#### Abstract

The purpose of this article is to calculate the generalized nonlocal conductivity tensor of a spherical particle made of isotropic and linear materials. The generalized conductivity tensor is a crucial element in the formulation of the mean-field theories of the electromagnetic response of random particulate systems. This is equivalent to what is called the T-Matrix in multiple scattering theories. Here, a new method is proposed for finding explicit expressions for this tensor directly from its definition 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 specific systems are analyzed and the results of some calculations are presented as a numerical example.

#### 1. Introduction

In a previous paper [1], a reference was made to the generalized nonlocal conductivity tensor, as the main physical concept in the development of a formalism for the calculation of the effective bulk electromagnetic response of randomly located discrete scatterers such as turbid colloids. This approach was later extended to the calculation of the reflection and transmission amplitudes of the average electric field of colloidal suspensions confined in a half-space [2]. The generalized conductivity tensor continued to play a major role in the framework of the theoretical description [3]. Colloidal systems are defined as two-phase systems in which a sparse phase is immersed within a continuous one [4]. While the continuous phase is usually called the matrix, the sparse phase is said to be composed of colloidal particles. When an external electromagnetic field is incident upon a colloidal system, it induces currents within the randomly located colloidal particles and these currents radiate electromagnetic fields which act back upon the particles themselves. The total electromagnetic field conformed in this way can be split into two components, an average component with a smooth spatial variation and traveling in a definite direction, called the coherent beam, and a fluctuating component with abrupt spatial variations and traveling in all different directions, called the diffuse field.

The work cited above [2], related to the reflection and transmission amplitudes from colloidal suspensions, dealt with the reflection and transmission amplitudes of the coherent beam for a system of randomly located identical spheres, within a homogeneous matrix with a flat interface occupying a half-space. The 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 average electric field is analogous to the one used in multiple-scattering-theory (MST), to solve similar problems. The 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 [5] 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 first recall that the T-matrix operator and the generalized nonlocal conductivity tensor are actually proportional to each other [1] obeying, essentially, the same integral equation. Then, 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. Therefore, the method described here poses an alternative to the calculation of the T-matrix operator, by simply finding the induced current inside a sphere given an arbitrary external electric field, yielding attractive closed-form expressions for its components. Finally, we want to point out that although the components of the T-matrix operator have been already calculated for the case of a sphere with no intrinsic magnetic response [5], here we provide new closed-form expressions for these components for the more general case of a sphere with an intrinsic magnetic local permeability different from the one of its surroundings, yielding, besides a bulk contribution, also a surface contribution coming from 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

The nonlocal conductivity tensor of an isolated sphere is defined as the linear response of the internal induced current to an external electric field; in its most general form it can be written aswhere the dependence on the frequency on a given quantity denotes the -component of the corresponding time Fourier transform. Here, is the total internal induced current, is the generalized nonlocal conductivity tensor, and is the external electric field; it could also be called the exciting field. The term generalized means that in all internal induced currents are included, even those that give rise to a magnetic response. In this sense Eq. (1) can be regarded as a generalized nonlocal Ohm’s law.

The total internal induced current can be calculated, also, by using the local relationshipwherewhere is the radius of the sphere, is its bulk local conductivity, and is the internal electric field, that is, the electric field within the sphere. But the electric field at any point is given byHere is the volume of the sphere and alsois the free Green’s function dyadic, , andTherefore, when one uses Eq. (2) to iterate Eq. (4), one can find the equation for , as defined by Eq. (1). First, since equation Eq. (4) also holds inside the sphere , then, if we multiply Eq. (4) by , we obtainSubstituting now Eq. (1) into Eq. (7), we have thatBy interchanging the order of integration and comparing the terms within the integral, we findNow, we recall the definition of the T-matrix through the following integral relation [6, Eq. 10.4.2]: Here corresponds to the external field within the sphere. is the electric field scattered by the sphere for . Since one can identify the scattered field with the second term in the right-hand side of Eq. (4), by replacing in the right-hand side of Eq. (4) with the expression given in Eq. (1) it follows immediately that Eq. (10) yieldsThis relation is valid for and . Therefore, the calculation of all the components of can be performed by solving the integral equation given in Eq. (9). This procedure has been already followed for the corresponding integral equation for T-matrix operator [5]. It is pertinent to point out that the definition of the T-matrix operator is given, sometimes, as the linear integral relationship between the scattered field and the external field [5]. In this case the Green’s function dyadic is incorporated in the definition of the T-matrix operator; thus its relationship with the generalized nonlocal conductivity has to be modified correspondingly.

#### 3. Calculation Procedure

Here we propose an alternative and simple way to evaluate the components of the nonlocal generalized conductivity tensor for a sphere, by taking advantage of its physical interpretation. This means that one has only to calculate the current density induced within the sphere by an arbitrary external electric field. In order to see how this is done, let us start by assuming that all the fields are in the frequency domain. Now we consider that an external electromagnetic plane wave is incident upon the sphere, and we write its electric field aswhere is the wave vector, is a unit vector along the direction of polarization, and we are assuming a unit amplitude. Then, according to Eq. (1), the current density induced within the sphere by this external plane wave is given bywhere as an argument in recalls the polarization of the external electric field. Now, we define the Fourier transform pair of asAnd by taking the Fourier transform in Eq. (13), one can writeFinally, the component of the above equation becomesThis means that the component of the nonlocal generalized conductivity tensor 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 .

But the induced current can be written in terms of the polarization and magnetization material fields, and , aswhereandHere and refer to the local electric permittivity and the local magnetic permeability of the sphere which are assumed to respond locally and linearly to the electric and magnetic fields ( and ) within the sphere. Here we have located the center of the sphere at the origin of the coordinate system, and while and are the local electric permittivity and the local magnetic permeability of the sphere, and denote the corresponding response functions of the medium outside the sphere; here we have a sharp discontinuity of these functions. Therefore, by plugging Eqs. (19) and (20) into Eq. (18), the induced current density can be written as where and are the electric and magnetic fields inside the sphere and the last term, on the right-hand side, corresponds to an induced surface current. Therefore, the induced current within the sphere is completely determined by the values of the internal electric and magnetic fields, and the calculation of is reduced to the calculation of and .

This means that the calculation of all the components can be set as a scattering problem, that is, as the problem of calculating the scattered and internal electromagnetic fields induced by an electromagnetic plane wave incident upon a sphere, for three different polarizations of the incident plane wave. This problem is very similar to the classical problem of the scattering of a plane wave by a sphere, solved first by Gustav Mie in 1908 [7]. The main difference between this problem and the classical Mie solution is that here the wave vector and the wave frequency of the incident plane wave are independent of each other, while, in the problem solved by Mie, the incident plane wave is a self-propagating wave whose wave vector and frequency are related through the dispersion relation of the medium surrounding the sphere. This also means that here the incident plane wave has to be generated by external currents capable of fixing its wavevector to any given value. For example, if are chosen as the cartesian unit vectors and the sphere is immersed in a medium with permittivity and permeability , then the external electric field must satisfy the Maxwell’s equationswhere one includes as a source the external current . Thus, the external electric field satisfies the vectorial equationand the source is chosen in order to obtain the desired external electric field, say, a plane wave propagating along the axis, with polarization taking each one of the polarizations . It can be shown that in this case becomesas it could be verified 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 field with a longitudinal component. Nevertheless, the calculation procedure of the scattered and internal fields follows the standard mathematical process by first expanding the electromagnetic fields on a spherical basis and then setting boundary conditions both at infinity and on the surface of the sphere.

#### 4. Spherical Basis and Boundary Conditions

The purpose of this section is to solve Maxwell’s equations for a sphere of radius located at the origin, and with the external currents given by Eqs. (25)-(27). We start by writing Maxwell’s equations aswhere we have assumed, as above, that all fields have an oscillatory time dependence , andThus we start by expanding the external plane wave in a vector spherical harmonic basis; that is, we write , , in the functional basis [8, p. 23]wherewhere is the wave number associated to the wave equation that satisfies the vector spherical harmonicsfor , , , , andfor . Here are the associated Legendre polynomials as defined in [9, p. 437] and denote either one of spherical Bessel functions:It can be shown by using the orthogonality property of the vector spherical harmonics that the expansion of a plane wave for the three different polarizations in a vector spherical harmonic basis is given by [10][pg. 420, Eq. ].where the second variable in the argument of denotes the polarization. The magnetic fields are calculated with Faraday’s lawWe include the result in Appendix A, in Eqs. (A.1), (A.2), and (A.3). Now, we proceed to solve the scattering problem by assuming that the electric field outside the sphere is the incident external electric field plus a scattered electric field , while inside the sphere is , and they must obey the electromagnetic boundary conditions at the surface of the sphere. Inside the sphere, the electric field must fulfill the inhomogeneous vectorial wave equation given by Eq. (28), whose solution is the sum of the homogeneous equation, expressed as an expansion of vector spherical harmonics, plus a particular solutionwhich can be written in terms of the external field . Here while Here we are using the notation meaning that the particular solution of the electric field depends on the polarization of the exciting field. The magnetic field associated to this electric field is given by Eqs. (A.4), (A.5), and (A.6) in Appendix A. Therefore, the electric field induced inside a sphere with radius can be written aswhere the coefficients are determined by the boundary conditions. The corresponding magnetic field is given by Eqs. (A.7), (A.8), and (A.9), where the magnetic field is given by (A.1)-(A.2), for the three different polarizations of the external electric field . We now assume that the scattered field can be written aswhere the index indicates that the functional dependence of the vector spherical harmonics is through the spherical Hankel function . The scattered magnetic field is a consequence of Faraday’s law and is given by Eqs. (A.10), (A.11), and (A.12) in Appendix A.

Now we apply the boundary conditions for the electromagnetic field that consist in the continuity of the parallel component, of both the electric and magnetic fields, at the surface of the sphere . In a spherical basis these boundary conditions can be written asandThe first set of boundary conditions given in Eqs. (52) yields the following expressions for the expansion coefficients:whereThe second set of boundary conditions given in Eqs. (53) yields, for an external electric field of the form , the following expressions for the expansion coefficients of the scattered field in Eq. (51) and the internal electric field in Eq. (48):Once the internal electric field has been calculated, we are able to calculate the current density induced within the sphere with the aid of Eq. (21) that we rewrite aswhere is given in Eqs. (A.7)-(A.9) for each polarization of the external electric field.

#### 5. The Induced Current

Now one substitutes the internal electric field for each polarization, as given in Eqs. (46), (47), and (48), into equation (62), to arrive at the following result.

The induced current inside the sphere for each polarization of the incident field with arbitrary wave vector iswhere are given byThis result allows us to express the conductivity tensor in a mixed Fourier representation asand one must remember that the amplitude of the incident plane wave has been taken as unity.

##### 5.1. Fourier Transform of the Induced Current

Once we have an expression for the internal induced current , we can take its Fourier transform and, with the aid of Eq. (16), we obtain an expression for the nonlocal conductivity tensor in Fourier space. Since is 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 [8, p. 27], we define the vectorsThen, the vector spherical harmonics can be written asand we recall that in our notationNext, it can be shown [8, p. 28] that the functions (72), (73), and (74) satisfy the following integral relations:whereand this integral relation will be used to evaluate the angular integration of the Fourier transform of (72), (73), and (74). Thus, by using Eq. (75) and taking its Fourier transform we obtainHere we have written separating the directional dependence from the magnitude. Also here indicates that the integration volume is the volume of a sphere of radius , and we have used the spherical basis for evaluating the integrals. In this basis, , where is the vector basis in which the spherical harmonics were defined, and is chosen along the wave vector of the incident plane wave.

Now we use Eqs. (82) and (75) to getThus, one can finally writewhere the argument of the function indicates the corresponding Fourier space, , which are the spherical coordinates in the Fourier space. Also we have definedTherefore, using this result, the Fourier transform of and can be written asFurthermore, it can also be shown [1] that the integral in Eq. (87) can be readily evaluated to yieldOne can also calculate the Fourier transform of using Eq. (76) and the same procedure as above. After some algebraic simplifications, one can writewhere we have definedandIn fact it is possible to evaluate , defined in Eq. (95), in closed form, as it will be seen later on. It turns out to be given bywhere , are the coefficients in the result (54), (57) calculated without magnetic response, i.e., .

Finally we need the Fourier transform of , which in our coordinate system, where the axis coincides with the propagation direction , could be written as , and it is given byHere indicates the Fourier transform operator. As particular case, if and is the angle between them, this implies that and the former expression can be written as

#### 6. Results

With these expressions it is now possible to calculate the Fourier transform of each vector spherical harmonic and each plane wave that appear in the expressions for the internal induced current given in Eqs. (63), (64), and (65). We split the nonlocal conductivity tensor aswhere corresponds to the conductivity associated to the induced currents in Eqs. (63), (64), and (65) located in the bulk of the sphere while corresponds 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. (16) we arrive at the main results of this work.

*Bulk Contribution*where are given by Eqs. (66), (67), and (68). We must remember that the basis is such that coincide with the direction. The angles are the spherical coordinates of with respect to ; see Figure 1.

Now we include the surface magnetization currents induced upon the sphere; for this we must include the contribution of the surface term (62) and take its Fourier transform.

Because the surface term (62) includes a Dirac delta for , then the Fourier transform with variable for the surface term isThe field is given by (A.7)-(A.9), and by the fact that the vector spherical harmonics can be expressed in terms of the functions by the expression<