)
dr
and V/k is the AWM approximated
jdrl*(r)*u,(r--,)@k(r).
+
1i /
I
Y,
(7) wavefunction.
dr’ G’(r,r’)q(r’-r,)w,(r’).
(4)
(8)
the term w,(r) is a weighting factor. In the zeroth order of the approximation it is taken to be the zeroth order (distorted) wavefunction. That this is a good choice is based on the fact that o,(r) = Ii/k(r) gives the optimal solution wk (r) = vk (r) [ 22,23 1. Application to model problems of gas-surface scattering and atom-diatom scattering [ 23,241 has that AWM is a useful time-saving shown approximation. We have done preliminary calculations on onedimensional fractal-like lattices with df =log 2/lag 3 and d,-= log 3/lag 5 for the case where there is no background potential. Though simplistic, this is done with the hope that some insight can be achieved into scattering from more realistic and larger fractal lattices. The T-matrix for a spherically symmetric gaussian potential at each atomic site, has the following form for the scattering of plane waves, 7‘(k’, k) x exp( -Ak’/2)
(3)
The interaction potential V(r) is usually a corrugation potential on a background hard wall-like potential. The @ in eqs. ( 1) and (2) should then be replaced by an appropriately distorted wave. In the first order Born approximation, the T-matrix element has the following form,
Vk’.k)=
II
A
(1)
Once the T-matrix is known, other quantities such as the differential cross section can easily be evaluated. The scattered wavefunction I+Y:” is related to the incident wavefunction dk by the Lippmann-Schwinger equation, w:‘”
LETTERS
x 2 l/‘,exp( iAk*r, ) .
erf(iAk/..‘2) (9)
where Ak=k’ -k, erf( 2) is the error function in Z, j; is defined in eq. (6 ) and r, is the atomic position as in eq. ( 3). The wavevectors k (incident) and k’ (scattered) are appropriately scaled with respect to the variance of the gaussian, which is taken to be identical at each site. We have also used h= 1 and m= 1 in the Schrodinger equation to derive eq. (9). The results presented in figs. l-3 show a clear de-
Volume
128, number
PHYSICS
6,7
LETTERS
A
11 April 1988
n =32; n =16; df = log 2/leg
d,=
log 2/1og3
3 ok:
r,’
a
k=.ol ,I
;.* :: .r. .a
. . . .
.
O-
O a
0
8
.
b
n
2ll
8
Fig. I Differential scattering cross section (do/dQ), in units of 4x*a2, plotted against the scattering angle (0). where a is the variance of thegaussian potential, v=exp( -r*/2az) at each atomic site, and the fractal dimension ofthe lattice is log Z/log 3. The number ofatoms is(a) 16(ak=0.001 to 10);and (b) 32(ak=O.l to 10).Theparameterakisthescaledwave-vector(~=l,m=1).
pendence of scattering cross section on the fractal dimensionality of a lattice. Figs. 1 and 2 show the differential cross section plotted as a function of the scattering angle for different generations (hence different scales, in the construction of a fractal) of the lattices. While figs. la and 2a show the results for 16 third generations) and 18 (&log 2/lag 3, (d,= log 3/lag 5, second generation ) atoms respectively, figs. lb and 2b are for 32 (&=log 2/lag 3, fourth generation) and 54 (&log 3/lag 5, third generation) respectively. The increasing number of peaks as one goes from lower to higher energies (ak=O.OOl to 10, in figs. la and 2a and ak=O.l to 10 in figs. 1b and 2b, where a is the variance of the gaussian potential at each atomic site) for incoming waves, is expected because of diffraction effects. The features seen here are expected to be more prominent for larger lattices. In fig. 3, we have shown a logarithmic plot of the integrated cross section,
[J( da/dQ) do] /47t”a2, against the scaled wavenumber of the incoming wave for the two different generations of two different fractal sets considered earlier. Immediately visible is a striking cross-over in the linear slope when ak= 1. While the linear slope confirms the fractal property of the cross-section data, the cross-over for the same value of ak in all the plots is significant. Cross-over from euclidean to noneuclidean dimensions have been seen before in the spin-lattice relaxation of proteins [25]. In the case of optical diffraction on fractals [ 261, the integrated structure factor (proportional to the scattering intensity) is seen to exhibit a similar power law dependence on the wavevector. There is qualitative agreement between our results and some of the first order Born calculations for X-ray scattering off a Menger sponge reported by Schmidt and Decai [ 271. The cross-over from a positive to a negative slope in fig. 3 remains puzzling. It is possible that because of 357
Volume 128. number
5.7
n = 18; df= ak=
PHYSICS
log ~/lo
LETTERS
.A
I I ‘April 1988
n = 54; df = log 3/109
5
268X104 ‘-\
.Ol ,’
\
dc; dn
O a Fig. 2. dn/dQ (see fig. 1). in umts of 4x’a’. plotted against (a) 18 (uk=O.OOl to IO):and (b) 54 (uk=O.l to IO).
8. The fractal dimewon
greater uncertainties in the position variable for ak-c 1, a completely different fractal behaviour arises in this regime. The near constancy of the slope in fig. 3 for the lattices of the same fractal dimension indicates that though we have considered a small number of atoms for the calculations, we can still reach conclusions that are more general in nature. In the regime ak> 1. we see that the lattices with different fractal dimensions give rise to different fractal behaviours in the scattering cross-section output. This is a significant result, though a rigorous justification for this needs to be worked out. Certainly a more realistic calculation would require an ensemble averaging over a reasonably large number of atoms. i.e., a much larger amount of computer storage and time. The results shown here were obtained using a gaussian elimination technique for matrix inversion and 4-point Gauss-Legendre quadrature for numerical integration. 3.58
of the lattice is log 3/lag 5. The number
of atoms is
In summary, we have shown here the results of a set of calculations for scattering from fractal-like lattices using the AWM approximation. We notice a clear dependence of scattering cross-section data on fractal behaviour of the lattice. A more explicit dependence on fractal dimension should appear in eq. (9 ). Further work is in progress. One of us (PKC) wishes to thank Shyamalava Mazumdar of the Tata Institute of Fundamental Research, Bombay, for his assistance in the computations. We also thank Professor B.M. Deb for his unending encouragement to continue this work in difficult circumstances. Thanks are also due to the encouraging and critical comments by the referees. that helped us to understand the problem better. A preliminary report of the theoretical ideas presented here has appeared elsewhere [ 281.
Volume
128. number
PHYSICS
6,7
LETTERS
1 I April 1988
A
I I
log (4
(4
log
i Fig. 3. Logarithm of integrated cross section (integrated over the scattering angle only) plotted against the logarithm of the wave-vector scaled with respect to the variance of the gaussian potential at each atomic site. (a) &= log 2/lag 3: slope at ski I= 1.75 (n = 16). 1.80 (n=32);slopeforak>l=-1.8 (n=l6), -1.72(~=32). (b)dr=log3/log5;slopeatak~l=l.75 (n=l8.54);slopeatak>l=-2.08 (n= 18), -2.1 (~~54). n is the number ofatoms in the lattice.
References [ 1 ] B.B. Mandelbrot, man, San Francisco,
The fractal 1983).
geometry
of nature
[ 141 J.S. Helman, A. Coniglio and C. Tsallis, Phys. Rev. Lett. 53 (1984) 1195. [ 151 S.H. Liu, Phys. Rev. B 32 (1985) 6094. [ 16 JM. Ausloos and D.H. Berman, Proc. R. Sot. A 400 ( 1985) 331.
(Free-
[ 21 L. Pietronero and E. Tosatti, eds., Fractals in physics (NorthHolland,
Amsterdam,
[3] S. Liu, ed., Kinetics Holland, Amsterdam,
[ 17 ] M. Berry and Z.V. Lewis, Proc. R. Sot. A 370 ( 1980) 459. [ 18 ] H.D. Bale and P.W. Schmidt, Phys. Rev. Lett. 53 (1984)
1986). of aggregation 1986).
and gelation
(North-
596.
[4] M. Ghil, R. Benzi and G. Parisi, eds., Turbulence and predictability of geophysical flows and climate dynamics (North-Holland, Amsterdam, 1985). [5] A. LeMehaute,
[ 6 ] L. Kadanoff,
J. Stat. Phys. 37 (1984) Phys. Today 39 ( 1986
[7] F. Pfeiferand
D. Avnir, J. Chem. Phys. 79 (1983)
[S] H. Singh and B.M. Deb, Pramana [9] J.C. Tully and M.J. Cardillo, [ lo] P. Meakin,
665.
) 6. 3558.
J. Phys. 27 (1986)
Science 223 (1984)
(1985)
1231 H. Singh, D.K. Dacol and H. Rabitz, (1986) 1852.
337.
445.
[24]H. Singh, D.K. (1987) 339.
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