Abstract

In this work we present an novel approach to the calculation of the nonlocal conductivity tensor of an isolated sphere (T-matrix operator) as an ordinary electromagnetic wave scattering problem through the use of ordinary boundary conditions. Exact closed expressions are found.
Research Article
Mie-Type Calculation of the Generalized Electromagnetic
Nonlocal Conductivity Tensor for a Sphere and Its Equivalence
to the T-Matrix Operator
E. Gutiérrez-Reyes ,1Rubén G. Barrera,2and A. Garc-a-Valenzuela3
1CONACYT-Centro de Investigaci´
on Cient´
ıca y de Educaci´
on Superior de Ensenada, Baja California, Unidad La Paz,
Miraores No. 334 e/Muleg´
e y La Paz, C.P. 23050, La Paz, BCS, Mexico
2Instituto de F´
ısica, Universidad Nacional Aut´
onoma de M´
exico, Apartado Postal 20-364, 01000 Ciudad de M´
exico, Mexico
3Instituto de Ciencias Aplicadas y Tecnolog´
ıa, Universidad Nacional Aut´
onoma de M´
exico, Apartado Postal 70-168,
04510 Ciudad de M´
exico, Mexico
Correspondence should be addressed to E. Guti´
errez-Reyes; edahigure@gmail.com
Received 13 September 2018; Accepted 23 December 2018; Published 6 February 2019
Academic Editor: Francesca Vipiana
Copyright ©  E. Guti´
errez-Reyes et al. is is an open access article distributed under the Creative Commons Attribution
License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly
cited.
e purpose of this article is to calculate the generalized nonlocal conductivity tensor of a spherical particle made of isotropic
and linear materials. e generalized conductivity tensor is a crucial element in the formulation of the mean-eld theories of the
electromagnetic response of random particulate systems. is is equivalent to what is called the T-Matrix in multiple scattering
theories. Here, a new method is proposed for nding explicit expressions for this tensor directly from its denition including the
magnetic response of the spheres. Its relation with the -Matrix in the theory of single scattering is stated as a generalization. Several
approximations and limit cases of possible interest in specic systems are analyzed and the results of some calculations are presented
as a numerical example.
1. Introduction
In a previous paper [], a reference was made to the gen-
eralized nonlocal conductivity tensor, as the main physical
concept in the development of a formalism for the cal-
culation of the eective bulk electromagnetic response of
randomly located discrete scatterers such as turbid colloids.
is approach was later extended to the calculation of the
reection and transmission amplitudes of the average electric
eld of colloidal suspensions conned in a half-space []. e
generalized conductivity tensor continued to play a major
role in the framework of the theoretical description [].
Colloidal systems are dened as two-phase systems in which a
sparse phase is immersed within a continuous one []. While
thecontinuousphaseisusuallycalledthematrix,thesparse
phaseissaidtobecomposedofcolloidalparticles.When
an external electromagnetic eld is incident upon a colloidal
system, it induces currents within the randomly located
colloidal particles and these currents radiate electromagnetic
elds which act back upon the particles themselves. e total
electromagnetic eld conformed in this way can be split
into two components, an average component with a smooth
spatial variation and traveling in a denite direction, called
the coherent beam, and a uctuating component with abrupt
spatial variations and traveling in all dierent directions,
called the diuse eld.
e work cited above [], related to the reection and
transmission amplitudes from colloidal suspensions, dealt
with the reection and transmission amplitudes of the coher-
ent beam for a system of randomly located identical spheres,
within a homogeneous matrix with a at interface occupying
a half-space. e procedure devised for this calculation
involved the solution of an integral equation posed in terms
of the generalized nonlocal conductivity tensor of an isolated
sphere and valid for a low concentration of spheres. Also,
it has been pointed out that this integral equation for the
Hindawi
Mathematical Problems in Engineering
Volume 2019, Article ID 1530821, 20 pages
https://doi.org/10.1155/2019/1530821
Mathematical Problems in Engineering
average electric eld is analogous to the one used in multiple-
scattering-theory (MST), to solve similar problems. e
integral equation in MST is usually written in terms of the so-
called T-matrix operator, whose calculation for the case of an
isolated sphere has already been performed [] by solving the
integral equation obeyed by this operator. Since a wide variety
of systems are usually modeled as a collection of spheres, the
calculation of the T-matrix for an isolated sphere becomes a
necessary ingredient in this type of calculations.
Here, we rst recall that the T-matrix operator and
the generalized nonlocal conductivity tensor are actually
proportional to each other [] obeying, essentially, the same
integral equation. en, we propose a quite straightforward
procedure to calculate the generalized nonlocal conductivity
tensor of an isolated sphere, making use of its physical
interpretation. Instead of solving the integral equation, our
procedure is based on a Mie-type scattering boundary-value
problem in the presence of external currents, yielding closed-
form expressions for all the components of the generalized
nonlocal conductivity tensor, or, equivalently, the T-matrix
operator. We consider the sphere to be characterized by
both, a local electric permittivity and a local magnetic
permeability. erefore, the method described here poses
an alternative to the calculation of the T-matrix operator,
by simply nding the induced current inside a sphere given
an arbitrary external electric eld, yielding attractive closed-
form expressions for its components. Finally, we want to point
outthatalthoughthecomponentsoftheT-matrixoperator
have been already calculated for the case of a sphere with no
intrinsic magnetic response [], here we provide new closed-
form expressions for these components for the more general
case of a sphere with an intrinsic magnetic local permeability
dierent from the one of its surroundings, yielding, besides
abulkcontribution,alsoasurfacecontributioncomingfrom
induced surface currents. It is also important to notice that
all this is possible due, essentially, to the simplicity of the
calculation procedure proposed here.
2. Basic Concepts
e nonlocal conductivity tensor of an isolated sphere is
dened as the linear response of the internal induced current
to an external electric eld; in its most general form it can be
written as
𝐼
;=𝑉𝑠
𝑁𝐿
,
󸀠;⋅
𝑒𝑥𝑡
󸀠;3󸀠()
where the dependence on the frequency on a given quantity
denotes the -component of the corresponding time Fourier
transform. Here,
𝐼is the total internal induced current,
𝑁𝐿 is the generalized nonlocal conductivity tensor, and
𝑒𝑥𝑡 is the external electric eld; it could also be called the
exciting eld. e term generalized means that in
𝐼all
internal induced currents are included, even those that give
rise to a magnetic response. In this sense Eq. () can be
regarded as a generalized nonlocal Ohm’s law.
etotalinternalinducedcurrent
𝐼can be calculated,
also, by using the local relationship
𝐼
;=𝑙𝑜𝑐
𝑠
;
𝐼
;()
where
𝑙𝑜𝑐
𝑠
;=
𝑠()<
0>, ()
where is the radius of the sphere, 𝑠() is its bulk local
conductivity, and 𝐼is the internal electric eld, that is, the
electric eld within the sphere. But the electric eld at any
point
is given by

;
=
𝑒𝑥𝑡
;
+0𝑉𝑠
0
,
󸀠;⋅
𝐼
󸀠;3󸀠
()
Here 𝑠is the volume of the sphere and also
0
,
󸀠;=
1+1
2
0∇∇
,
󸀠;()
isthefreeGreensfunctiondyadic,0=/,and

,
󸀠;=exp 0
−
󸀠
4
−
󸀠.()
erefore, when one uses Eq. () to iterate Eq. (), one can
nd the equation for
𝑁𝐿, as dened by Eq. (). First, since
equation Eq. () also holds inside the sphere |
|<,then,if
we multiply Eq. () by 𝑠,weobtain
𝐼
;
=𝑉𝑠3󸀠
−
󸀠𝑙𝑜𝑐
𝑠
󸀠;
𝑒𝑥𝑡
󸀠;
+0𝑙𝑜𝑐
𝑠
;
𝑉𝑠
0
,
󸀠;⋅
𝐼
󸀠;3󸀠.
()
Substituting now Eq. () into Eq. (), we have that
𝑉𝑠
𝑁𝐿
,
󸀠;⋅
𝑒𝑥𝑡
󸀠;3󸀠
=𝑉𝑠3󸀠𝑙𝑜𝑐
𝑠
󸀠;
−
󸀠
⋅
𝑒𝑥𝑡
󸀠;
+0𝑠
𝑉𝑠
0
,
󸀠󸀠;
𝑉𝑠
𝑁𝐿
󸀠󸀠,
󸀠;
𝑒𝑥t
󸀠;3󸀠3󸀠󸀠.
()
Mathematical Problems in Engineering
By interchanging the order of integration and comparing the
terms within the 3󸀠integral, we nd
𝑁𝐿
,
󸀠;=𝑙𝑜𝑐
𝑠
;
−
󸀠
+00𝑉𝑠
0
,
󸀠󸀠;
𝑁𝐿
󸀠󸀠,
󸀠;3󸀠󸀠.
()
Now, we recall the denition of the T-matrix through the
following integral relation [, Eq. ..]:
𝑆
;=𝑉𝑠3󸀠𝑉𝑠3󸀠󸀠
0
,
󸀠;

󸀠,
󸀠󸀠;⋅
𝑒𝑥𝑡
󸀠󸀠;. ()
Here
𝑒𝑥𝑡 corresponds to the external eld within the sphere.
𝑆is the electric eld scattered by the sphere for |
|>.
Since one can identify the scattered eld
𝑆with the second
term in the right-hand side of Eq. (), by replacing
𝐼in the
right-hand side of Eq. () with the expression given in Eq. ()
it follows immediately that Eq. () yields
0
𝑁𝐿
,
󸀠;=

,
󸀠;. ()
isrelationisvalidfor|
| ≤ and |
󸀠|≤. ere-
fore, the calculation of all the components of
𝑁𝐿 can
be performed by solving the integral equation given in
Eq. (). is procedure has been already followed for the
corresponding integral equation for T-matrix operator [].
ItispertinenttopointoutthatthedenitionoftheT-
matrix operator is given, sometimes, as the linear integral
relationship between the scattered eld and the external eld
[]. In this case the Greens function dyadic is incorporated in
the denition of the T-matrix operator; thus its relationship
with the generalized nonlocal conductivity has to be modied
correspondingly.
3. Calculation Procedure
Here we propose an alternative and simple way to evaluate
the components of the nonlocal generalized conductivity
tensor
𝑁𝐿 for a sphere, by taking advantage of its physical
interpretation. is means that one has only to calculate the
current density induced within the sphere by an arbitrary
external electric eld. In order to see how this is done, let
us start by assuming that all the elds are in the frequency
domain. Now we consider that an external electromagnetic
planewaveisincidentuponthesphere,andwewriteits
electric eld as
𝑒𝑥𝑡 =
𝛽𝑖󳨀
𝑝⋅󳨀
𝑟()
where
isthewavevector,
𝛽is a unit vector along
the direction of polarization, and we are assuming a unit
amplitude. en, according to Eq. (), the current density
𝐼
induced within the sphere by this external plane wave is given
by
𝐼
,
;
𝛽=𝑉𝑠
𝑁𝐿
,
󸀠⋅
𝛽𝑖󳨀
𝑝⋅󳨀
𝑟󸀠3󸀠.()
where
𝛽as an argument in
𝐼recalls the polarization of the
external electric eld. Now, we dene the Fourier transform
pair of
𝑁𝐿 as
𝑁𝐿
,
󸀠= 1
(2)6
R3R3
𝑁𝐿
,
󸀠𝑖󳨀
𝑝⋅󳨀
𝑟−𝑖󳨀
𝑝󸀠󳨀
𝑟󸀠3󸀠3. ()
𝑁𝐿
,
󸀠
=R3R3
𝑁𝐿
,
󸀠−𝑖󳨀
𝑝⋅󳨀
𝑟𝑖󳨀
𝑝󸀠󳨀
𝑟󸀠3󸀠3. ()
And by taking the Fourier transform in Eq. (), one can write
𝐼
󸀠,
;
𝛽=
𝑁𝐿
󸀠,

𝛽.()
Finally, the component of the above equation becomes
𝐼𝛼
󸀠,
;
𝛽≡
𝛼
𝐼
󸀠,
;
𝛽
=
𝛼
𝑁𝐿
󸀠,

𝛽
≡𝑁𝐿
𝛼𝛽
󸀠,

()
is means that the component of the nonlocal general-
ized conductivity tensor
𝑁𝐿(
󸀠,
)has a straightforward
physical interpretation: it is equal to the component of the
󸀠Fourier transform of the current density induced within
the sphere by an electromagnetic plane wave with wavevector
and polarization
𝛽.
But the induced current
𝐼can be written in terms of the
polarization and magnetization material elds,
and
,as
𝐼
=

+∇×

 ()
where

=𝑠−0(1−(−))
𝐼
 ()
and

=(1−(−))1
01
𝑠
𝐼
 ()
Here 𝑠and 𝑠refer to the local electric permittivity and the
local magnetic permeability of the sphere which are assumed
torespondlocallyandlinearlytotheelectricandmagnetic
Mathematical Problems in Engineering
elds (
𝐼and
𝐼)withinthesphere.Herewehavelocatedthe
centerofthesphereattheoriginofthecoordinatesystem,and
while 𝑠and 𝑠are the local electric permittivity and the local
magnetic permeability of the sphere, 0and 0denote the
corresponding response functions of the medium outside the
sphere;herewehaveasharpdiscontinuityofthesefunctions.
erefore, by plugging Eqs. () and () into Eq. (), the
induced current density can be written as
𝐼
=−𝑠−0
𝐼
+1
01
𝑠∇
×
𝐼
−1
01
𝑠(−)
𝑟×
𝐼.()
where
𝐼and
𝐼are the electric and magnetic elds
inside the sphere and the last term, on the right-hand side,
corresponds to an induced surface current. erefore, the
induced current within the sphere is completely determined
by the values of the internal electric and magnetic elds, and
the calculation of
𝐼is reduced to the calculation of
𝐼and
𝐼.
is means that the calculation of all the components
𝑁𝐿
𝛼𝛽 (
󸀠,
) can be set as a scattering problem, that is,
as the problem of calculating the scattered and internal
electromagnetic elds induced by an electromagnetic plane
wave incident upon a sphere, for three dierent polarizations
𝛽of the incident plane wave. is problem is very similar
to the classical problem of the scattering of a plane wave
by a sphere, solved rst by Gustav Mie in  []. e
main dierence between this problem and the classical
Miesolutionisthatherethewavevector
and the wave
frequency of the incident plane wave are independent of
each other, while, in the problem solved by Mie, the incident
planewaveisaself-propagatingwavewhosewavevector
and frequency are related through the dispersion relation of
the medium surrounding the sphere. is also means that
here the incident plane wave has to be generated by external
currents capable of xing its wavevector
to any given value.
For example, if
𝛽are chosen as the cartesian unit vectors
and the sphere is immersed in a medium with permittivity
0and permeability 0, then the external electric eld
𝑒𝑥𝑡
must satisfy the Maxwell’s equations
∇×
𝑒𝑥𝑡 =0
𝑒𝑥𝑡 ()
∇×
𝑒𝑥𝑡 =
𝑒𝑥𝑡 0
𝑒𝑥𝑡 ()
where one includes as a source the external current
𝑒𝑥𝑡.
us, the external electric eld satises the vectorial equation
∇×∇×
𝑒𝑥𝑡 +200
𝑒𝑥𝑡 =0
𝑒𝑥𝑡 ()
and the source
𝑒𝑥𝑡 is chosen in order to obtain the desired
external electric eld, say, a plane wave
𝛽𝑖𝑝𝑧 propagating
along the axis, with polarization
𝛽taking each one of the
polarizations {
𝑥,
𝑦,
𝑧}.Itcanbeshownthatinthiscase
𝑒𝑥𝑡
becomes
𝑒𝑥𝑡
,
;
𝑥= 1
02−2
0𝑖𝑝𝑧
𝑥()
𝑒𝑥𝑡
,
;
𝑦= 1
02−2
0𝑖𝑝𝑧
𝑦()
𝑒𝑥𝑡
,
;
𝑧=− 1
02
0𝑖𝑝𝑧
𝑧,()
as it could be veried by direct substitution. Note that in
order to solve this problem in its most general way one
has to accept the presence of longitudinal currents that will
give rise to an external eld with a longitudinal component.
Nevertheless, the calculation procedure of the scattered and
internal elds follows the standard mathematical process by
rst expanding the electromagnetic elds on a spherical basis
and then setting boundary conditions both at innity and on
the surface of the sphere.
4. Spherical Basis and Boundary Conditions
epurposeofthissectionistosolveMaxwellsequations
for a sphere of radius located at the origin, and with the
external currents
𝑒𝑥𝑡 given by Eqs. ()-(). We start by
writing Maxwell’s equations as
∇×∇×
+2𝑠
;𝑠
;
 =0
𝑒𝑥𝑡 ()
where we have assumed, as above, that all elds have an
oscillatory time dependence −𝑖𝜔𝑡,and
𝑠
;=
𝑠()<
0()>
and 𝑠
;=
𝑠()<
0()>.
()
us we start by expanding the external plane wave in a vector
spherical harmonic basis; that is, we write 𝑖𝑝𝑧
𝑥,𝑖𝑝𝑧
𝑦,𝑖𝑝𝑧
𝑧
in the functional basis [, p. ]
𝑒𝑛𝑚 =∇×
𝑒𝑛𝑚,
𝑜𝑛𝑚 =∇×
𝑜𝑛𝑚, ()
𝑒𝑛𝑚 =1
∇×
𝑒𝑛𝑚,
𝑜𝑛𝑚 =1
∇×
𝑜𝑛𝑚,()
𝑒𝑛𝑚 =1
∇𝑒𝑛𝑚,
𝑜𝑛𝑚 =1
∇𝑜𝑛𝑚,()
Mathematical Problems in Engineering
where
𝑒𝑛𝑚 =cos 𝑚
𝑛(cos )𝑛,()
𝑜𝑛𝑚 =sin (cos )𝑛,()
where isthewavenumberassociatedtothewaveequation
that satises the vector spherical harmonics
∇×∇×
−2
=0, ()
for
𝑒𝑛𝑚,
𝑜𝑛𝑚,
𝑒𝑛𝑚,
𝑜𝑛𝑚,and
∇∇⋅
+2
=0, ()
for
𝑒𝑛𝑚,
𝑜𝑛𝑚.Here𝑚
𝑛are the associated Legendre poly-
nomials as dened in [, p. ] and 𝑛denote either one of
sphericalBesselfunctions:
𝑛=
2𝑛+1/2  ()
𝑛=
2𝑛+1/2 . ()
It can be shown by using the orthogonality property of the
vector spherical harmonics that the expansion of a plane
wave for the three dierent polarizations {
𝑥,
𝑦,
𝑧}in a
vector spherical harmonic basis is given by [][pg. , Eq.
(36)-(38)].
𝑒𝑥𝑡
;
𝑥=𝑖𝑝𝑧
𝑥
=
𝑛=1 2+1
(+1)𝑛
𝑜𝑛1 ,
−
𝑒𝑛1 ,
()
𝑒𝑥𝑡
;
𝑦=𝑖𝑝𝑧
𝑦
=
𝑛=1 (2+1)
(+1)𝑛
𝑒𝑛1 ,
+
𝑜𝑛1 ,
()
𝑒𝑥𝑡
;
𝑧=𝑖𝑝𝑧
𝑧=
𝑛=0 (2+1)𝑛−1
𝑒𝑛0 ,
 ()
wherethesecondvariableintheargumentof
𝑒𝑥𝑡 denotes the
polarization.emagneticeldsarecalculatedwithFaradays
law
𝑒𝑥𝑡 =1
0∇×
𝑒𝑥𝑡.()
We include the result in Appendix A, in Eqs. (A.), (A.),
and(A.).Now,weproceedtosolvethescatteringproblem
by assuming that the electric eld outside the sphere is the
incident external electric eld
𝑒𝑥𝑡 plus a scattered electric
eld
𝑠, while inside the sphere is
𝐼, and they must obey
the electromagnetic boundary conditions at the surface of the
sphere. Inside the sphere, the electric eld
𝐼must fulll the
inhomogeneous vectorial wave equation given by Eq. (),
whose solution is the sum of the homogeneous equation,
expressed as an expansion of vector spherical harmonics, plus
aparticularsolution
𝑝𝑎𝑟𝑡
;
𝑥=𝑠
02−2
0
2−2
𝑠
𝑒𝑥𝑡
;
𝑥
=𝑠
02−2
0
2−2
𝑠𝑖𝑝𝑧
𝑥,()
𝑝𝑎𝑟𝑡
;
𝑦=𝑠
02−2
0
2−2
𝑠
𝑒𝑥𝑡
;
𝑦
=𝑠
02−2
0
2−2
𝑠𝑖𝑝𝑧
𝑦,()
𝑝𝑎𝑟𝑡
;
𝑧=0
𝑠
𝑒𝑥𝑡
;
𝑧=0
𝑠𝑖𝑝𝑧
𝑧,()
which can be written in terms of the external eld
𝑒𝑥𝑡.Here
2
0=
200while 2
𝑠=
2𝑠𝑠.Here we are using the
notation
𝑝𝑎𝑟𝑡(
;
𝑥)meaning that the particular solution
of the electric eld depends on the polarization
𝑥of the
exciting eld. e magnetic eld associated to this electric
eldisgivenbyEqs.(A.),(A.),and(A.)inAppendixA.
erefore, the electric eld induced inside a sphere with
radius can be written as
𝐼
;
𝑥=
𝑛=1 2+1
(+1)𝑛𝑛
𝑜𝑛1 𝑠,

−𝑛
𝑒𝑛1 𝑠,
+𝑠
02−2
0
2−2
𝑠𝑖𝑝𝑧
𝑥
()
𝐼
;
𝑦=
𝑛=1 (2+1)
(+1)𝑛𝑛
𝑒𝑛1 𝑠,

+𝑛
𝑜𝑛1 𝑠,
+𝑠
02−2
0
2−2
𝑠𝑖𝑝𝑧
𝑦
()
𝐼
;
𝑧=
𝑛=0 (2+1)𝑛−1𝐿
𝑛
𝑒𝑛0 𝑠,
+0
𝑠
⋅𝑖𝑝𝑧
𝑧
()
where the coecients 𝑛,𝑛,𝐿
𝑛are determined by the bound-
ary conditions. e corresponding magnetic eld is given by
Eqs. (A.), (A.), and (A.), where the magnetic eld
𝑒𝑥𝑡
is given by (A.)-(A.), for the three dierent polarizations
of the external electric eld
𝑒𝑥𝑡. We now assume that the
scatteredeldcanbewrittenas
𝑠
;
𝑥=
𝑛=1 2+1
(+1)𝑛𝑛
(3)
𝑒𝑛1 0,

−𝑛
(3)
𝑜𝑛1 0,
, ()
Mathematical Problems in Engineering
𝑠
;
𝑦=
𝑛=1 (2+1)
(+1)𝑛𝑛
(3)
𝑜𝑛1 0,

+𝑛
(3)
𝑒𝑛1 0,
, ()
𝑠
;
𝑧=
𝑛=1 (2+1)𝑛−1𝐿
𝑛
(3)
𝑒𝑛0 0,
, ()
where the index (3)indicates that the functional dependence
of the vector spherical harmonics is through the spherical
Hankel function 𝑛(0). e scattered magnetic eld is a
consequence of Faraday’s law and is given by Eqs. (A.),
(A.), and (A.) in Appendix A.
Now we apply the boundary conditions for the electro-
magnetic eld that consist in the continuity of the parallel
component, of both the electric and magnetic elds, at the
surface of the sphere =.Inasphericalbasisthese
boundary conditions can be written as
𝑒𝑥𝑡𝜃+
𝑠𝜃=
𝐼𝜃
and
𝑒𝑥𝑡𝜃+
𝑠𝜃=
𝐼𝜃,()
and
𝑒𝑥𝑡𝜙+
𝑠𝜙=
𝐼𝜙
and
𝑒𝑥𝑡𝜙+
𝑠𝜙=
𝐼𝜙.()
e rst set of boundary conditions given in Eqs. () yields
the following expressions for the expansion coecients:
𝑛,0,𝑠= (1−)
𝑛𝑠/𝑠𝑛𝑠1/0𝑛
𝑛𝑠
𝑛0𝑠/𝑠𝑛𝑠0/0𝑛0
𝑛𝑠 ,()
𝑛,0,𝑠= (1−)𝑛𝑠/𝑠
𝑛𝑠1/0
𝑛𝑛𝑠
𝑛0𝑠/𝑠
𝑛𝑠0/0
𝑛0𝑛𝑠 ,()
𝑛,0,𝑠= 𝑛01/0
𝑛0/0
𝑛0(1−)𝑛
𝑛0𝑠/𝑠
𝑛𝑠0/0
𝑛0𝑛𝑠 ,()
𝑛,0,𝑠=
𝑛01/0𝑛0/0𝑛0(1−)
𝑛
𝑛0𝑠/𝑠𝑛𝑠0/0𝑛0
𝑛𝑠 ,()
where
𝑛= 1
𝑛,
𝑛= 1
𝑛, ()
=𝑠
02−2
0
2−2
𝑠,
=2−2
0
2−2
𝑠.()
e second set of boundary conditions given in Eqs. ()
yields, for an external electric eld of the form
𝑧𝑖𝑝𝑧,the
following expressions for the expansion coecients of the
scattered eld in Eq. () and the internal electric eld in Eq.
():
𝐿
𝑛,0,𝑠= 0
𝑠𝑠
0𝑛𝑠
𝑛0𝐿
𝑛, ()
𝐿
𝑛,0,𝑠
=10/𝑠𝑛/𝑛0
𝑛𝑠𝑛00/𝑠𝑠/0𝑛𝑠
𝑛0.()
Once the internal electric eld has been calculated, we are
able to calculate the current density
𝐼inducedwithinthe
sphere with the aid of Eq. () that we rewrite as
𝐼
=−𝑠−0
𝐼
+𝑠
0−1∇
×
𝐼
+1−𝑠
0(−)
𝑟
×
𝐼
,
()
where
𝐼is given in Eqs. (A.)-(A.) for each polarization of
the external electric eld.
5. The Induced Current
Now one substitutes the internal electric eld for each polar-
ization, as given in Eqs. (), (), and (), into equation
(), to arrive at the following result.
e induced current inside the sphere for each polariza-
tion {
𝑥,
𝑦,
𝑧}of the incident eld with arbitrary wave vector
is
𝐼
;
𝑥
=𝑘
𝑛=1 2+1
(+1)𝑛𝑛
𝑜𝑛1 𝑠,

Mathematical Problems in Engineering
−𝑛
𝑒𝑛1 𝑠,
+𝑝𝑠
02−2
0
2−2
𝑠𝑖𝑝𝑧
𝑥
+1−𝑠
0(−)
𝑟×
𝐼
;
𝑥,
()
𝐼
;
𝑦=𝑘
𝑛=1 (2+1)
(+1)𝑛𝑛
𝑒𝑛1 𝑠,

+𝑛
𝑜𝑛1 𝑠,
+𝑝𝑠
02−2
0
2−2
𝑠𝑖𝑝𝑧
𝑦
+1−𝑠
0(−)
𝑟×
𝐼
;
𝑦,
()
𝐼
;
𝑧=𝑘
𝑛=0 (2+1)𝑛−1𝐿
𝑛
𝑒𝑛0 𝑠,

+𝑝𝑧 0
𝑠𝑖𝑝𝑧
𝑧+1−𝑠
0(−)
𝑟×
𝐼
;
𝑧,
()
where 𝑘,𝑝,𝑝𝑧 are given by
𝑘=1
02
𝑠−2
0, ()
𝑝=1
020𝑠−0+1−0
𝑠2, ()
𝑝𝑧 =1
020𝑠−0. ()
is result allows us to express the conductivity tensor in a
mixed Fourier representation as
𝑁𝐿
,

𝑥=
𝐼
;
𝑥, ()
𝑁𝐿
,

𝑦=
𝐼
;
𝑦, ()
𝑁𝐿
,

𝑧=
𝐼
;
𝑧, ()
and one must remember that the amplitude of the incident
planewavehasbeentakenasunity.
5.1. Fourier Transform of the Induced Current. Once we have
an expression for the internal induced current
𝐼,wecan
take its Fourier transform and, with the aid of Eq. (), we
obtain an expression for the nonlocal conductivity tensor
𝑁𝐿(
󸀠,
) in Fourier space. Since
𝐼is given as an
expansion in vector spherical harmonics, the calculation of
its Fourier transform requires the calculation of the Fourier
transform of each spherical harmonic, and this is what we do
next.
First, following [, p. ], we dene the vectors
𝑛𝑚 ,=𝑚
𝑛(cos )𝑖𝑚𝜙
𝑟,()
𝑛𝑚 ,
=
𝑚
𝑛(cos )
𝜃+𝑚
𝑛(cos )
sin
𝜙𝑖𝑚𝜙,()
𝑛𝑚 ,
=𝑚
𝑛(cos )
sin
𝜃
𝑚
𝑛(cos )
𝜙𝑖𝑚𝜙.()
en, the vector spherical harmonics can be written as
𝑛𝑚 ,,=𝑛
𝑛𝑚 ,, ()
𝑛𝑚 ,,=(+1)𝑛
𝑛𝑚 ,
+1
𝑛
𝑛𝑚 ,, ()
𝑛𝑚 ,,=
𝑛
𝑛𝑚 ,
+𝑛
𝑛𝑚 ,, ()
andwerecallthatinournotation
𝑛𝑚 ,,=
𝑒𝑛𝑚 ,,+
𝑜𝑛𝑚 ,,,()
𝑛𝑚 ,,=
𝑒𝑛𝑚 ,,+
𝑜𝑛𝑚 ,,,()
𝑛𝑚 ,,=
𝑒𝑛𝑚 ,,+
𝑜𝑛𝑚 ,,. ()
Next, it can be shown [, p. ] that the functions (), (),
and()satisfythefollowingintegralrelations:
𝑛𝑚 ,,
=𝑛−1
4 2𝜋
0𝜋
0−𝑖𝑘𝑟̂
𝑟⋅̂
𝑟󸀠
𝑛𝑚 󸀠,󸀠sin 󸀠󸀠󸀠,()
𝑛𝑚 ,,
=𝑛
42𝜋
0𝜋
0−𝑖𝑘𝑟̂
𝑟⋅̂
𝑟󸀠
𝑛𝑚 󸀠,󸀠sin 󸀠󸀠󸀠,()
𝑛𝑚 ,,
=𝑛−1
4 2𝜋
0𝜋
0−𝑖𝑘𝑟̂
𝑟⋅̂
𝑟󸀠
𝑛𝑚 󸀠,󸀠sin 󸀠󸀠󸀠,()
where
⋅
󸀠=sin sin 󸀠cos 󸀠+cos cos 󸀠,()
Mathematical Problems in Engineering
and this integral relation will be used to evaluate the angular
integration of the Fourier transform of (), (), and ().
us, by using Eq. () and taking its Fourier transform we
obtain
𝑉𝑎
𝑛𝑚 𝑠−𝑖󳨀
𝑝󸀠󳨀
𝑟3=2𝜋
0𝜋
0𝑎
0𝑛𝑠
𝑛𝑚 ,−𝑖𝑝󸀠𝑟(
̂
𝑝󸀠̂
𝑟) sin 2
=𝑎
0𝑛𝑠2𝜋
0𝜋
0
𝑛𝑚 ,−𝑖𝑝󸀠𝑟(
̂
𝑝󸀠̂
𝑟)
sin 2.
()
Here we have written
󸀠
=
󸀠(
󸀠
) separating
the directional dependence from the magnitude. Also here
𝑎indicates that the integration volume is the volume of
a sphere of radius ,andwehaveusedthesphericalbasis
{󸀠,󸀠
𝑝,󸀠
𝑝}for evaluating the integrals. In this basis,
󸀠=
󸀠sin 󸀠
𝑝cos 󸀠
𝑝
𝑥+
󸀠sin 󸀠
𝑝sin 󸀠
𝑝
𝑦+
󸀠cos 󸀠
𝑝
𝑧,where
{
𝑥,
𝑦,
𝑧}is the vector basis in which the spherical harmonics
were dened, and
𝑧is chosen along the wave vector
of the
incident plane wave.
NowweuseEqs.()and()toget
𝑉𝑎
𝑛𝑚 𝑠−𝑖󳨀
𝑝󸀠󳨀
𝑟3
=𝑎
0𝑛𝑠
𝑛𝑚 󸀠,󸀠
𝑝,󸀠
𝑝2
=4
𝑛𝑎
0𝑛𝑠𝑛󸀠2
𝑛𝑚 󸀠
𝑝,󸀠
𝑝.
()
us, one can nally write
𝑛𝑚 󸀠,󸀠
𝑝,󸀠
𝑝
=43
𝑛𝑛,𝑛 𝑠,󸀠,2
𝑛𝑚 󸀠
𝑝,󸀠
𝑝, ()
where the argument of the function indicates the corre-
sponding Fourier space, 󸀠,󸀠
𝑝,󸀠
𝑝, which are the spherical
coordinates in the Fourier space. Also we have dened
𝑛,𝑛󸀠1,2,1
0𝑛1𝑛󸀠2𝑁. ()
erefore, using this result, the Fourier transform of
𝑒𝑛𝑚
and
𝑜𝑛𝑚 can be written as
𝑒𝑛𝑚 󸀠,󸀠
𝑝,󸀠
𝑝
=43
𝑛𝑛,𝑛 𝑠,󸀠,2Re
𝑛𝑚 󸀠
𝑝,󸀠
𝑝, ()
𝑜𝑛𝑚 󸀠,󸀠
𝑝,󸀠
𝑝
=43
𝑛𝑛,𝑛 𝑠,󸀠,2Im
𝑛𝑚 󸀠
𝑝,󸀠
𝑝. ()
Furthermore,itcanalsobeshown[]thattheintegral
𝑛,𝑛(𝑠,󸀠,2)inEq.()canbereadilyevaluatedtoyield
𝑛,𝑛 1,2,2= 1
2
1−2
22𝑛1𝑛−1 2
−1𝑛−1 1𝑛2. ()
OnecanalsocalculatetheFouriertransformof
𝑛𝑚 using Eq.
()andthesameprocedureasabove.Aersomealgebraic
simplications, one can write
𝑒𝑛𝑚 󸀠,󸀠
𝑝,󸀠
𝑝=43
𝑛−1 (+1)1,𝑠,󸀠
Re
𝑛𝑚 󸀠
𝑝,󸀠
𝑝+2,𝑠,󸀠
Re
𝑛𝑚 󸀠
𝑝,󸀠
𝑝,
()
𝑜𝑛𝑚 󸀠,󸀠
𝑝,󸀠
𝑝=43
𝑛−1 (+1)1,𝑠,󸀠
Im
𝑛𝑚 󸀠
𝑝,󸀠
𝑝+2,𝑠,󸀠Im
𝑛𝑚
⋅󸀠
𝑝,󸀠
𝑝,
()
where we have dened
1,𝑠,󸀠= 1
3𝑎
0𝑛𝑠
𝑠
𝑛𝜌=𝑝󸀠𝑟
+1
𝑛𝜌=𝑘𝑠𝑟𝑛
𝜌=𝑝󸀠𝑟2
=𝑛𝑠
𝑠𝑛󸀠
󸀠,
()
and
2,𝑠,󸀠1
3𝑎
0(+1)𝑛𝑠
𝑠
𝑛󸀠
󸀠+1
𝑛𝜌=𝑘𝑠𝑟
1
𝑛𝜌=𝑝󸀠𝑟2.
()
In fact it is possible to evaluate 2,denedinEq.(),inclosed
form,asitwillbeseenlateron.Itturnsouttobegivenby
2,𝑠,󸀠
=
󸀠𝑠2−󸀠2
𝑛󸀠,𝑠
𝑛󸀠,𝑠,()
Mathematical Problems in Engineering
where
𝑛(󸀠,𝑠),
𝑛(󸀠,𝑠) are the coecients in the
result (), () calculated without magnetic response, i.e.,
𝑠=0.
Finally we need the Fourier transform of 𝑖𝑝𝑧,whichin
our coordinate system, where the axis coincides with the
propagation direction
,couldbewrittenas𝑖󳨀
𝑝⋅󳨀
𝑟,anditis
given by
𝑖󳨀
𝑝⋅󳨀
𝑟=431
−
󸀠
−
󸀠.()
Here
indicates the Fourier transform operator. As partic-
ular case, if |
| = |
󸀠|=
0and is the angle between
them, this implies that |
−
󸀠|=20sin(/2)and the former
expression can be written as
1
−
󸀠
−
󸀠=120sin (/2)
20sin (/2).()
6. Results
With these expressions it is now possible to calculate the
Fourier transform of each vector spherical harmonic and
eachplanewavethatappearintheexpressionsfortheinternal
induced current given in Eqs. (), (), and (). We split the
nonlocal conductivity tensor as
𝑁𝐿
󸀠,
=
𝑉
󸀠,
+
Σ
󸀠,
, ()
where
𝑉corresponds to the conductivity associated to the
induced currents in Eqs. (), (), and () located in the
bulk of the sphere while
Σcorresponds to the conductivity
associated to the induced currents located at the surface of
the sphere, that is, the term with ().
6.1. Nonlocal Conductivity Tensor. Finally, using Eq. () we
arrive at the main results of this work.
Bulk Contribution
𝑉
󸀠,

𝑥=43𝑘
𝑛=1 2+1
(+1)𝑛
⋅𝑛,𝑛 𝑠,󸀠,2Im
𝑛1 󸀠
𝑝,󸀠
𝑝+𝑛
⋅(+1)1,𝑠,󸀠Re
𝑛1 󸀠
𝑝,󸀠
𝑝
+2,𝑠,󸀠Re
𝑛1 󸀠
𝑝,󸀠
𝑝+43𝑝𝑠
0
2−2
0
2−2
𝑠
1
−
󸀠
−
󸀠
𝑥,
()
er=
p
e
e
ey
ex
ez=
p
F : Figure of the coordinate systems employed.
𝑉
󸀠,

𝑦=43𝑘
𝑛=1 2+1
(+1)𝑛
⋅𝑛,𝑛 𝑠,󸀠,2Re
𝑛1 󸀠
𝑝,󸀠
𝑝−𝑛(
+1)1,𝑠,󸀠Im
𝑛1 󸀠
𝑝,󸀠
𝑝
+2,𝑠,󸀠Im
𝑛1 󸀠
𝑝,󸀠
𝑝+43𝑝𝑠
0
2−2
0
2−2
𝑠
1
−
󸀠
−
󸀠
𝑦,
()
𝑉
󸀠,

𝑧=43𝑘
𝑛=1 (2+1)𝐿
𝑛(
+1)1,𝑠,󸀠Re
𝑛0 󸀠
𝑝,󸀠
𝑝()
+2,𝑠,󸀠Re
𝑛0 󸀠
𝑝,󸀠
𝑝+43𝑝𝑧 0
𝑠
1
−
󸀠
−
󸀠
𝑧,()
where 𝑘,𝑝,𝑝𝑧 are given by Eqs. (), (), and (). We
must remember that the basis {
𝑥,
𝑦,
𝑧}is such that
𝑧
coincide with the
direction. e angles 󸀠
𝑝,󸀠
𝑝are the
spherical coordinates of
󸀠with respect to
;seeFigure.
Now we include the surface magnetization currents
induced upon the sphere; for this we must include the
contribution of the surface term () and take its Fourier
transform.
Because the surface term () includes a Dirac delta for
=, then the Fourier transform with variable
󸀠for the
surface term is
Σ
󸀠,

=21𝑠
02𝜋
0𝜋
0
𝑟×
𝐼−𝑖𝑝󸀠𝑎̂
𝑝󸀠̂
𝑟sin . ()
 Mathematical Problems in Engineering
e eld
𝐼is given by (A.)-(A.), and by the fact that the
vector spherical harmonics can be expressed in terms of the
functions
𝑚𝑛,
𝑚𝑛,
𝑚𝑛 by the expression
𝑛𝑚 ,,=𝑛
𝑛𝑚 ,, ()
𝑛𝑚 ,,=(+1)𝑛
𝑛𝑚 ,
+1
𝑛
𝑛𝑚 ,. ()
Also we have that
𝑟×
𝑚𝑛 ,=0, ()
𝑟×
𝑚𝑛 ,=
𝑚𝑛 ,, ()
𝑟×
𝑚𝑛 ,=
𝑚𝑛 ,, ()
and by (), () we get that
2𝜋
0𝜋
0
𝑟×
𝑛𝑚𝑖𝑝󸀠𝑎̂
𝑝󸀠̂
𝑟sin 
=−4
𝑛
𝑛𝑚 󸀠,󸀠
𝑝,󸀠
𝑝, ()
2𝜋
0𝜋
0
𝑟×
𝑛𝑚𝑖𝑝󸀠𝑎̂
𝑝󸀠̂
𝑟sin 
=4
𝑛−1
𝑛𝑚 󸀠,󸀠
𝑝,󸀠
𝑝. ()
With all this we have the next result.
e surface term given by () associated to the surface
current and by the application of the integration formulas
()-() we nd that its Fourier transform, when we consider
that the external electric eld has a unitary amplitude, gives
the part of the nonlocal conductivity tensor associated to
these surface currents; consider the following.
Surface Contribution
Σ
󸀠,

𝑥=421𝑠
0
𝑛=1 2+1
(+1)
⋅𝑠
𝑠𝑛
𝑛𝑠+
0
𝑛
𝑜𝑛1 󸀠,󸀠
𝑝,󸀠
𝑝
− 𝑠
𝑠𝑛𝑛𝑠+
0𝑛
𝑒𝑛1 󸀠,󸀠
𝑝,󸀠
𝑝
()
Σ
󸀠,

𝑦=−421𝑠
0
𝑛=1 2+1
(+1)
⋅𝑠
𝑠𝑛
𝑛𝑠+
0
𝑛
𝑒𝑛1 󸀠,󸀠
𝑝,󸀠
𝑝
+ 𝑠
𝑠𝑛𝑛𝑠+
0𝑛
𝑜𝑛1 󸀠,󸀠
𝑝,󸀠
𝑝
()
Σ
󸀠,

𝑧=420
𝑠−1𝑠
0
𝑛=0 (2+1)
⋅𝐿
𝑛𝑛𝑠
𝑒𝑛0 󸀠,󸀠
𝑝,󸀠
𝑝()
where
=2−2
0
2−2
𝑠.()
erefore, we have arrived to the main result of our work, the
complete nonlocal generalized conductivity tensor written
as the addition of the bulk conductivity and the surface
conductivity
𝑁𝐿
󸀠,
=
𝑉
󸀠,
+
Σ
󸀠,
.()
We must remember thatthe base {
𝑥,
𝑦,
𝑧}is such that vector