1

Effective medium methods for optical properties of nanoparticles

H. Y. Chung1, P. T. Leung2,3 and D. P. Tsai1,4,5

1Reearch Center for Applied Sciences, Academia Sinica, Taipei 115, Taiwan, Republic of China

2Institute of Optoelectronic Sciences, National Taiwan Ocean University, Keelung, Taiwan, Republic of China

3Department of Physics, Portland State University, P. O. Box 751, Portland, Oregon 97207, U.S.A.

4Department of Physics, National Taiwan University, Taipei 106, Taiwan, Republic of China

5Graduate Institute of Applied Physics, National Taiwan University, Taipei 106, Taiwan, Republic of China

Abstract

Optical properties of nanoparticles, especially those of metallic nanoparticles, are known

to be intriguing which can deviate significantly from those of bulk materials, leading to

important novel applications in areas ranging from biosensing to enhancement of device

performance. These properties can be controlled to cover a wide range of possibilities via

manipulation of the geometry, size, and material of these particles. Hence theoretical

understanding of them is important in the development of nanoparticle technology. In this

review, we elaborate a powerful approach based on effective medium models which has been

found to be efficient for the characterization of these particles with sufficient accuracy. These

models can be formulated to account for different physical effects which are significant in

different ranges of particle size as compared to the optical wavelengths. Formulations and

numerical illustrations of this approach will be presented based mainly on several of our recent

works.

2

Introduction

While empirically it has been known for many centuries that colored glass can be made

with the doping of fine metallic particles, the recent surge in the modern optical science of

nanoparticles (NP) – especially metallic nanoparticles (MNP) – has been due mainly to the

advances in various nanotech fabrication techniques, as well as in computational power in the

simulation of the optical properties of these particles.1-7 With the more precise characterization

and control in the making of these particles, many novel applications of them have been

developed in time – in various diversified areas from biosensing8-17 and enhanced spectroscopy18-

27 to improving the performance of optoelectronic device such as light emitting device.28-37

In the recent developments in the plasmonics of these MNP’s,38 researchers have focused

on the collective motion of the free electrons in these particles which has led to hugely-enhanced

local fields on resonance. These fields then provide enhancements for various optical and

spectroscopic processes in the vicinity of these MNP’s.18, 38 Since this resonance has sensitive

dependence on the geometry of the particles, various designs have been proposed for its control –

beyond that of a bare spherical or spheroidal particle. These include, for example, spherical

nanoshells,39-45 nanorice,46-54 and multilayered nanomatryoshka.55-58 These structures provide

great tunability for the plasmon resonance frequency as well as the enhanced field magnitude via

the control of the respective geometrical parameters such as the shell thickness and the aspect

ratio.

Besides experimental and technological developments, significant progress in the

theoretical understanding of the optical and spectroscopic properties of these MNP’s has also

taken place in recent time. For particle size greater than ~ 10 nm, classical electrodynamics will

3

be accurate enough to account for most of these properties. One of these known as hybridization

theory (HT) was developed by the Rice group,55, 59-61 which has been found to be powerful in the

prediction of the coupled resonance modes in these complicated structures. However, the

enhanced fields are usually more complicated to be described by HT and most often numerical

methods have been adopted for their calculations. For particle size below ~ 10 nm, more

complicated quantum mechanical effects can manifest and the theoretical treatment becomes

more challenging. Although density functional theory is powerful for this treatment, it has been

limited to rather small particles (say, below a few nm) due to its high demand of computation

power. It is partially for this reason that nonlocal electrodynamics has been introduced to enable

the conventional electrodynamic theories to be applicable to smaller size particles with partial

account of the quantum effects.62-70

For particles of greater sizes, most of the microscopic features of the free electrons in the

MNP can be ignored and classical optics is sufficient for their characterization. Moreover,

although exact electrodynamic solutions or numerical methods are usually available depending

on the geometry of the MNP, these solutions are often of high complexity to be implemented.

For sizes within 5% to 20% of the wavelength of light, a simpler approach known as the

modified long wavelength approximation (MLWA) can be adopted. This was first introduced by

Meier and Wokaun in 198371 and contains first order corrections to the quasi-static theory which

has enough accuracy for particle size below ~ 20% of the wavelength. Although the original

theory71 was formulated for a single spherical particle, we shall see in the following review that

this approach can be extended to NP’s with more complicated structures and still retains its

simplicity and efficiency compared to the exact or numerical solutions.

4

The main theme of our present review is to demonstrate the powerful treatment using

various effective medium models for single NP’s of complicated structure. While there have

many comprehensive reviews recently published on the science of nanoparticles,1, 8, 18, 28, 72 few

of them has focused on the application of effective medium theory (EMT) for the

characterization of these particles. We will see in the following that these effective methods are

indeed very powerful for the description of heterogeneous systems, as well as neutral and non-

neutral systems.

As for non-neutral systems, it turns out that it is not uncommon for the NP’s under study

to carry a net charge (mostly surface charges).73-79 This is the case for many naturally-occurring

particles such as cosmic dusts and ice crystals in stormy atmosphere; as well as particles in the

laboratory such as those found in colloidal solutions. Hence it is of relevance to account for

these charge effects on the optical response of these NP’s. For this purpose, we shall treat both

non-metallic NP’s and MNP’s in our following review, and shall see that although such charge

effects in general are small, they can be subtle to be elaborated.

With the formulation of various EMT’s, we shall illustrate their application for the

description of both optical and spectroscopic phenomena in the presence of these NP’s. These

will include the study of various extinction and scattering from these particles, as well as

different molecular spectroscopy observed in the vicinity of these MNP’s. This later

phenomenon refers to, in particular, the well-known surface enhanced Raman scattering

(SERS)80-87 and molecular fluorescence near MNP’s with emphasis on our own recent studies

which have looked at some novel aspects of these phenomena. Our review will hence be

organized as follows. In section 2, an account of the various effective medium theories will be

presented while applications of these theories to various optical and spectroscopic phenomena

5

will be given in section 3. Discussion and a summary of the future prospects of the nanoparticle

sciences will be given to conclude our review.

Effective medium theory (EMT)

The optical properties of a nanoparticle can be in principle obtained via solving the

Maxwell’s equations with the appropriate boundary conditions. One typical example will be the

interaction of a plane wave with a solid sphere which was solved by G. Mie in 1908.88, 89

However, if the nanoparticle is inhomogeneous, such as a core-shell particle or a graded index

particle, the analytical calculation of such problems can be very complicated. The goal of the

EMT here is to seek an effective homogeneous nanoparticle which provides the same optical

responses of the original inhomogeneous nanoparticle, thus simplifying the calculation.

To introduce EMT let us first consider a case of core-shell particle, see Fig. 1(a). The

dielectric functions of the core, shell and surrounding medium are

EMT for core-shell and core-multishell particles

1 2,ε ε , and hε , respectively.

Our goal is to seek an effective particle of the same shape as that of the original core-shell

particle but with the homogeneous dielectric function effε , see Fig.1 (b), such that the optical

responses of the effective nanoparticle are identical to the original core-shell particle. Since in

quasistatic model, all the optical responses of a particle are represented by its polarizability α ,

the effective particle is required to have the same α of the original core-shell particle.

To determine effε , we replace the surrounding medium by 2ε as illustrated in Figs. 1(c)

and (d). In this situation the surrounding medium is the same material of the shell, so the

original core-shell particle is reduced to a solid sphere with a radius equal to that of the core.

6

Because the equivalence relation between the original particle and the effective particle should

hold with any host medium, we have

2 12 1 2( ) (, ),S eff Sα ε α εε ε= , (1)

the effective dielectric function can hence be solved from this relation. Note that α is known if

the boundaries 1S and 2S are of certain simple shapes such as spheres or spheroids.

To apply this EMT process to case of the core-shell spheroid (known as “nanorice” ), we

apply the result for the polarizability of spheroid:

2 1

( ) ( )h

m mm h m

a CA B

ε εαε ξ ε ξ

+ −=

−

, (2)

where ξ and a are spheroidal coordinate and foci, and the coefficients mA

, mB

, and mC

are

expressed in terms of the associated Legendre functions. The effective dielectric function can

thus be obtained in the following form:

[ ] [ ][ ] [ ]

1 1 2 1 22

12 2

2

1 1 2

( ) ( ) ( ) ( )( ) ( ) ( ) ( )

m m m mm

m m m

eff

m

A B B BA A B A

ε εε ε

ξ ξ ξ ξξ ξ ξ ξε ε

− − −=

− − −

. (3)

(see Ref. 90 for more details).

For a core-multishell spheroid, a nanomatryoshka, the above EMT leads to the following

iterative formula for the polarizability:

( )( ) ( )( )

( )( ) ( )( )

1 1 1 1 1 12 1

1 1 1 1 1 1

n n n n n n n nn h h lm lm n lm lm h lm n lm n lm h lmn l

lm lm n n n n n n n n n nn lm h lm h lm lm n lm lm lm lm h lm n lm n h

A B B Aa C

A B A B A B

ε ε ε η ε ζ ε η ε ζ ε εα

ε ε ε η ε ζ ε η ε ζ ε ε

− − − − − −+

− − − − − −

− − − − −=

− − − − −,

(4)

where n is the number of shells (see Ref. 90 for more details).

7

The above result can be generalized to the case of a graded index particle. For such a

particle of radius

EMT for graded index particles

a and a radial varying dielectric function ( )rε , the effective dielectric function

of the -th multipole mode satisfies the following equation:91, 92

,, ,( ) ( ) ( )

( )( ) ( 1) ( )eff ef

ff

ef

d r r rdr r

r rr

εε

εε ε

ε−

= − + +

. (5)

Here the effective dielectric function means that for the -th multipole mode, the original graded

sphere is equivalent to a homogeneous sphere of the same radius with a uniform dielectric

function , ( )eff aε

, i.e. the -th polarizability of the graded sphere can be expressed by the

effective dielectric function as follows:

, 2

,

1

( 1)( )

( )heff

ef hf

aa

aα

ε εεε

+−=

+ +

. (6)

Hence the homogenization of a graded index particle amounts to solving the differential

equations as given in (5).

The EMTs introduced above are based on the quasistatic theory, which is much simpler

than the full electrodynamic theory and is accurate when the particle size is much smaller than

the wavelength of the incident wave. However, such accuracy starts to degrade when the surface

plasmon resonance takes place. The contribution of dynamic effects becomes prominent around

the resonance leading to the suppression of the resonance strength predicted by quasistatic theory.

Modified long-wavelength approximation (MLWA)

Since the full dynamic theory will require complicated solutions of a boundary-value

problem involving both the electric and magnetic fields, Meier and Wokaun71 had introduced a

modified long-wavelength approximation (MLWA) to improve on the quasistatic model. Unlike

the quasistatic model in which the whole particle is treated as an oscillating dipole, MLWA treats

8

the particle as a system composed of small dipole elements, and each dipole element is described

by the full dynamic theory. According to MLWA the quasistatic dipole polarizability is

modified and for a sphere, one has

32 3

12 ( 1)( 2 / 3)ML a

x i xεα

ε ε−

=+ − − +

, (7)

where the 2x term represents the dynamic depolarization and the 3x term is the radiation-

damping correction to the quasistatic model. Just by replacing the original polarizablity with this

modified result, the validity of the quasistatic model can be extended to larger particles of radius

from about 1% to 10% of the wavelength around the surface plasmon resonance region.

Note that though the MLWA provides a convenient route for correcting the quasistatic

model, it only considers the correction of the dipole oscillation, and hence is more useful for

studying the far field properties such as scattering, absorption, and extinction. For description of

near field properties in which the higher multipole modes may play a more important role, the

MLWA is hence more limited.

For studying the core-shell or core-multishell particles, the dynamic corrections can thus

be introduced into the EMT via replacing the polarizability in Eq. (1) by MLα in (7). Also for

graded-index particles, the MLWA modifies Eq. (5) to take the following form (for 1= ):91, 92

2 2 2 2( ) ( ) ( 1 11 ( ) 2 ( )3 3

)( )

eff eff

fef

d r r rdr r

k r r kr

r rε ε ε

εε ε − + +

− = −

. (8)

The classical Drude model is widely adopted for the study of the optical properties of

metallic particles. However, it has been known that if the size of the particle is smaller than

about 10nm, the classical theory has to be corrected to account for quantum effects. Since the

full quantum mechanical calculations, such as density functional theory, is strictly limited by the

Nonlocal effects for ultrasmall particles

9

speed of computer, an alternative approach is desired. Nonlocal theory is one of the simple

methods that brings the quantum mechanical corrections into a classical system, via using

nonlocal dielectric functions which were derived from quantum many body theories93 and

therefore accounts for the quantum behavior.

In principal, to study the nonlocal effects, one has to start from the relation

3( ) ( , ) ( )d xε ′ ′ ′= ∫D x x x E x . (9)

However, for the case of solid sphere, Fuchs and Claro62, 94 had shown that under certain

assumptions, the nonlocal effects can be taken into account if one considers the following

effective dielectric function

12

, 0

( )2(2 )( ,

1)eff

j kaa dkk

επ ε ω

−∞

=+

∫

, (10)

where a is the radius of the sphere and j

is the spherical Bessel function of order .

One of the simplest nonlocal model is the hydrodynamic model:

2

2 2( , ) 1( )

p

i kk

ωω

ω ω γ βε

+ −= − , (11)

where 3 / 5 Fvβ = and Fv is the Fermi velocity. This model can be treated as the first order

approximation of the Lindhard-Mermin model.93, 95-98 Using this model in Eq. (10), one

obtains62

1

1/2 1

2

, /21 (2 ) ( )1 ( )e

pff

aI

uu K uε

εωβ + +

−

+

= +

, (12)

where 2 ( /)p iu a ω ω ω γ β− += , I

and K

are modified Bessel functions, and ε is the local

dielectric function obtained from Eq. (11) with 0β = .

10

By using EMT, the result in Eq. (10) can be generalized to the core-multishell particle

case.99

Besides applying to neutral particles, the concept of EMT can also be used to study

particles with extraneous surface charges. Bohren and Hunt had solved the full dynamics of the

scattering problem of a charged sphere by considering the boundary conditions:73

EMT for charged spheres

( )2 1 0−× =n EE , (13)

and

( )2 1−× =n H H K , (14)

where K is the surface current caused by the extraneous surface charges. By solving the

boundary value problems, Bohren and Hunt had derived the modified Mie scattering coefficients,

thus extending the Mie scattering theory to the charged sphere case.

Though Bohren and Hunts’ theory is an exact full dynamic theory, it is useful to derive

its quasistatic version which will be less complicated. By considering the long-wavelength limit

and by comparing with the quasistatic model, one can find that the charged sphere can be

effectively replaced by a neutral sphere of the same size with an effective dielectric function

given by100-101

, ( /)1eff i xε ε τ+= +

, (15)

where τ is proportional to the number of extraneous charges on the sphere and x is the size

parameter of the sphere.

Applications

To demonstrate the usefulness of the above EMT, we here present some examples.

11

Figure 2 shows the extinction cross section of a hollow silver spherical shell of inner

radius 60 nm and outer radius 70 nm placed in vacuum. The curve predicted by the Mie theory

is treated as a standard to verify the validity of the EMT models. As we can see that all of the

quasistatic models, including static, MLWA and IMLWA (“Improved MLWA”102-104) modify

only the dipole resonances. Each model has two resonances, which are the consequences of the

coupling between the dipole resonances of the inner and outer spheres. According to the

hybridization model,55, 59 the low frequency mode is referred to the bonding mode, while the high

frequency mode is referred to the antibonding mode.

By comparing with the results from the Mie theory, it is clearly seen that the dynamic

effects contained in the MLWA significantly modify the predictions from the static model. The

unreasonably high resonance peak predicted by the static model is due to the assumption of the

perfectly coherent oscillation of the electrons, this happens when the retardation effects and the

radiative loss are ignored as are accounted for in MLWA.

Figure 3 shows the corresponding results for core-multishell particles. Similar to the

core-shell particle case shown in Fig. 2, the four resonance modes are due to the coupling of the

dipole resonance at the four metal-dielectric interfaces. Again, the results predicted by the Mie

theory are treated as a standard. The MLWA again provides significant dynamic corrections to

the static model.

Besides application to far field scattering from a NP, the EMT can also be applied to

study near field interactions. Figure 4 shows some results from the study of the energy transfer

between a donor and an acceptor molecule, a process known as Forster resonance energy transfer

(FRET).105-107 The two molecules are placed in the vicinity of a core-shell spheroid (a nanorice)

to achieve enhancement in the energy transfer rate, which is defined by

12

2

,AD AD ind

AD

U UR

U+

= , (16)

where ADU and ,AD indU are associated with the energy transfer between the donor and the

acceptor with and without the nanorice, respectively. Unlike the far field situation, the

contribution from the high multipole resonance modes are clearly present. For each case the

resonance modes split into the low frequency bonding and high frequency antibonding modes,

with the splitting more pronounced as the shell gets thinner due to stronger coupling between the

interface plasmons.

Figure 5 shows the application of the EMT to a graded-index sphere. The dielectric

function of the sphere is assumed to have the following form: 0( ) ( ) /Agr r rε ωε= , where

0 30 nmr = . The results are compared with the Mie theory obtained by approximating the sphere

as a stratified system of many layers. Again, the MLWA results are much closer to the Mie

results than the static model. Also, from Figure 5(b) we can see that the MLWA is more

significant for the larger particles since the dynamic effects become more important as the

particle size increases. By comparing with the Mie theory this figure shows that for the graded-

index spheres, the results from Eq. (8) which was formulated based on the concept of EMT are

reliable. In other words, the EMT can also be applied to inhomogeneous systems with a

continuously varying dielectric profile.

Figure 6 shows the results of the nonlocal effects for small core-multishell particles. It is

clearly seen that the dipole resonance in Figure 6(a) is manifested in the four coupling modes at

the four metal-dielectric interfaces. Similarly, in Figure 6(b), the three coupling modes at the

three metal-dielectric interfaces are manifested. The small blueshift of each resonance caused by

nonlocal effects is clearly observed.

13

Figure 7 shows the application of the EMT to the study of the surface enhanced Raman

scattering (SERS) from a molecule near a MNP with extraneous surface charges. It is clear that

the extraneous surface charges lead to blueshifts in the resonance frequency. This can be

understood from the effective dielectric function in Eq. (15), from which if one adopts the Drude

model for the metal, the modified resonance frequency for the n -th multipole resonance can be

obtained in the form:

2

21 ( 1)2 1

sn p

p

n nn

ωω ωω

= + ++

, (17)

which can be interpreted as an effective increase in plasma frequency. From this formula one

can also note that the blueshift effect is more significant for higher multipoles.

In addition to the shifts in the resonance frequencies, the surface charge can also lead to

enhancement of the resonance scattering amplitude as well. This is due to the assumption of the

small damping in the motion of the surface charges via /s BTkγ = , with Bk T being the thermal

energy. This then leads to a reduction of the average damping for the whole system, resulting in

an enhanced resonance scattering amplitude.

Another molecular process of interest other than SERS is fluorescence. Figure 8 shows

the normalized fluorescence rate of a molecule in the vicinity of a charged sphere as a function

of distance from the sphere. Similar to Figure 7, the extraneous surface charges can enhance the

resonance amplitude for the rate. It can be seen that when the molecule is too close to the

particle, the resonance will be quenched. This quenching effect is due to the increase in

nonradiative loss. The results show that the optimal distance for maximizing the fluorescence

rate can be controlled by varying the amount of extraneous surface charges on the MNP.

14

Conclusion

In this brief review, we have demonstrated, based mainly on our recent works, the

powerful role the EMT can play in the description and understanding of the various optical

properties of nanoparticles, especially MNP’s. We have seen, in order to extend the tunability of

the resonance frequency and the enhanced field amplitude in the vicinity of the MNP, particles of

richer geometrical content are desired. These then led to the developments of various nano-

structures in the past two decades to include systems like the metallic nanoshell, nanorice,

nanomatryoshka, and graded-index particles. However, the theoretical formulation of the optics

of these systems is far more complicated than that for a homogeneous spherical particle since the

electrodynamic equations for these heterogeneous systems (both in geometry and in material)

can be extremely difficult to solve.

We have hence shown how the EMT can account successfully for the properties of these

nano systems with enough accuracy, and can account effectively for various important optical

effects such as dynamical and nonlocal effects for particles with larger and smaller size,

respectively. In conclusion, we believe that even with the availability of various powerful

numerical software which can solve Maxwell’s equations for a highly heterogeneous and

inhomogeneous system efficiently, the EMT formalism is still of value as it can provide insight

into the physical origins of the various observed effects which are not always obvious from

numerical simulations.

Acknowledgement

The authors gratefully acknowledge the financial support of the National Science Council of

Taiwan (NSC 102-2745-M-002-005-ASP, 102-2911-I-002-505, 101-2911-I-002-107, 100-2923-

15

M-002-007-MY3, 101-2112-M-002-023-, 100-2112-M-019 -003 -MY3). They are also grateful

to National Center for Theoretical Sciences, Taipei Office, Molecular Imaging Center of

National Taiwan University, National Center for High-Performance Computing, Taiwan, and

Research Center for Applied Sciences, Academia Sinica, Taiwan for their support

.

16

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22

Figure 1. Illustration of the EMT, see text. [Adapted from Ref. 90, with permission from the

publisher].

23

300 400 500 600 700 800 900 10000.0

0.2

0.4

0.6

0.8

1.0

1.2

1.4

1.6

1.8

2.0

Extin

ctio

n cr

oss

sect

ion

(105 n

m2 )

Wavelength (nm)

Static MLWA IMLWA Mie

Figure 2. Comparison of the extinction cross sections obtained in the static limit, from MLWA,

and from IMLWA against the exact Mie theory results. The results are shown for a spherical

nanoshell with inner radius 60 nm and outer radius 70 nm. The silver nanoshell is hollow inside

and placed in vacuum. [Adapted from Ref. 90, with permission from the publisher].

24

10-2

10-1

100

101

102

0.0 0.2 0.4 0.6 0.8 1.0 1.210-2

10-1

100

101

102

103

Ext

inct

ion

Effic

ienc

y

static MLWA Mie

(a)

Extin

ction

Effi

cienc

y

ω /ωp

(b)

Figure 3. Comparison of the extinction efficiencies according to the LWA, MLWA, and the

exact Mie results for a four-layer nanomatryushka composed of : glass/Ag/glass/Ag of

dimensions (a) 10 nm/15 nm/20 nm/25 nm, and (b) 10 nm/20 nm/30 nm/40 nm, respectively.

[Adapted from Ref. 99, with permission from the publisher].

25

Figure 4. Comparison of the enhancement factor ( )R ω for different geometries of the nanorice.

The foci and the outer (surface) aspect ratio are fixed at 20 nm and 2/3, respectively. The inner

aspect ratios are set for three different values, which are 0.2, 0.4, and 0.6. The nanorice has a

hematite core ( 1ε = 9.5 ) with the silver shell embedded in the vacuum. The donor and the

acceptor are located at the two “poles” at (0, 0, 30 nm) and (0, 0, 30 nm)− in Cartesian

coordinates and aligned along the z direction. [Adapted from Ref. 54, with permission from the

publisher].

26

Figure 5. Comparison of the extinction efficiencies computed using the quasi-static (dotted), the

MLWA (dashed) and the stratified Mie (solid) models, respectively, for a graded sphere with

radius of (a) 15 nm and (b) 30 nm. The dielectric function of the sphere is taken to have the form

0( ) ( ) /Agr r rε ωε= , where 0 30 nmr = . [Adapted from Ref. 91, with permission from the

publisher].

27

0.0

0.2

0.4

0.6

0.8

1.0

0.0 0.2 0.4 0.6 0.8 1.0 1.20.0

0.1

0.2

0.3 (b)

Ex

tincti

on E

fficie

ncy Nonlocal

Local(a)

Ext

incti

on E

fficie

ncy

ω /ωp

Figure 6. Extinction efficiency for a four layer shell with the radii 1 1r = nm, 2 2r = nm, 3 3r =

nm, 4 4r = nm. The shell is made of (from core) glass/ silver/ glass/ silver for case (a), and

28

silver/glass/ silver/ glass for case (b). [Adapted from Ref. 99, with permission from the

publisher].

Figure 7. The SERS enhancement ratio R of (a) radial and (b) tangential molecules near a silver

sphere of the radius 5 nma = with surface charges: 11

65 10 Cq −×= , 162 8.33 10 Cq −×= ,

153 1.67 10 Cq −×= and 15

4 1 5 C2 0.q −= × . The distance between the molecule and the sphere is

fixed at 5 nmd = . [Adapted from Ref. 100, with permission from the publisher].

29

Figure 8. Normalized fluorescence rate as a function of distance from the silver sphere of radius

5nm. The molecule is along (a) radial and (b) tangential directions. The emission frequencies

are set at each of the dipole resonance for the cases 0 0q = , 161 1.67 10 Cq −×= , and

162 10 5 Cq −×= , with values equal to 0.57 pω , 0.58 pω , and 0.58 pω , respectively. [Adapted from

Ref. 101, with permission from the publisher].

.