Visible region polarization spectroscopic studies of template-synthesized gold nanoparticles oriented in polyethylene

Article abstract of DOI:10.1021/jp972869i

We have obtained visible range polarization spectra of template-synthesized gold nanoparticles oriented in polyethylene (PE). The plasmon resonance extinction bands observed with the incident electric field polarized parallel, perpendicular, and at interm

Full text of DOI:10.1021/jp972869i

J. Phys. Chem. B 1998, 102, 361-371
361
Visible Region Polarization Spectroscopic Studies of Template-Synthesized Gold
Nanoparticles Oriented in Polyethylene
Nathir A. F. Al-Rawashdeh,† Marie L. Sandrock, Carolyn J. Seugling, and Colby A. Foss, Jr.*
Department of Chemistry, Georgetown UniVersity, Washington, District of Columbia 20057
ReceiVed: September 3, 1997; In Final Form: NoVember 6, 1997
We have obtained visible range polarization spectra of template-synthesized gold nanoparticles oriented in
polyethylene (PE). The plasmon resonance extinction bands observed with the incident electric field polarized
parallel, perpendicular, and at intermediate angles to the direction of friction orientation are consistent with
the long axis of the particles being aligned with the gross orientation axis. For all particle sizes considered
(radii 16, 38, and 60 nm) the degree of linear dichroism increases with the amount of gold deposited in the
template synthesis step prior to extraction and orientation. The experimental spectra agree with the predictions
of the Rayleigh, Maxwell-Garnett, and dynamical Maxwell-Garnett theories only qualitatively. All of these
treatments fail to predict the dependence of the spectral extinction intensities on the polarization angle θ.
T-matrix scattering calculations suggest that the contribution from electric quadrupole modes cannot be ignored
in the 38 and 60 nm radius particles. The calculated θ-dependence of the extinction intensity does not resemble
experiment for 16 nm radius particles, but the theory-experiment comparison is more favorable for the larger
radius systems. Some possible models for these observations are discussed.
Introduction
to those predicted by the small particle limit theories. In other
words, the extinction in parallel polarization was observed to
The anodic aluminum oxide-based template synthesis method
has been applied in numerous studies to prepare micro- and
nanoscopic metal particles.1-11 The parallel arrays of cylindrical
pores in certain anodic aluminas allow for the preparation of
uniformly oriented rod- or needlelike metal structures. Fur-
thermore, since the oxide material is transparent over large
segments of the visible and infrared spectral regions, the
characterization of the optical properties of the metal nano-
structures has also been quite straightforward.1-3,5,6,8-10
A few years ago, it was shown that template-synthesized gold
particles can be removed from their oxide host and oriented in
polymer matrixes.8 Recently, we reported the results of a study
of the optical spectra of oriented Au particle/polyethylene (PE)
composite films, where both the size and the shape of the Au
particles were varied.10 In that study, we found that the gold
particle plasmon resonance extinction maxima for incident fields
be lower than in perpendicular polarization.
In this paper, we detail the procedures for preparing oriented
Au particle/PE films and the polarization spectra of these
composites. We then discuss the observed results in the context
of a more rigorous scattering treatment for nonspherical particles
developed by Barber and Hill.14 We also discuss the relative
virtues of the so-called dynamical Maxwell-Garnett model6
which attempts to incorporate particle size effects into a quasi-
static framework and has been used in the past to model
experimental data.6,9,15
Theoretical Background
For dilute solutions of particles, the transmittance T of the
composite medium can be expressed as12,16
T ≈ exp(-NCextd)
(1)
polarized parallel to the direction of particle orientation (λmax
-
(0)) were red-shifted relative to the extinction maxima for
perpendicular polarization (λmax(90)). In general qualitative
accord with simple scattering and effective medium theories,
the difference between λmax(0) and λmax(90) was found to
increase with the particle aspect ratio.
where N is the concentration of particles (m-3), Cext is the
extinction cross section (in m2), and d is the optical path length
(in m). The extinction cross section is the sum of losses arising
from absorption and scattering:
However, we were unable to explain the observed polarization
angle dependence of the extinction intensities. Small particle
limit treatments such as the Rayleigh scattering12 and Maxwell-
Garnett13 theories predict that the extinction intensity for an
incident electric field polarized parallel to the long particle axis
will be larger than for the perpendicular case.10 We found this
to be qualitatively correct in experiment only for the smallest
diameter (32 nm) gold particles. Larger diameter systems
exhibited extinction intensity-polarization angle trends opposite
Cext ) Cabs + Csca
(2)
For particles much smaller than the incident wavelength λ
(e.g., radius a e 0.1 λ),13 the absorption and scattering cross
sections can be related to the particle polarizability R via the
expressions12,16
Cabs ) k Im{R}
Csca ) (k4/6π)|R|
(3a)
(3b)
2
* Corresponding author.
† Current address: Department of Chemistry, Jordan University of
Science and Technology, Irbid, Jordan.
where k is the wavevector ()2p/λ), “Im” specifies the imaginary
S1089-5647(97)02869-1 CCC: $15.00 © 1998 American Chemical Society
Published on Web 01/08/1998

362 J. Phys. Chem. B, Vol. 102, No. 2, 1998
Al-Rawashdeh et al.
2
part of R, and |R| signifies the square modulus of R. The
nates. However, as the particle increases in size relative to λ,
the magnetic dipole and higher order electric and magnetic
multipole terms become important.
The calculation of extinction cross sections for nonspherical
particles is much more difficult. For ellipsoids of revolution
or otherwise axisymmetric particles, the treatment for Cext is
similar in spirit to that of spheres, namely14,20
polarizability is a function of the dielectric functions of the
particle (ꢀm) and host medium (ꢀ0) and also the size and shape
of the particle. In the small particle (or Rayleigh scattering)
limit, the complex polarizability of the particle is given by17
ꢀm - ꢀ0
V
R )
(4)
3q ꢀm + κꢀ0
2
∞
∞
Cext ) (2π/k2)
D
σmn Re{fσmn + gσmn
}
(9)
where V is the particle volume. The depolarization factor q
and screening factor κ are a function of particle shape. The
depolarization factor q for a general ellipsoid with semiaxes a,
b, and c can be found by solution of Dirichlet’s integral,18 but
a very close numerical approximation is the inverse axes ratio12
∑∑∑
σ)1m)0n)m
where fσmn and gσmn are the scattering coefficients for the
magnetic and electric fields, respectively, and σ, m, and n are
the expansion indices of vector spherical harmonic functions.16
Dσmn is a normalization constant.
1/x
qx ≈
(5)
1/a + 1/b + 1/c
The calculation of fσmn and gσmn is not trivial and has been
done for ellipsoids of revolution by the “separation-of-variables”
method20,21 and the so-called “T-matrix” method.14,22 For our
studies, we chose the latter because the source code is
commercially available14 and relatively easy to execute. The
details of the T-matrix theory are far beyond the scope of this
paper, and the reader is directed to refs 14 and 22. The key
point is that this general scattering treatment allows us to
consider the size dependence of the electric dipole resonance
and the contributions from higher multipoles that Rayleigh-limit
treatments do not address.
Mie scattering theory can be used to describe size and shape
effects for single particles but does not account for interparticle
interactions. On the other hand, effective medium type treat-
ments such as Maxwell-Garnett theory address the effects of
interparticle interactions via the volume fraction term fm but
assume that the particles are infinitely small relative to λ. In
an attempt to incorporate particle size effects into Maxwell-
Garnett theory, we previously adopted an expression for a size-
where x is equal to a, b or c, and qx is the depolarization factor
along the given axis of interest. The screening factor along a
given axis is then simply κx ) (1/qx - 1).13
For prolate ellipsoids of revolution, where a and b are the
semimajor and semiminor axes, respectively, the depolarization
factors of relevance are those parallel and perpendicular to the
axis of revolution, and are given by
1
q| ≈
(6a)
(6b)
1 + 2(a/b)
1
q ≈
2 + b/a
For composite systems where the particle density is high and
interparticle electromagnetic interactions can occur, it is con-
venient to define an average complex dielectric constant ꢀ, which
is a function of the dielectric constants of the particles and their
host medium, as well as the volume fraction (fm) and shape of
the particles. Perhaps the most familiar theory is that of
Maxwell-Garnett, where ꢀ can be found by solution of the
equation13,19
23,24
dependent depolarization factor qeff
qeff ) q - 1/3k2b2 - i2/9k3b3
(10)
where b is either the spherical radius or some characteristic
dimension of the particle.6 The second term on the right-hand
side of eq 10 is the dynamic depolarization factor. The third
term accounts for damping associated with the emitted fields
of the oscillating particle.23.24
When qeff is incorporated into Rayleigh or Maxwell-Garnett
theory (via an effective screening factor keff ) 1/qeff - 1), an
increase in particle size engenders a red shift and damping of
the particle plasmon resonance band. While such size effects
are seen in experiment, calculated spectra based on eq 10 and
Maxwell-Garnett theory tend to overestimate the red shift with
increasing particle size.6 It should also be emphasized that eq
10 attempts only to address the size dependence of the electric
dipole response. Magnetic and higher order electric terms are
ignored.
ꢀ - ꢀ0
ꢀ + κꢀ0
ꢀm - ꢀ0
) f
(7)
m ꢀm + κꢀ0
The refractive index n and absorption coefficient k are related
to the average dielectric constant via ꢀ ) (n + ik)2. Equation
7 does not explicitly address particle size, but like the Rayleigh
expressions for Cext, the underlying assumption is that the
particles are negligibly small relative to the incident wave-
length.19
Since the gold particles examined in our studies are not
negligibly small, it is necessary to consider more rigorous
scattering theories which apply to particles of any size. For
dilute collections of spheres, the Mie theory expression for the
extinction cross section is appropriate:12,16
∞
Cext ) (2π/k2) (2n +1) Re{a + b }
(8)
Experimental Section
∑
n
n
n)1
Porous Anodic Alumina Film Preparation. Prior to
anodization, aluminum plates (7 cm × 7 cm × 1.00 mm,
Aldrich, 99.999%) were electropolished at ca. 0.15 A cm-2
in a 75-80 °C mixture of sulfuric and phosphoric acid (60:40
v/v mixture of 96% w/v sulfuric acid (Mallinckrodt) and
85% w/v phosphoric acid (Fisher ACS)). The currents were
typically applied for 3 min, followed by prompt removal and
where an and bn are the scattering coefficients for the electric
and magnetic fields, respectively, and the index n indicates the
type of induced oscillation (n ) 1 for dipole, n ) 2 for
quadrupole, etc.). The coefficients an and bn can be calculated
in a straightforward manner using Riccati-Bessel functions.12,16
For very small particles, the electric dipole term (a1) predomi-

Gold Nanoparticles Oriented in Polyethylene
J. Phys. Chem. B, Vol. 102, No. 2, 1998 363
rinsing of the Al plates in distilled water. Alumina films
containing 38 nm radius pores were prepared by anodizing
the polished aluminum plates at 50 V in ca. 17 L of 6%
oxalic acid (Mallinckrodt Analytical Reagent). The cathode
was a 9 cm × 5 cm × 1 mm lead foil. Voltage was supplied
by a Sorensen DCRB-300 dc power source. Within the cell,
stainless steel coils connected to an external circulating refrig-
erator bath (1:1 ethylene glycol/water mix) maintained the
anodizing cell temperature between 0 and 3 °C. The cell
solution was stirred gently with a helical glass stirrer. Sixty
nanometer pore films were prepared in a similar manner, except
that the applied voltage was 90 V, and the electrolyte was 2%
oxalic acid. Anodization for 12-14 h yielded films of 40-50
µm total thickness. The films were removed from their Al
substrate by voltage reduction25 and immersion in 15% phos-
phoric acid.
Alumina films containing 16 nm radius pores were prepared
under the same conditions described above, except that the
electrolyte was 4% sulfuric acid, and the applied voltage was
20 V. In this case, the dc voltage was from a Kepco ABC-
25-4 dc power supply. Also, the voltage reduction was followed
by immersion of the anodized Al plates in 25% sulfuric acid.
Figure 1. Extraction and orientation of template-synthesized metal
particles. (A) Impregnation with polyethylene (PE) under heat and
pressure. (B) metal particle/PE composite structure after dissolution
of alumina host in aqueous NaOH. (C) Friction orientation of metal
and PE nanostructures.
The template synthesis of gold particles within the pores of
the anodic alumina films has been described in detail
elsewhere.3-6 Briefly, we used an Anatech Hummer 10 to
sputter the barrier side of the porous films with a silver layer
ca. 40-50 nm thick. The sputtered films were then clamped
between an Al foil-covered glass plate and an O-ring glass joint
(2.0 cm diameter) which served as the plating cell base. The
Al foil served as the working electrode contact. The reference
and counter electrodes were an SSCE26 and platinum gauze strip,
respectively. The plating solutions were stirred by gentle
bubbling of Ar or nitrogen. Ag and Au were deposited
potentiostatically. An EG&G PARC 273 potentiostat was used
to supply the voltage and to monitor the amount of charge passed
during deposition. A silver foundation was first prepared by
depositing 2-4 C from a silver thiocyanate plating solution27
potentiostatically at -0.6 V vs SSCE. The silver plating
solution was then removed and the cell rinsed thoroughly. Gold
structures were then prepared by potentiostatic deposition of
gold from an Au(I) cyanide plating solution (Technic, Inc.,
Orotemp 24). Once the desired amount of gold was deposited,
the silver foundation was removed by placing the composite
film in concentrated nitric acid (Mallinckrodt, 92%). The Au/
alumina composite films were then rinsed and allowed to dry
in air.
Figure 2. Polarization spectroscopy setup. The polarization angle θ
is defined as 0° when the electric field is polarized parallel to the particle
orientation direction. θ ) 90° when the electric field is perpendicular
to the particle orientation direction.
thick layer of polyethylene (Polysciences) is placed on the
barrier side of the Au particle/alumina composite film. (The
Au particles are closer to the barrier side.) The two films are
then sandwiched between microscope slides and placed on a
heating plate. A wood block and ca. 2 kg of weight are placed
on the microscope slide/film sandwich. Heat is then applied
sufficient to melt the polyethylene and drive it into the pores to
contact the Au particles (Figure 1A). After ca. 30 min of
heating, the Au/alumina/PE composite is detached from the
microscope slides and immersed in 1-2 M aqueous sodium
hydroxide. This base solution dissolves the aluminum oxide,
leaving fibrils of PE containing the Au particles (Figure 1B).
The Au particles are then oriented by brushing the fibrous
surface with a soft rubber blade (Figure 1C).
Visible polarization spectra were measured using a Hitachi
U-3501 spectrometer equipped with a 210-2130 polarizer
accessory. Figure 2 shows a schematic of the optical setup.
All spectra were obtained at normal incidence with the polariza-
tion angles defined as θ ) 0° when the incident electric field is
parallel to the direction of particle orientation and θ ) 90° when
the field is perpendicular to the particle main axes.
To characterize the dimensions of the radius template-
synthesized gold particles prepared in the 20 V alumina films,
we removed the particles from their host oxide. Small sections
of the Au/alumina composites were placed on a track-etched
polycarbonate filter membrane (Nuclepore 0.03 µm pores).
Dilute aqueous NaOH was introduced to the composite film
dropwise while the filter membrane was under suction. When
the aluminum oxide material appeared to be dissolved, distilled
water was added dropwise to rinse the particles. With the
suction off, portions of the liberated particles were transferred
to Formvar-coated Cu grids (200 mesh, EM Sciences) by adding
to, and then removing from, the filter surface small drops of
distilled water. The Cu grids were then allowed to dry at room
temperature. Transmission electron microscope (TEM) images
were obtained using a JEOL 1200 EX.
The procedure for removing the template-synthesized particles
from their oxide host and their orientation in the polyethylene
matrix are summarized schematically in Figure 1. A 20 µm

364 J. Phys. Chem. B, Vol. 102, No. 2, 1998
Al-Rawashdeh et al.
TABLE 1: Gold Nanoparticle Dimensions from
Transmission Electron Microscopic Measurements
host film
anodization
voltage (V)
gold deposition
amount
particle
radiusb
(nm)
particle
length
2a (nm)
aspect
ratio
(a/b)
(coulombs)a
20
20
20
20
50
50
50
90
90
90
0.20
0.30
0.50
1.20
0.30
0.50
0.70
0.30
0.50
1.20
16 ( 0.5
16 ( 0.5
16 ( 0.5
17 ( 2
38b
33 ( 1
40 ( 5
67 ( 7
220 ( 20
120d
1.0
1.3
2.2
6.3
1.6
2.6
3.4
2.1
3.4
8.2
38b
200d
38 ( 2c
60b
256c
250d
60b
410d
60b
980d
a Deposited over a 3.14 cm2 area. b Radius taken from ref 6 TEM
measurements on particles grown with different Au loadings, but in
porous hosts prepared under identical conditions. c Radius or length
taken from ref 6 for particles grown under identical conditions for both
host film preparation and Au loading. d Particle length estimated from
least-squares analysis of data in ref 6.
Results
In Table 1 we summarize the measured dimensions of the
template-synthesized Au particles as determined by TEM
measurements. For the 16 nm radius particles prepared in our
lab, the results are based on the observation of particles filtered
onto polycarbonate membranes. The data for particles grown
in the 50 and 90 V anodic aluminas are taken from ref 6 or
from interpolation of plots of aspect ratio versus the number of
coulombs deposited.28 From these TEM data, it is clear that
we have a systematic relationship between the amount of Au-
(I) deposited into the porous oxides and the aspect ratio of the
resulting Au particles. Unfortunately, we have not been
successful in obtaining TEM images of the Au particles within
the PE host. Since we are aware that the PE extraction and
orientation procedure may cause breakage of the particles, we
will refer to Au deposition amounts rather than absolute particle
dimensions in the discussion below.
Figure 3 shows a series of polarization spectra of oriented
Au particle/PE composites, where the Au particles were prepared
in 20 V anodic films. The “0.10 C”, “0.30 C”, etc., pertain to
the amount of Au(I) electrodeposited over the 3.14 cm2 working
electrode area in the template synthesis step. The numbers on
the spectral curves are the polarization angles (θ) as defined in
Figure 2. At the lowest level of Au deposition (0.10 C, Figure
3A), the polarization dependence of both λmax and the extinction
intensity is weak. However, as the amount of deposited gold
increases, the dependence of both the peak position and intensity
on the incident polarization becomes more pronounced (Figure
3B,C). At the largest Au deposition amount (1.0 C, Figure 4)
the difference in λmax between θ ) 0° and θ ) 90° polarizations
cannot be determined with certainty because of the working
range of the polarizer.29 However, it is clear that the λmax for
θ ) 0° has been shifted into the near-infrared. All of the
polarization spectra exhibit an isosbestic point whose wavelength
increases with the amount of Au deposited during the particle
preparation step.
Similar trends are seen for Au/PE composites prepared from
larger radius Au particles. Figure 5 shows the polarization
spectra of 38 nm radius Au particles oriented in PE. At low
Au loadings (0.3 C, Figure 5A) the polarization effects are weak.
At higher deposition amounts, the dependence of λmax on θ
becomes stronger (B and C). Unlike the 16 nm radius systems
where the extinction is much greater for parallel polarization,
the extinction intensities for the 38 nm radius particles at θ )
Figure 3. Polarization spectra of 16 nm radius Au particles oriented
in PE. The aspect ratios (a/b) are estimated from TEM images (see
text). Numbers on curves are the polarization angle θ. (A) 0.10 C Au-
(I). (B) 0.30 C Au(I). (C) 0.50 C Au(I). All spectra are measured against
air. Features at ca. 850 nm are detector change artifact.
0° and θ ) 90° are similar. The bands at θ ) 0° are also much
broader than in the case of the 16 nm radius Au particle
composites.
Finally, Figure 6 shows polarization spectra of 60 nm radius
Au particles oriented in PE. For small Au deposition amounts,
there is a blue shift and decrease in extinction as the polarization
angle changes from 0 to 90 (Figure 6A). However, at larger
deposition amounts, the parallel polarization spectra show
extinction intensities lower than in the case of perpendicular
polarization (Figures 6B,C).
Discussion
For all of the oriented Au particle/PE composite systems
considered here, we observe an increase in the degree of

Gold Nanoparticles Oriented in Polyethylene
J. Phys. Chem. B, Vol. 102, No. 2, 1998 365
Figure 4. Polarization spectra of 16 nm radius Au particles in PE, at
1.2 C Au(I) loading. Numbers on curves have same meanings as in
Figure 3. Aspect ratios (a/b) are estimated from TEM images. Dashed
line on θ ) 0° spectrum is probable profile beyond the polarizer
working range.
dichroism as the amount of deposited gold in the template
synthesis step is increased. This is consistent with an increase
in the Au particle aspect ratio. For the smallest radius systems
(16 nm), the extinction intensity is higher at θ ) 0° relative to
θ ) 90° for all aspect ratios. This intensity dependence on θ
is diminished in the 38 nm radius system, where the extinctions
at θ ) 0° and 90 are nearly the same. For the largest radius
systems (60 nm), the θ ) 0° intensity is greater than the θ )
90° case only for the smallest Au deposition amounts. For larger
deposition amounts and particle aspect ratios, there is an
apparent extinction trend reversal.
The dependence of the extinction maximum on polarization
angle can be understood qualitatively in terms of eq 4 and the
particle dipolar plasmon resonance condition Re{ꢀm} ) -Re-
{κꢀ0}.24 Since the real component of the gold dielectric constant
ꢀm is negative and decreases with wavelength in the visible
spectrum, the value of κ determines the wavelength of maximum
extinction. When the incident electric field is polarized parallel
to the main axis of a rod or prolate ellipsoid, the corresponding
screening parameter κ is large and leads to a long wavelength
extinction maximum. On the other hand, when the field is
perpendicular to the particle axis, κ is small (and approaches
unity for infinitely long cylinders), thus leading to an extinction
maximum at shorter wavelengths.
Figure 5. Polarization spectra of 38 nm radius Au particles oriented
in PE. Numbers on curves are the polarization angle θ. Aspect ratios
(a/b) are estimated from TEM images. (A) 0.30 C Au(I). (B) 0.50 C
Au(I). (C) 0.70 C Au(I). All spectra are measured against air. Features
at ca. 850 nm are detector change artifact.
The isosbestic point seen in all of the experimental data
(except Figure 5A) is highly suggestive of a polarization angle
dependence of Cext such as
taken from Johnson and Christy.30 For the polyethylene host,
we assume a refractive index and absorption coefficient of 1.40
and 0, respectively. The experimental refractive index of PE
is actually closer to 1.5,31 but the process of extraction and
orientation probably lead to a less dense host medium for the
Au particles.
Comparison with Rayleigh and Maxwell-Garnett Theo-
ries. Figures 7 and 8 show polarization spectra calculated using
Rayleigh and Maxwell-Garnett theories, both of which inherit
eq 4’s assumptions regarding particle size (a, b , λ) and polar
response (dipole only). In Figure 7, the Rayleigh calculations
(plotted as Qext ) Cext/πaˆ2)32 show that the extinction maximum
for incident fields polarized along the particle axis (θ ) 0°)
occurs at a longer wavelength than the maximum for perpen-
dicular polarization (θ ) 90°). The greater the particle aspect
Cext(θ) ) Cext(θ)0) cos2 θ + Cext(θ)90) sin2 θ (11)
which can be derived from the general expressions for small
ellipsoids discussed by Bohren and Huffman.16 Equation 11
not only predicts an isosbestic point, but predicts that a spectrum
of system of nonuniformly oriented particles should show
contributions from both long- and short-axis polarization modes
even at θ ) 0° and θ ) 90°. Figures 3C and 4 show such
contributions, demonstrating the limitations of our particle
extraction/orientation procedure.
To understand the dependencies of λmax and the total
extinction on the polarization angle θ, we calculated extinction
spectra using Rayleigh, Maxwell-Garnett, and T-matrix scat-
tering treatments. In all simulations, the gold optical data are

366 J. Phys. Chem. B, Vol. 102, No. 2, 1998
Al-Rawashdeh et al.
Figure 7. Rayleigh theory calculations (eqs 2-4) for Au ellipsoids
with aspect ratios 1.5, 2.0, and 3.0. Solid curves are extinction curves
for θ ) 0° . Dashed curves correspond to θ ) 0° . The host medium
is assumed to be nonabsorbing, with n ) 1.4.
Figure 8. Maxwell-Garnett calculations of extinction spectra for Au
ellipsoids with the same aspect ratios as given in Figure 7. The metal
volume fraction fm is equal to 0.10, and the optical path length is 0.10
µm. The solid and dashed curves are spectra for θ ) 0° and 90°,
respectively.
experimentally observed combination of large changes in λmax
and moderate changes in extinction with θ.
The first suspect in the consideration of experiment-theory
discrepancies is particle size. As discussed in the Theory
section, both Rayleigh and MG theories assume that the
scattering particle is much smaller than the incident wavelength.
Indeed, the radius of the template synthesized gold particles is
much smaller than λ in the visible spectrum, but the long axis
in many cases is not (see Table 1). As a first step in examining
the possible relationships between polarization angle-dependent
intensities and particle size, we simulated extinction spectra
using a dynamic Maxwell-Garnett model where qeff is given
by eq 10. Figure 9 A shows a series of calculated polarization
spectra for 16 nm radius Au particles and the same series of
aspect ratios considered in Figures 7 and 8. While there is a
slight decrease in the ratio Amax(θ)0°)/Amax(θ)90°) relative to
the simple MG results, the comparison with experiment is still
poor.
Figure 6. Polarization spectra of 60 nm radius Au particles oriented
in PE. Numbers on curves are the polarization angle θ. Aspect ratios
(a/b) are estimated from TEM images. (A) 0.30 C Au(I). (B) 0.50 C
Au(I). (C) 1.2 C Au(I). All spectra are measured against air. Features
at ca. 850 nm are detector change artifact.
ratio, the greater the difference in the spectra at θ ) 0° and θ
) 90°. The trends in λmax with θ and a/b are the same for
spectra calculated using Maxwell-Garnett (MG) theory (Figure
8). At a volume fraction fm ) 0.10, interparticle interactions
do not affect the peak positions greatly. However, the bands
are significantly broadened relative to the Rayleigh calcula-
tions.
The Rayleigh and MG calculated spectra are similar to
experiment only with general regard to the polarization depen-
dence of λmax. Significantly, the extinction intensity ratios (Amax
-
(θ)0° )/Amax(θ)90°) or Qext,max(θ)0°)/Qext,max(θ)90°)) are
consistently much higher in theory than in experiment. Using
either model, we are unable to find an aspect ratio (or
combination of a/b and fm in the MG case) that yields the
Since the particle size adjustments according to eq 10 were
found to overestimate the red shifts in λmax as the particle radius
is increased,6 we also consider another form of qeff, in which
the dynamic depolarization factor is omitted and the radiation-

Gold Nanoparticles Oriented in Polyethylene
J. Phys. Chem. B, Vol. 102, No. 2, 1998 367
Figure 10. Normalized extinction coefficient spectrum calculated for
16 nm particles using the T-matrix method. Numbers on curves are
polarization angles. (A) Aspect ratio ) 1.3. (B) Aspect ratio ) 2.2.
The host medium is assumed to be nonabsorbing, with n ) 1.40.
Figure 9. Polarization spectra calculated using dynamic Maxwell-
Garnett treatments for 16 nm radius Au ellipsoids with aspect ratios as
given in Figures 7 and 8. The metal volume fraction fm is equal to
0.10, and the optical path length is 0.10 µm. The solid and dashed
curves are spectra for θ ) 0° and 90°, respectively. (A) Spectral
calculations based on eq 10 definition of qeff, with b ) the particle
semimajor axis for θ ) 0° and b ) 16 nm for θ ) 90°. (B) Calculations
based on radiation damping adjustment only (see text).
cylinders. The experimental particles are neither cylinders nor
ellipsoids, but we chose the latter geometry because the
computations are somewhat easier. Furthermore, the calculated
spectra for the two geometries (holding radius and aspect ratio
constant) seem to be very similar, except that the extinction
maxima for cylinders is somewhat red-shifted relative to the
case of ellipsoids. All T-matrix calculations were performed
on a Dell OptiPlex GXMT 5133 personal computer. Calcula-
tions of Qext were done at 20 nm intervals of the visible
spectrum, and smoothed curves through these points were
generated with Microsoft Excel version 7.0 for Windows.
Figure 10 shows a set of polarization spectra calculated for
16 nm radius gold ellipsoids in a nonabsorbing host medium
of refractive index n ) 1.4. In Figure 10A, the results for an
ellipsoid of aspect ratio 1.3 are shown. Compared to Figure
3B, which is a set of spectra for particles whose aspect ratios
may be as large as 1.3, the theoretical spectra show a smaller
change in λmax between θ ) 0° and θ ) 90° and a larger
difference in extinction intensity. However, the change in
extinction with θ is much less than predicted by the Rayleigh
or simple MG treatments. For an aspect ratio of 2.2, the
calculated spectra in Figure 10B are very close to the experi-
mental results (Figure 3C) in terms of λmax. The extinction
intensity change with θ is still more severe than seen in
experiment.
damping term is simply related to particle the volume:33
qeff ) q - i(4π2V/3λ3)
(12)
As shown in Figure 9B, the omission of the dynamic
depolarization term reduces the λmax values for the θ ) 0°
polarization spectra of a given particle size and aspect ratio but
does not bring the relative intensities in the two polarizations
closer to the experimental results.
T-Matrix Scattering Calculation Results. Spectra calcu-
lated using the Maxwell-Garnett model modified for particle
size effects do not resemble experimental data, even for the
smallest particle systems which should be the most amenable
to such approximate treatments. Thus, we consider the predic-
tions of the more rigorous T-matrix scattering treatment. As
discussed in the Theory section, the full scattering treatment
allows us to consider the size dependence of electric dipole
modes, as well as contributions of other electric and magnetic
polarization modes. We should note that, in our calculation
work, we were unable to achieve numerical convergence for
aspect ratios greater than about 2.6 for the larger radius systems.
However, the calculated spectra for particles that deviate only
moderately from the spherical geometry are quite revealing.
Also, we note that the T-matrix routines written by Barber and
Hill are amenable to both ellipsoids of revolution and finite
In Figure 11, the calculated polarization spectra for 38 nm
radius Au ellipsoids with aspect ratios 1.3, 1.56, and 2.6 are
shown. For the 1.3 and 1.56 aspect ratio systems (Figures
11A,B), the spectra are dominated by particle size and θ-de-
pendent dipolar resonances which are much broader than in the

368 J. Phys. Chem. B, Vol. 102, No. 2, 1998
Al-Rawashdeh et al.
5B) compare more favorably with the calculated results for a/b
) 1.3 (Figure 11B). While we were not able to obtain reliable
results for an aspect ratio of 3.6, the experimental spectra in
Figure 5C may in fact arise from shorter particles (i.e., a/b )
1.6-2). These results may be consistent with the template-
synthesized Au particles being broken during the extraction and
orientation procedure.
Another plausible interpretation of the experimental data for
38 nm radius Au particles in Figures 5B,C is that the long axis
resonance component is actually a mixture of electric dipole
and quadrupole modes, with a predominance of the latter. In
the T-matrix calculation for an aspect ratio of 2.6, the quadrupole
resonance maximum occurs at approximately the same wave-
length (640 nm) as the λmax at θ ) 0° in the experimental
spectrum (Figure 5B). It is also significant that the intensity
of the quadrupolar resonance is smaller than a dipole mode in
the same wavelength range (i.e., for a particle of identical radius
but lower aspect ratio); this may explain the lower extinction
intensity seen in the experimental spectra at θ ) 0°.
A
distribution of aspect ratios, originating in the template synthesis
step and possibly exacerbated in the extraction/orientation
procedure, would then account for the experimental plasmon
resonance bands being much broader than the calculated results.
In Figure 12, we show the T-matrix results for Au ellipsoids
of radius 60 nm. For particles with a/b ) 1.3, the θ ) 0°
resonance consists of a broad dipole band centered at ca. 760
nm and a quadrupole mode at ca. 550 nm (Figure 12A). The
θ ) 90° spectrum shows a very broad dipole mode centered at
about 625 nm and a quadrupole mode at ca. 540 nm. As the
aspect ratio increases to 1.6 and 2, the quadrupole contribution
to the θ ) 90° extinction vanishes, and the dipole mode shifts
to shorter wavelengths (540-550 nm). At the same time, the
dipole mode in the θ ) 0° maxima red shifts into the near-
infrared with increasing aspect ratio, and a weak quadrupole
band persists near 600 nm (Figures 12B,C).
At first inspection of the experimental spectra of 60 nm radius
particles in Figure 6A, it would seem that the slight change in
λ
max and extinction intensity with polarization angle is indicative
of a particle aspect ratio near unity. However, as can be seen
from the calculated spectra in Figure 12A, where the particle
aspect ratio is 1.3, we should expect the θ ) 90° maximum to
appear as much broader band centered at a longer wavelength
(ca. 625 nm) than is seen in experiment (550 nm). To examine
this further, we have also calculated the spectra for a 60 nm
radius particle of aspect ratio 1.1 and find that the θ ) 90°
maximum is ca. 610 nm, a value still well above the experi-
mental λmax. Based on the position of the experimental λmax
for the θ ) 90° polarization spectrum shown in Figure 6A, it
would appear that the particles are on average distinctly
nonspherical. The θ ) 0° experimental spectrum, whose visible
region maximum occurs at about 610 nm, may then be due to
the quadrupole mode of a more elongated particle.
Figure 11. Normalized extinction coefficient spectrum calculated for
38 nm particles using the T-matrix method. Numbers on curves are
polarization angles. (A) Aspect ratio ) 1.3. (B) Aspect ratio ) 1.56.
(C) Aspect ratio ) 2.60. The host medium is assumed to be
nonabsorbing, with n ) 1.40.
case of the 16 nm radius systems. The extinction intensity
differences between the θ ) 0° and θ ) 90° spectra are also
considerably diminished, though still greater than seen in
experiment. At an aspect ratio of 2.6, the a-axis dipolar
resonance is severely red-shifted, and a second (electric quad-
rupole) resonance appears at ca. 640 nm (Figure 11C).
In comparing the T-matrix results for 38 nm radius Au
particles with the experimental data (Figure 5), it would seem
that the average aspect ratios of the Au particles in the PE matrix
are lower than indicated by the TEM data given in Table 1.
For example, the experimental spectra in Figure 5A pertain to
particles whose aspect ratios are supposedly ca. 1.6. Yet, the
slight change in λmax with θ suggests more spherelike particles.
Similarly, the experimental data for oriented Au particles whose
aspect ratios are, according to the TEM data, ca. 2.6 (Figure
The experimental spectra for higher gold loadings (Figures
6B,C) compare reasonably well with the calculated spectra in
Figure 12B,C. The experimental θ ) 90° maxima occur
between 530 and 550 nm. The experimental θ ) 0° spectra
show a lower extinction intensity than their θ ) 90° counter-
parts. For example, in Figure 6C, there is no observed extinction
maximum in the visible range for θ ) 0° polarization. Rather,
there is simply a sloping plateau between 600 and 850 nm (the
upper limit for the polarizer). In the T-matrix results for aspect
ratios 1.6 and 2, the θ ) 90° maxima occur at wavelengths
near 550 nm, and we see long axis dipole resonances red-shifted
into the near-IR, leaving a broad region of lower extinction

Gold Nanoparticles Oriented in Polyethylene
J. Phys. Chem. B, Vol. 102, No. 2, 1998 369
) 0° maxima are still somewhat more intense than the θ ) 90°
maxima. For the largest (60 nm) radius particles, even modest
aspect ratios of 1.6 and 2 lead to θ ) 0° dipole resonances
whose maxima occur well into the near-IR, thus leading to a
large region of low extinction in the visible and an overall
pattern that is quite like the experimental results.
Scattering Contributions from the Host Medium and
Other Effects. The spectral calculations to this point have
addressed only the extinction from the particles themselves.
Since the scattering calculations do not completely resolve the
questions regarding the polarization dependence of the extinction
intensities (especially for the 16 nm radius systems), it is useful
to consider the contributions from the polyethylene matrix. A
thin PE film of uniform thickness does not absorb significantly
in the visible spectrum, but the Au particle extraction and
orientation procedure certainly imparts roughness to the polymer
host which can lead to wavelength-dependent scattering losses.
Since the scattering coefficient for a simple dielectric material
like polyethylene will increase with decreasing wavelength, we
consider the possibility that the theory-experiment discrepancy
regarding the θ-dependence of the extinction intensities may
be due to background scattering. The extinction intensity of
the θ ) 90° bands, which occur at lower wavelengths, would
be enhanced by background scattering more than the θ ) 0°
bands, which occur at longer wavelengths.
While we know that portions of the PE host film is rendered
fibrous by the extraction procedure and that these fibers are
oriented during the Au particle orientation step, we do not
observe polarization angle-dependent background scattering in
the visible region. (The spectra at different θ’s seem to converge
in the higher energy region above the plasmon resonance bands.)
Thus, we assume that the background scattering is due to spheres
whose radii correspond to that of the aluminum oxide pores.
The refractive index and absorption coefficient were assumed
to be constant at 1.50 and 0, respectively, over the 400-900
nm spectral interval.
Figure 13A shows the θ ) 0° and θ ) 90° spectra for a 16
nm radius Au ellipsoid of aspect ratio 2.2. We assume an Au
particle number density of 880 µm-3 and an Au/PE composite
path length of 50 nm. To elevate the q ) 90 extinction band
to an appreciable extent, the PE sphere number density is taken
as 105 µm-3 and path length 7.7 µm. From these calculated
spectra, it is clear that background scattering may explain the
overall increase in extinction with decreasing wavelength seen
in experiment. However, the θ ) 90° resonance band is also
quite distorted, and increasing the contribution from background
scattering to bring this band closer in intensity to the θ ) 0°
resonance only results in washing out the θ ) 90° band
completely.
Figure 12. Normalized extinction coefficient spectrum calculated for
16 nm particles using the T-matrix method. Numbers on curves are
polarization angles. (A) Aspect ratio ) 1.3. (B) Aspect ratio ) 1.6.
(C) Aspect ratio ) 2.0. The host medium is assumed to be nonabsorb-
ing, with n ) 1.40.
Figure 13B shows a similar calculation for a 38 nm radius
Au ellipsoid of aspect ratio 1.56. The Au particle density is
assumed to be 93 µm-3 and the composite path length 100 nm.
The PE sphere number density is taken as 2600 µm-3, with a
path length of 0.85 µm. The addition of background scattering
brings the overall appearance of the spectra closer to experiment.
However, the relative intensities of the θ ) 0° and θ ) 90°
bands cannot be brought into closer agreement with experiment
simply by increasing background contributions without severe
distortion of the spectra.
between ca. 600 and 800 nm. While we did not calculate spectra
for aspect ratios greater than 2, the contribution from the dipole
mode can be expected to diminish in the visible range as the
aspect ratio is increased further.
A key trend in the T-matrix calculated spectra is that the
difference in extinction intensity between the θ ) 0° and θ )
90° polarization cases decreases as the particle radius increases.
Spectra calculated for 16 nm radius gold ellipsoids do not
resemble the experimental results and, indeed, are not very
different from spectra simulated using the Rayleigh and
Maxwell-Garnett theories. For 38 nm radius particles, the
calculated extinction intensities in the θ ) 0° and θ ) 90°
polarizations become more similar in magnitude, though the θ
In a recent note, we described the preparation and optical
properties of dichroic films prepared via friction orientation and
pyrolysis of alkanethiol-coated gold particles on Teflon-coated
glass slides.34 In that study, we found a similar discrepancy
between the θ-dependent extinction seen in experiment and that

370 J. Phys. Chem. B, Vol. 102, No. 2, 1998
Al-Rawashdeh et al.
incorporates both ET and particle size effects into an effective
mean free lifetime; as the probability of electron transfer
increases, the mean free lifetime decreases. As the particle size
decreases, the influence the surface ET process on the measured
spectrum increases.
The connection between Persson’s theory and our experi-
mental results can be seen when one considers the wavelength
and particle size dependence of surface electromagnetic en-
hancement. If ET occurs between the metal particle and the
host medium, then it is likely to be subject to local field
enhancements. It is well-known that electromagnetic enhance-
ments for gold are large below ca. 2 eV (620 nm) but are
diminished above this energy.37-39 In the case of our template-
synthesized Au particles, surface electromagnetic enhancements
would be strongest for θ ) 0° polarization where the plasmon
resonance maximum occurs at longer wavelengths. Considering
only the polarization of the gold particle (as we did in all of
the simulated spectra discussed above), we would always expect
the θ ) 0° dipole resonance to show a higher extinction than
the θ ) 90° case. However, if the former engenders a stronger
surface-enhanced ET, then its extinction will be self-limiting.
While we currently are able to consider its implications in
only in a qualitative way, the surface-enhanced ET model is
quite reasonable when one considers the comparison of experi-
mental and T-matrix spectra for Au particles of different sizes.
With regard to the θ-dependence of the extinction intensities,
the greatest discrepancy between theory and experiment is seen
for the smallest (16 nm radius) particles. Considering the
electrodynamic effects discussed by Meier and Wokaun23 and
Zeman and Schatz,24 it is our smallest radius particles that should
exhibit the highest surface enhancements. As the particle size
increases, radiation damping attenuates electromagnetic en-
hancements.23,24 Consistent with this expectation, as the particle
radius increases, the comparison between the experimental
polarization spectra and the T-matrix calculations improves; the
calculated extinction intensity difference between the θ ) 0°
and θ ) 90° dipolar resonances is diminished, and electric
quadrupole bands that arise for the higher aspect ratio particles
are quite weak. Thus, for the largest particles the θ-dependence
of the extinction intensities is accounted for by a rigorous
scattering treatment (the T-matrix method). The smaller
particles, which produce the higher surface electromagnetic
enhancements, may have a more complex relationship with their
host medium.
Figure 13. T-matrix scattering calculations for Au ellipsoids with
contributions from host medium scattering. Numbers on curves pertain
to polarization angle θ. (A) 16 nm radius Au ellipsoids of aspect ratio
2.2, with contributions from 16 nm radius spheres of refractive index
n ) 1.50. (B) 38 nm radius Au ellipsoids of aspect ratio 1.56, with
contributions from 38 nm radius spheres of refractive index n ) 1.50.
See text for details.
predicted by effective medium theory when the host medium
was assumed to be nonabsorbing. However, when the host
medium was modeled as a Lorentz oscillator with a strong
resonant mode near the plasmon resonance bands of the gold
structures, the θ-dependence of the extinction intensity was
subdued, and the simulated spectra better resembled experiment.
In that study, the nature of the host medium after flame treatment
was unknown, and an absorbing host model seemed appropriate.
For the composite films described in this paper, where the host
medium is polyethylene, such an interpretation would be difficult
to justify.
Conclusions
We have prepared dichroic composite materials by removing
template-synthesized gold nanoparticles from their alumina hosts
and friction-orienting them in polyethylene. The extent of
dichroism, as evaluated by the polarization angle dependence
of the plasmon resonance maxima (λmax) and extinction inten-
sites, increases systematically with the amount of gold deposited
into the porous anodic alumina hosts during the template
synthesis step. For particles that are distinctly rodlike (i.e., with
aspect ratios greater than unity), the extinction maximum for
incident electric fields polarized along the long axis occurs at
a longer wavelength than the λmax for polarization along the
particle radius.
The observed relationships between λmax, the polarization
angle θ, and the particle aspect ratio are in qualitative accord
with the predictions of the Rayleigh scattering and Maxwell-
Garnett theories. However, the observed dependence of the
extinction intensities on θ cannot be explained by these simple
treatments which assume that the metal particle is much smaller
We also considered the possibility that the gold optical data30
we used in the modeling studies were somehow inappropriate
for our nanoscopic structures. For example, the electron mean
free lifetime τ decreases with particle size because of surface
scattering.24 To examine these effects, we adjusted the free-
electron component of the experimental optical data using the
procedure discussed by Granqvist and Hunderi35 and then used
the adjusted data to model the extinction spectra. Lowering τ
damped the extinction intensity of both the θ ) 0° and θ )
90° bands but did not significantly change their relative
intensities.
An intriguing possibility may be inferred from the work of
Persson,36 who discusses the damping of plasmon resonance
bands by photon-induced electron-transfer (ET) processes
between the Fermi level of the metal and unfilled orbitals of
the molecules comprising the host medium. Persson’s theory

Gold Nanoparticles Oriented in Polyethylene
J. Phys. Chem. B, Vol. 102, No. 2, 1998 371
(14) Barber, P. W.; Hill, S. C. Light Scattering by Particles: Compu-
tational Methods; World Scientific: River Edge, NJ, 1990.
(15) Hornyak, G. L.; Patrissi, C. J.; Martin, C. R.; Valmalette, J.-C.;
Dutta, J.; Hofman, H. Nanostruct. Mater. 1997, 9, 575.
(16) Bohren, C. F.; Huffman, D. R. Absorption and Scattering of Light
by Small Particles; Wiley: New York, 1983.
(17) Equation 4 can be derived from the expression in ref 11 (p 70),
where Lj is the same as qx in this paper, and κ ) 1/q - 1.
(18) (a) Fricke, H. J. Appl. Phys. 1953, 24, 644. (b) Stoner, E. C. Philos.
Mag. 1945, 36, 803.
than the incident wavelength. Incorporation of particle size
effects into Maxwell-Garnett theory does not improve experi-
ment-theory comparisons as they regard extinction intensity-θ
relationships.
The spectra obtained using the T-matrix calculation are in
poor agreement with the experimental data for the smallest
radius particles (16 nm). As the particle radius increases to 38
and 60 nm, the agreement in terms of the θ-dependence of
spectral extinction improves. We cannot account for these
findings in terms of background scattering from the PE matrix
or particle size effects on the metal dielectric properties.
However, the unexpectedly low experimental θ ) 0° extinction
intensities may result from reductions in the mean free lifetime
of electrons in the metal phase arising from metal particle-to-
host medium electron transfer.
(19) Maxwell-Garnett, J. C. Philos. Trans. R. Soc. London 1904, 302,
385.
(20) Asano, S. Appl. Opt. 1979, 18, 712.
(21) Asano, S.; Yamamoto, G. Appl. Opt. 1975, 14, 29.
(22) Varadan, V. K.; Varadan, V. V. Acoustic, Electromagnetic and
Elastic WaVe Scattering: Focus on the T-Matrix Approach; Pergamon: New
York, 1980.
(23) Meier, M.; Wokaun, A. Opt. Lett. 1983, 8, 851.
(24) Zeman, E. J.; Schatz, G. C. In Proceddings of the 17th Jerusalem
Symposium; Pullman, B., Jortner, J., Nitzan, A., Gerber, B., Eds.; Reidel:
Dordrecht, Holland, 1984; p 413.
Acknowledgment. The authors are grateful to the National
Science Foundation for its support of this work (Grant DMR
9625151). Acknowledgment is also made to the donors of the
Petroleum Research Fund, administered by the American
Chemical Society, for partial support of this work. Electron
microscopy support was provided by the Lombardi Cancer
Center Microscopy and Imaging Shared Resource (U.S. Public
Health Service Grant 2P30-CA-51008). N.R. also thanks the
Jordan University of Science and Technology (JUST) for
financial support during his stay at Georgetown.
(25) Furneaux, R. C.; Rigby, W. R.; Davidson, A. P. U.S. Patent No.
4,687,551, Aug 18, 1987.
(26) The silver-silver chloride reference electrode (SSCE) consisted
of an AgCl-coated silver wire immersed in 4 M KCl saturated with AgCl.
The reference solution was separated from the plating solution phase by a
porous glass junction (Vycor).
(27) Shumilova, N. A.; Zutaeva, E. V. In Encyclopedia of Electrochem-
istry of the Elements; Bard, A. J., Ed.; Marcel-Dekker: New York, 1978;
Vol. 8.
(28) The aspect ratios for the 38 and 60 nm radius particles were
estimated from one-parameter least-squares analysis of the a/b (versus
number of coulombs) data in ref 6 (the deposition area is the same for the
two studies). For the 38 nm radius system, a/b ) (5.2 ( 0.2) × #C. For
the 60 nm radius system, a/b ) (6.8 ( 0.5) × #C.
References and Notes
(29) The spectrum of two crossed polaroids shows zero transmittance
between 330 and 700 nm, less than 0.02% T between 700 and 782 nm, and
less than 0.05% between 782 and 790 nm. We do not consider the data
above 790 nm quantitatively meaningful.
(1) Preston, C. K.; Moskovits, M. J. Phys. Chem. 1988, 92, 2957.
(2) Preston, C. K.; Moskovits, M. J. Phys. Chem. 1993, 97, 8405.
(3) Tierney, M. J.; Martin, C. R. J. Phys. Chem. 1989, 93, 2878.
(4) Foss, C. A., Jr.; Tierney, M. J.; Martin, C. R. J. Phys. Chem. 1992,
96, 9001.
(5) Foss, C. A., Jr.; Hornyak, G. L.; Stockert, J. A.; Martin, C. R. J.
Phys. Chem. 1992, 96, 7497.
(6) Foss, C. A., Jr.; Hornyak, G. L.; Stockert, J. A.; Martin, C. R. J.
Phys. Chem. 1994, 98, 2963.
(7) Martin, C. R. Science 1994, 266, 1961.
(8) Foss, C. A., Jr.; Hornyak, G. L.; Stockert, J. A.; Martin, C. R. Proc.
Mater. Res. Soc. 1993, 286, 431.
(30) Johnson, P. B.; Christy, R. W. Phys. ReV. B 1972, 6, 4370.
(31) See for example: Ashok, J.; Varaprasad, P. L. H.; Birch, J. R. In
Handbook of Optical Constants of Solids II; Palik, E., Ed.; Academic
Press: New York, 1991.
(32) For spheres, the quantity aˆ is simply the sphere radius. For ellipsoids
of revolution with semimajor axis a and semiminor axis b, aˆ is taken as
equal to a(a/b)2/3. See ref 14, p 103.
(33) Wokaun, A.; Gordon, J. P.; Liao, P. F. Phys. ReV. Lett. 1982, 48,
957.
(34) Lu, A. H.; Lu, G. H.; Kessinger, A. M.; Foss, C. A., Jr. J. Phys.
Chem., accepted.
(9) Hornyak, G. L.; Patrissi, C. J.; Martin, C. R. J. Phys. Chem. B
1997, 101, 1548.
(10) Al-Rawashdeh, N.; Foss, C. A., Jr. Nanostructured Mater. 1997,
9, 383.
(35) Granqvist, C. G.; Hunderi, O. Phys. ReV. B 1977, 16, 122.
(36) Persson, B. N. J. Surf. Sci. 1993, 281, 153.
(37) Schatz, G. C. In Surface Enhanced Raman Scattering; Chang, R.
K., Furtak, T. E., Eds.; Plenum Press: New York, 1982.
(38) Allen, C. S.; Schatz, G. C.; van Dyne, R. P. Chem. Phys. Lett.
1980, 75, 201.
(11) Huber, C. A.; Huber, T. E.; Sadoqi, M.; Lubin, J. A.; Manalis, S.;
Prater, C. B. Science 1994, 263, 800.
(12) van de Hulst, H. C. Light Scattering by Small Particles; Dover:
New York, 1980.
(13) Aspnes, D. E. Thin Solid Films 1982, 89, 249.
(39) Chase, B.; Parkinson, B. J. Phys. Chem. 1991, 95, 7810.