- Email: [email protected]
Calculation of the aggregation and electrodynamic effects in granular systems
*-_ -_ lfi!z
s
PINSBA 1
ELSEVIER
Physica
Calculation
A 207 (1994)
123-130
of the aggregation effects in granular
and electrodynamic systems
L.F. Fonseca, M. Gomez, L. Cruz’ Department
of Physics, University of Puerto Rico, Rio Piedras, Puerto Rico 00931
Abstract A calculation based on a multiple scattering model is presented in order to obtain the absorption of aggregated granular metals. The’ model is based on a T-matrix formalism in which the scattering units are clusters of nearest-neighbor metallic particles and the behavior of the scattering units is described by the orientational-averaged T-matrix of the clusters. The remainder of the system containing these scattering units will be considered in an effective medium calculation. The formalism is applicable to systems in which the sizes of the particles require the consideration of electrodynamic effects.
1. Introduction
The Maxwell-Garnett and Bruggeman theories are two basic effective medium models that describe the optical properties of systems containing small metallic particles dispersed in a dielectric matrix. The applicability of these two models is restricted to systems whose metallic regions are small enough with respect to the wavelength of the incident radiation and in which the long-range electrostatic dipolar
electric
interaction
gives
the main
contribution.
This
condition
applies
neither to systems whose metallic particles become aggregated nor to those whose metallic particles are too large for the electrostatic limit to apply. Even though several theories have been developed recently that improve on the simple arguments of the MG and B models, the research is still in progress. Recently, the effects of aggregation have been considered as a possible explanation for the difference between standard effective medium theories and the observed far infrared absorption of some granular metal systems (see, for example [l]). Also, aggregation has been proposed to explain the appearance of more than one absorption band in aggregated fractal colloids [2]. In several cases aggregation can 1 Present
address:
Department
of Physics,
037%4371/94/$07.00 0 1994 Elsevier SSDI 0378-4371(93)E0541-L
Room
Science
6-222, MIT,
B.V. All rights
Cambridge,
reserved
MA 02139,
USA.
124
L.F.
Fonseca et al. I Physica A 207 (1994) 123-130
lead to systems in which the particles fuse together to form a bigger region with metallic characteristics but with a reduced electron mean free path as in the case of heat-treated composites [3]. In this work a multiple scattering calculation [4] based on the T-matrix formalism [5] is used to obtain the electromagnetic response of the different systems mentioned above.
2. The theory The starting point in our calculation is the approximation in which we can consider the short- and long-range interactions separately. Under this assumption the multipolar interactions between nearest neighbors are considered using a multiple scattering formalism where the scattering unit is the metallic particle surrounded by its nearest neighbors. The contribution from the rest of the system will be considered within an effective medium calculation. Following Varadan et al. [4], the electric field E at any point in the medium is the sum of the incident field E, plus the fields scattered by these units, E”, E; (r - ri) ,
E(r) = E,(r) + c
(1)
where ri is the position of the ith scattering unit. The field interacting with the ith unit is EF(r)=E,(r)+CE;‘(r-r,),
dc(r-ri(<2d.
(2)
j#i
The parameter d is the radius of the smallest sphere that inscribes the scattering unit. The solutions of the vector Helmholtz equation {4} are chosen as base set given by
where 7 = 1, 2, u = even (e) or odd (o), II = 1, 2, . . . , m = 0, 1, . . . , n, and Y,,,(F) = cos(m+) Pz(cos 0) ,
Y,,,(F) = sin(m+) Pz(cos 0) .
The index T = 1, 2 describes magnetic or electric excitations respectively. The parameter E, is the Neumann symbol defined as .sO= 1, E, = 2 otherwise; II is the order of the multipole, and c gives the parity of the elementary functions. The regular form of the basis functions is obtained by replacing the Hankel by the Bessel functions. Upon expansion of the electric fields Es and Er in terms of the set {I/J}, the coefficients {b} of the exciting fields and the coefficients {B} of the scattered fields are related by (3)
where Ti,,, I denotes the elements of the T-matrix due to the ith scattering unit.
L.F. Fonseca et al. I Physica
125
A 207 (1994) 123-130
The calculation of the elements of the T-matrix follow the formalism by Waterman [5] which applies to single particles as well as a particle surrounded by nearest neighbors as the scattering units. The calculation of the T-matrix elements for more than one scatterer can be performed by using the formalism of Peterson and Strom [6]. Inserting Eq. (3) into the field equation a relation between the scattered and incident field expansion coefficients {A} is obtained,
where S,.,,. are the matrix elements describing the translation properties of the base functions. Finally, a configurational average of the above equation is performed assuming (B,)ij=(B,)j,
(5)
which states that the averages taken with one or two fixed units are approximately the same. Considering no correlation between the scattering units other than the impenetrability condition we obtain
(6) where V is the volume of the sample and V’ is V minus a spherical volume of radius 2d that takes into account impenetrability. Using an effective medium approach, (BL) i = X,, exp(ik,,,
* ri)
,
(7)
where keff is the effective propagation wave vector, a final system of coupled equations is obtained for the unknowns X,. From Eq. (6) the dispersion relation is obtained by finding the relevant root of the determinant of the matrix with elements
where v is the number of scattering units per unit volume, and Znnll=
I
]S,Ak,r)
% exp(i&
. r)
-
exp(ik,,,
* r)
a,s,,.(k,~)]
ds .
(9)
r=Zd
When the scattering units have no rotational symmetry and they are randomly oriented in the aggregate, the T-matrix elements in Eq. (8) should be the elements of the orientational averaged T-matrix.
126
L.F.
3. Calculations 3.1.
Low-density
Fomeca
et al.
I Physica A 207 (1994) 123-130
and results aggregates
The case of low-density
aggregates
is a good starting
point for the application
of
the formalism because the scattering unit can be selected as an array of a few nearest neighbors. A system containing a low metallic concentration is the case of fractal colloidal aggregates [7]. In such systems the number of nearest-neighbor particles is low and the above formalism can be applied considering as scattering units arrays of only a few particles. A solution of gold colloids in water is an example of this type of systems that has been widely studied. Measurements of the optical absorption for such systems show a broadening of the absorption resonance peak due to single isolated particles. Also, they exhibit a new absorption band whose shift towards lower frequencies depends on the size of the aggregates that is related with the aggregation time. Theoretical calculations using different approaches have been applied to these structures [8,9]. Because the fractal pattern of these gold aggregates reduces the number of nearest neighbors that a particle could have, we consider as scattering units clusters of only two nearest neighbor spherical gold particles. The fact that the absorption spectrum of the orientationally averaged isolated cluster of two small gold particles shows two resonant bands justifies the choice of this simple scattering unit. The T-matrix T, of the chosen scattering unit is given in terms of the T-matrix, T, of the individual spherical particles as [6] T2 = R(d){ T[ 7 - a(-2d)T~+f)T]~‘[ + R(-d){T[I
- c@d)Tu(-2d)T]-‘[l
7 + a(-2d)T/?(2d)]}R(-d) + a@d)TR(-2d)]}R(d),
(10)
where 2d is the distance vector between the centers of the two spheres. The symbols R and u correspond to the translational matrices, as defined in ref. [6]. The calculation of the roots of the determinant described by Eq. (8) is performed by obtaining first the orientational average of T2. The computational time was reduced by approximating the orientational averaging process by assuming a random distribution along the three principal axes X, y, z when the incident wave vector is along the z-axis. Once the effective index of refraction of the aggregated system is calculated, each colloidal aggregate is considered as “uniform” regions immersed in the external medium (water) so that the standard Mie theory is used to calculate the optical absorption of the entire system under the assumption that the aggregates are far apart. The calculation of the real and imaginary parts of the effective index of refraction assuming the gold fractal aggregate composed of scattering units of two spheres is shown in Fig. 1. The gold spheres are 15 nm in diameter which corresponds to the average diameter reported by Weitz et al. [7]. The separation
L.F.
127
Fonseca et al. 1 Physica A 207 (1994) 123-130
3
1
0
0.8
0.6 Y 0.4
0.2
0 300
500
700
wavelength (nm)
Fig.
1.
900
o400
500
ml
700
800
SoI
wavelength (nm)
Fig. 2.
Fig. 1. Real (N) and imaginary (K) parts of the effective index of refraction of a low density aggregate of gold colloids in water solution calculated considering as scattering units the orientational average of a pair of two nearest gold particles. Fig. 2. Comparison between the observed absorption of a fractal colloidal gold system as reported ref. [2] (solid line) and the calculated absorption using the proposed model (dashed line).
in
between their centers was chosen as d = 15.16 nm which corresponds approximately to one molecular bond length. The external medium is water with dielectric function 1.77. The results for the effective index of refraction of such systems show two peaks that yield two absorption bands in the optical spectrum. The appearance of these two peaks are due to the short-range multipolar interaction between nearest-neighbor gold spheres. The calculations of Fig. 1 were performed considering up to the 17th multipolar order. Convergence is obtained already for multipolar terms of order 14. The sizes of the fractal aggregates increase with aggregation time. Figure 2 shows the comparison between the calculated absorption using the presented formalism and the experimental observation [2] at an aggregation time of 25 min.
128
L.F.
Fonseca et al. I Physica A 207 (1994) 123-130
A distribution of sizes in the aggregates was assumed with more abundance of the aggregates of 250nm. This aggregate size corresponds to the average size reported in ref. [7]. Even though the actual structure of the studied fractal system is not considered in detail and a high simplification has been done for the calculation of the absorption properties, the comparison between the calculations and the experimental results validates the original assumption that, for the calculation of the absorption, the short-range multipolar interaction between the nearest neighbors is the principal source for the determination of the absorption resonances. The long-range interactions exert a smaller influence on the resonances and are treated in this formalism in an effective-medium way. These conclusions have been proposed by other authors for the case of fractal aggregates [2,8]. Our calculation produces a better fit of the absorption band widths and relative strengths than previous works [8].
3.2. Electrodynamic effects Another type of aggregation process can produce fused clusters as in the case of heat-treated composites. These systems can be considered as large metallic clusters with a reduced electronic mean free path [3]. In the visible region, the size of such systems could be beyond the reach of the electrostatic approximation. Because the presented model is based on an electrodynamic formalism, retardation and electrodynamic effects are already incorporated and the formalism can be used to study the coherent wave propagation in such systems [4]. Figure 3 shows the comparison between the MG results and those obtained using the present model for gold fused clusters with different sizes. The calculations were performed using the bulk dielectric function for gold [lo]. The average sizes of the particles used in the proposed model are 60 nm and 100 nm. The increase of the size of the particles to values where the electrostatic approximation is no longer applicable produces the increase of the absorption at lower energies, as expected [ 111.
4. Conclusions
A procedure to obtain the electromagnetic response of granular aggregated metals is presented based on a multiple scattering T-matrix formalism. The procedure has intrinsically electrodynamic effects and multipolar interactions between the metallic particles and can be used to study non-fused and fused aggregated systems.
L.F. Fonseca et al. I Physica
1
1.5
2 energy
A 207 (1994) 123-130
2.5
3
129
3.5
@V)
Fig. 3. Log (Yas a function of the energy, where Q = 47rKlh cm-’ and K is the imaginary part of the effective index of refraction. Simple dashed curve is the standard MG calculation. The double-dashed and solid curves represent calculations using particles 60 nm and 100 nm in size, respectively.
For the first type of aggregated systems, the approach considers short-range and long-range interactions. The short-range interactions are taken into account by obtaining the T-matrix of a metallic particle surrounded by its neighbors, and the long-range interactions are considered performing an effective-medium calculation. Fractal colloidal aggregates are complex arrays of low-density aggregates. The calculation of the absorption resonances of such systems shows that even though a complete description of the optical properties of fractal aggregates requires a much more detailed theory, the absorption behavior is in fact dominated by the short-range interaction between nearest neighbors. Improvement on previous results [12] has been obtained by performing an orientational average of the T-matrix of each scattering unit. Recently many systems have been produced experimentally in which the aggregation is induced and controlled [1,13]. Many of such systems have a higher density of metallic particles than the fractal colloidal aggregates and consequently the average number of nearest neighbors is larger. Even though we obtained good results for the optical absorption of a gold fractal aggregate by using a simple pair of particles as the average scattering unit, a more complex array should be used to describe the experimental results for the higher-density aggregates. The presented formalism can be used for such systems but the scattering unit should be chosen as an array of more than two particles. The number of particles composing the scattering unit is only limited by the computational capabilities available. For the case of fused aggregates the formalism can be applied for the description of the coherent wave propagation when the averaged size of the aggregates is beyond the applicability of the electrostatic theories.
130
L.F.
Fonseca et al. I Physica A 207 (1994) 123-130
Acknowledgement
The support of EPSCoR-NSF
grant EHR-9108775 is acknowledged.
References [l] R.P. Devaty and A.J. Sievers, Phys. Rev. B 41 (1990) 7421. [2] H.M. Lindsay, M.Y. Lin, D.A. Weitz, P. Sheng, Z. Chen, R. Klein and P. Meakin, Faraday Discuss. Chem. Sot. 83 (1987) 153. [3] W.A. Curtin and N.W. Ashcroft, Phys. Rev. B 31 (1985) 3287 and references. [4] V.K. Varadan, V.N. Bringi and VV. Varadan, Phys. Rev. D 19 (1979) 2480. [5] P.C. Waterman, Phys. Rev. D 3 (1971) 8525. [6] B. Peterson and S. Strom, Phys. Rev. D 8 (1973) 3661. (71 D.A. Weitz, M.Y. Lin and C.J. Sandroff, Surface Sci. 158 (1985) 147. (81 Z. Chen and Ping Sheng, Phys. Rev. B 39 (1989) 9816. [9] F. Claro and R. Fuchs, Phys. Rev. B 44 (1991) 4109, and references therein. [lo] P.B. Johnson and R.W. Christy, Phys. Rev. B 6 (1972) 4370. [ll] U. Kreibig, M. Quinten and D. Schoenauer, Phys. Ser. T 13 (1986) 84. [12] L. Fonseca, L. Cruz, W. Vargas and M. Gbmez, Condensed Mat. Theor. 8 (1993) 561. [13] D. Schonauer and U. Kreibig, Surface Sci. 156 (1985) 100.
s
PINSBA 1
ELSEVIER
Physica
Calculation
A 207 (1994)
123-130
of the aggregation effects in granular
and electrodynamic systems
L.F. Fonseca, M. Gomez, L. Cruz’ Department
of Physics, University of Puerto Rico, Rio Piedras, Puerto Rico 00931
Abstract A calculation based on a multiple scattering model is presented in order to obtain the absorption of aggregated granular metals. The’ model is based on a T-matrix formalism in which the scattering units are clusters of nearest-neighbor metallic particles and the behavior of the scattering units is described by the orientational-averaged T-matrix of the clusters. The remainder of the system containing these scattering units will be considered in an effective medium calculation. The formalism is applicable to systems in which the sizes of the particles require the consideration of electrodynamic effects.
1. Introduction
The Maxwell-Garnett and Bruggeman theories are two basic effective medium models that describe the optical properties of systems containing small metallic particles dispersed in a dielectric matrix. The applicability of these two models is restricted to systems whose metallic regions are small enough with respect to the wavelength of the incident radiation and in which the long-range electrostatic dipolar
electric
interaction
gives
the main
contribution.
This
condition
applies
neither to systems whose metallic particles become aggregated nor to those whose metallic particles are too large for the electrostatic limit to apply. Even though several theories have been developed recently that improve on the simple arguments of the MG and B models, the research is still in progress. Recently, the effects of aggregation have been considered as a possible explanation for the difference between standard effective medium theories and the observed far infrared absorption of some granular metal systems (see, for example [l]). Also, aggregation has been proposed to explain the appearance of more than one absorption band in aggregated fractal colloids [2]. In several cases aggregation can 1 Present
address:
Department
of Physics,
037%4371/94/$07.00 0 1994 Elsevier SSDI 0378-4371(93)E0541-L
Room
Science
6-222, MIT,
B.V. All rights
Cambridge,
reserved
MA 02139,
USA.
124
L.F.
Fonseca et al. I Physica A 207 (1994) 123-130
lead to systems in which the particles fuse together to form a bigger region with metallic characteristics but with a reduced electron mean free path as in the case of heat-treated composites [3]. In this work a multiple scattering calculation [4] based on the T-matrix formalism [5] is used to obtain the electromagnetic response of the different systems mentioned above.
2. The theory The starting point in our calculation is the approximation in which we can consider the short- and long-range interactions separately. Under this assumption the multipolar interactions between nearest neighbors are considered using a multiple scattering formalism where the scattering unit is the metallic particle surrounded by its nearest neighbors. The contribution from the rest of the system will be considered within an effective medium calculation. Following Varadan et al. [4], the electric field E at any point in the medium is the sum of the incident field E, plus the fields scattered by these units, E”, E; (r - ri) ,
E(r) = E,(r) + c
(1)
where ri is the position of the ith scattering unit. The field interacting with the ith unit is EF(r)=E,(r)+CE;‘(r-r,),
dc(r-ri(<2d.
(2)
j#i
The parameter d is the radius of the smallest sphere that inscribes the scattering unit. The solutions of the vector Helmholtz equation {4} are chosen as base set given by
where 7 = 1, 2, u = even (e) or odd (o), II = 1, 2, . . . , m = 0, 1, . . . , n, and Y,,,(F) = cos(m+) Pz(cos 0) ,
Y,,,(F) = sin(m+) Pz(cos 0) .
The index T = 1, 2 describes magnetic or electric excitations respectively. The parameter E, is the Neumann symbol defined as .sO= 1, E, = 2 otherwise; II is the order of the multipole, and c gives the parity of the elementary functions. The regular form of the basis functions is obtained by replacing the Hankel by the Bessel functions. Upon expansion of the electric fields Es and Er in terms of the set {I/J}, the coefficients {b} of the exciting fields and the coefficients {B} of the scattered fields are related by (3)
where Ti,,, I denotes the elements of the T-matrix due to the ith scattering unit.
L.F. Fonseca et al. I Physica
125
A 207 (1994) 123-130
The calculation of the elements of the T-matrix follow the formalism by Waterman [5] which applies to single particles as well as a particle surrounded by nearest neighbors as the scattering units. The calculation of the T-matrix elements for more than one scatterer can be performed by using the formalism of Peterson and Strom [6]. Inserting Eq. (3) into the field equation a relation between the scattered and incident field expansion coefficients {A} is obtained,
where S,.,,. are the matrix elements describing the translation properties of the base functions. Finally, a configurational average of the above equation is performed assuming (B,)ij=(B,)j,
(5)
which states that the averages taken with one or two fixed units are approximately the same. Considering no correlation between the scattering units other than the impenetrability condition we obtain
(6) where V is the volume of the sample and V’ is V minus a spherical volume of radius 2d that takes into account impenetrability. Using an effective medium approach, (BL) i = X,, exp(ik,,,
* ri)
,
(7)
where keff is the effective propagation wave vector, a final system of coupled equations is obtained for the unknowns X,. From Eq. (6) the dispersion relation is obtained by finding the relevant root of the determinant of the matrix with elements
where v is the number of scattering units per unit volume, and Znnll=
I
]S,Ak,r)
% exp(i&
. r)
-
exp(ik,,,
* r)
a,s,,.(k,~)]
ds .
(9)
r=Zd
When the scattering units have no rotational symmetry and they are randomly oriented in the aggregate, the T-matrix elements in Eq. (8) should be the elements of the orientational averaged T-matrix.
126
L.F.
3. Calculations 3.1.
Low-density
Fomeca
et al.
I Physica A 207 (1994) 123-130
and results aggregates
The case of low-density
aggregates
is a good starting
point for the application
of
the formalism because the scattering unit can be selected as an array of a few nearest neighbors. A system containing a low metallic concentration is the case of fractal colloidal aggregates [7]. In such systems the number of nearest-neighbor particles is low and the above formalism can be applied considering as scattering units arrays of only a few particles. A solution of gold colloids in water is an example of this type of systems that has been widely studied. Measurements of the optical absorption for such systems show a broadening of the absorption resonance peak due to single isolated particles. Also, they exhibit a new absorption band whose shift towards lower frequencies depends on the size of the aggregates that is related with the aggregation time. Theoretical calculations using different approaches have been applied to these structures [8,9]. Because the fractal pattern of these gold aggregates reduces the number of nearest neighbors that a particle could have, we consider as scattering units clusters of only two nearest neighbor spherical gold particles. The fact that the absorption spectrum of the orientationally averaged isolated cluster of two small gold particles shows two resonant bands justifies the choice of this simple scattering unit. The T-matrix T, of the chosen scattering unit is given in terms of the T-matrix, T, of the individual spherical particles as [6] T2 = R(d){ T[ 7 - a(-2d)T~+f)T]~‘[ + R(-d){T[I
- c@d)Tu(-2d)T]-‘[l
7 + a(-2d)T/?(2d)]}R(-d) + a@d)TR(-2d)]}R(d),
(10)
where 2d is the distance vector between the centers of the two spheres. The symbols R and u correspond to the translational matrices, as defined in ref. [6]. The calculation of the roots of the determinant described by Eq. (8) is performed by obtaining first the orientational average of T2. The computational time was reduced by approximating the orientational averaging process by assuming a random distribution along the three principal axes X, y, z when the incident wave vector is along the z-axis. Once the effective index of refraction of the aggregated system is calculated, each colloidal aggregate is considered as “uniform” regions immersed in the external medium (water) so that the standard Mie theory is used to calculate the optical absorption of the entire system under the assumption that the aggregates are far apart. The calculation of the real and imaginary parts of the effective index of refraction assuming the gold fractal aggregate composed of scattering units of two spheres is shown in Fig. 1. The gold spheres are 15 nm in diameter which corresponds to the average diameter reported by Weitz et al. [7]. The separation
L.F.
127
Fonseca et al. 1 Physica A 207 (1994) 123-130
3
1
0
0.8
0.6 Y 0.4
0.2
0 300
500
700
wavelength (nm)
Fig.
1.
900
o400
500
ml
700
800
SoI
wavelength (nm)
Fig. 2.
Fig. 1. Real (N) and imaginary (K) parts of the effective index of refraction of a low density aggregate of gold colloids in water solution calculated considering as scattering units the orientational average of a pair of two nearest gold particles. Fig. 2. Comparison between the observed absorption of a fractal colloidal gold system as reported ref. [2] (solid line) and the calculated absorption using the proposed model (dashed line).
in
between their centers was chosen as d = 15.16 nm which corresponds approximately to one molecular bond length. The external medium is water with dielectric function 1.77. The results for the effective index of refraction of such systems show two peaks that yield two absorption bands in the optical spectrum. The appearance of these two peaks are due to the short-range multipolar interaction between nearest-neighbor gold spheres. The calculations of Fig. 1 were performed considering up to the 17th multipolar order. Convergence is obtained already for multipolar terms of order 14. The sizes of the fractal aggregates increase with aggregation time. Figure 2 shows the comparison between the calculated absorption using the presented formalism and the experimental observation [2] at an aggregation time of 25 min.
128
L.F.
Fonseca et al. I Physica A 207 (1994) 123-130
A distribution of sizes in the aggregates was assumed with more abundance of the aggregates of 250nm. This aggregate size corresponds to the average size reported in ref. [7]. Even though the actual structure of the studied fractal system is not considered in detail and a high simplification has been done for the calculation of the absorption properties, the comparison between the calculations and the experimental results validates the original assumption that, for the calculation of the absorption, the short-range multipolar interaction between the nearest neighbors is the principal source for the determination of the absorption resonances. The long-range interactions exert a smaller influence on the resonances and are treated in this formalism in an effective-medium way. These conclusions have been proposed by other authors for the case of fractal aggregates [2,8]. Our calculation produces a better fit of the absorption band widths and relative strengths than previous works [8].
3.2. Electrodynamic effects Another type of aggregation process can produce fused clusters as in the case of heat-treated composites. These systems can be considered as large metallic clusters with a reduced electronic mean free path [3]. In the visible region, the size of such systems could be beyond the reach of the electrostatic approximation. Because the presented model is based on an electrodynamic formalism, retardation and electrodynamic effects are already incorporated and the formalism can be used to study the coherent wave propagation in such systems [4]. Figure 3 shows the comparison between the MG results and those obtained using the present model for gold fused clusters with different sizes. The calculations were performed using the bulk dielectric function for gold [lo]. The average sizes of the particles used in the proposed model are 60 nm and 100 nm. The increase of the size of the particles to values where the electrostatic approximation is no longer applicable produces the increase of the absorption at lower energies, as expected [ 111.
4. Conclusions
A procedure to obtain the electromagnetic response of granular aggregated metals is presented based on a multiple scattering T-matrix formalism. The procedure has intrinsically electrodynamic effects and multipolar interactions between the metallic particles and can be used to study non-fused and fused aggregated systems.
L.F. Fonseca et al. I Physica
1
1.5
2 energy
A 207 (1994) 123-130
2.5
3
129
3.5
@V)
Fig. 3. Log (Yas a function of the energy, where Q = 47rKlh cm-’ and K is the imaginary part of the effective index of refraction. Simple dashed curve is the standard MG calculation. The double-dashed and solid curves represent calculations using particles 60 nm and 100 nm in size, respectively.
For the first type of aggregated systems, the approach considers short-range and long-range interactions. The short-range interactions are taken into account by obtaining the T-matrix of a metallic particle surrounded by its neighbors, and the long-range interactions are considered performing an effective-medium calculation. Fractal colloidal aggregates are complex arrays of low-density aggregates. The calculation of the absorption resonances of such systems shows that even though a complete description of the optical properties of fractal aggregates requires a much more detailed theory, the absorption behavior is in fact dominated by the short-range interaction between nearest neighbors. Improvement on previous results [12] has been obtained by performing an orientational average of the T-matrix of each scattering unit. Recently many systems have been produced experimentally in which the aggregation is induced and controlled [1,13]. Many of such systems have a higher density of metallic particles than the fractal colloidal aggregates and consequently the average number of nearest neighbors is larger. Even though we obtained good results for the optical absorption of a gold fractal aggregate by using a simple pair of particles as the average scattering unit, a more complex array should be used to describe the experimental results for the higher-density aggregates. The presented formalism can be used for such systems but the scattering unit should be chosen as an array of more than two particles. The number of particles composing the scattering unit is only limited by the computational capabilities available. For the case of fused aggregates the formalism can be applied for the description of the coherent wave propagation when the averaged size of the aggregates is beyond the applicability of the electrostatic theories.
130
L.F.
Fonseca et al. I Physica A 207 (1994) 123-130
Acknowledgement
The support of EPSCoR-NSF
grant EHR-9108775 is acknowledged.
References [l] R.P. Devaty and A.J. Sievers, Phys. Rev. B 41 (1990) 7421. [2] H.M. Lindsay, M.Y. Lin, D.A. Weitz, P. Sheng, Z. Chen, R. Klein and P. Meakin, Faraday Discuss. Chem. Sot. 83 (1987) 153. [3] W.A. Curtin and N.W. Ashcroft, Phys. Rev. B 31 (1985) 3287 and references. [4] V.K. Varadan, V.N. Bringi and VV. Varadan, Phys. Rev. D 19 (1979) 2480. [5] P.C. Waterman, Phys. Rev. D 3 (1971) 8525. [6] B. Peterson and S. Strom, Phys. Rev. D 8 (1973) 3661. (71 D.A. Weitz, M.Y. Lin and C.J. Sandroff, Surface Sci. 158 (1985) 147. (81 Z. Chen and Ping Sheng, Phys. Rev. B 39 (1989) 9816. [9] F. Claro and R. Fuchs, Phys. Rev. B 44 (1991) 4109, and references therein. [lo] P.B. Johnson and R.W. Christy, Phys. Rev. B 6 (1972) 4370. [ll] U. Kreibig, M. Quinten and D. Schoenauer, Phys. Ser. T 13 (1986) 84. [12] L. Fonseca, L. Cruz, W. Vargas and M. Gbmez, Condensed Mat. Theor. 8 (1993) 561. [13] D. Schonauer and U. Kreibig, Surface Sci. 156 (1985) 100.