# A study of anisotropic compact star with MIT bag EoS in $f(\Re,T)$ modified
theory

H. D. Singh<EMAIL_ADDRESS>J. Kumar<EMAIL_ADDRESS>Department of Mathematics, Central University of Jharkhand, Ranchi-835205,
India

###### Abstract

In the current study, we investigated a specific model of anisotropic strange
stars specially Her X-1, in the background of modified $f(\Re,T)$ gravity by
choosing $f(\Re,T)=\Re+2\xi T$, where $\Re$ is Ricci scalar, $T$ is the trace
of the energy- momentum tensor and $\xi$ is a coupling constant. To obtained
the solution for the modified field equations, we apply Buchdahl metric to our
equations. We consider the case, when the matter is governed by MIT bag model
equation of state as $P_{r}=\frac{1}{3}(\rho-4B)$, where $B$ is bag constant.
We calculate the values of unknown parameters using Schwarzschild interior
space-time followed by choosing the appropriate values of parameter $\xi$,
$K$, $\beta$ and also tabulated different values of bag constant for different
$\xi$. We examine the physical validity of our model by performing tests such
as energy conditions, equilibrium of the forces, the adiabatic index, redshift
and some more. The observational results showed that the proposed $f(\Re,T)$
model satisfies all these tests and are quite acceptable.

###### keywords:

Perfect fluids; Modified theory of relativity; Compact stars; MIT bag EoS;
General relativity.

##  1 Introduction

Einstein’s general theory of relativity has been continued to be very helpful
in uncovering hidden mystery of nature. Investigating the structural
properties of high density compact stars specially neutron stars has
interested many researchers both from mathematics and physics society. Pulsar
stars including high magnetic spinning stars were also considered as some of
high density compact stars in realm of astrophysics. Because of lack of
knowledge about these compact stars, it is believed that the material content
of such stars are subatomic particles. From predicting mercury’s orbit and
deflection of starlight because of some massive gravitational objects to
detecting gravitational waves and black holes in space, GR has been remarkably
used in concluding the phenomena of Cosmic Microwave Background Radiation [1],
Planck data [2], $\wedge$ Cold Dark Matter [3]. Still this theory has been
challenged by the presence of late-time acceleration of the universe and the
presence of the dark matter [4][5][6]. As a result of which, many
gravitational modified theory have been proposed such as $f(\Re)$ theory where
$\Re$ is the Ricci Scalar, $f(T,\tau)$ and where $T$ and $\tau$ are energy-
momentum tensor trace and torsion scalar, Gauss-Bonnet theory, Cartan gravity,
Rosen bimetric, etc. At present, Harko and his collaborators [7] look forward
and designed $f(\Re,T)$ gravity theory to work against the accelerated
expansion of the universe and dark energy matter[9] in the framework of
$f(\Re,T)$ gravity theory. Many studies have been done [10][11][12] in context
of anisotropic charged and uncharged models. Pretel and et.al studied the
static configuration of compact stars [13] with context of $f(\Re,T)$. The
models were constructed for $f(\Re,T)=\Re+2f(T)$ and
$f(\Re,T)=f_{1}(\Re)+f_{2}(T)$. Many authors, work in $f(\Re,T)$ of FLRW with
modified Chaplygin gas [14], Little Rip model [15] that explains present
stages of our universe. Santos and Ferst [16] studied the problem of casuality
in $f(\Re,T)$ modified gravity. Zubair and et.al [17] studied static spherical
symmetric of wormholes in framework of $f(\Re,T)$ gravity. Gravitational
decoupling by minimal geometric deformation model was studied by Azmat abd
Zubair [18]. Shabani and Farhoudi [19] uses dynamical procedure and found
solutions for $f(\Re,T)$ theory by considering parameters like Hubble
parameter and EoS parameter.

Now, different approaches were made to find analytic solutions of GR
equations. One is by embedding approach of four-dimensional spacetime. Such
type of approach is successful in acquisition of exact models [20]. In
numerous literatures, researchers showed solutions through embedding class-I
and class-II, solutions are also obtained by assuming various metric functions
[21][22]. Karmarkar condition has also been used in many model of compact
stars in the frame of different modified theories [23]. Kumar and et.al[24,
25, 26, 27, 28, 29] and many others[30, 31, 32, 33] used Buchdahl metrics
function in the framework of modified gravity . Zubair and Abbas [34] studied
interior model for compact stars in the framework of $f(\Re,T)$ gravity using
Krori and Barua solution. Moraes [35] studied stellar equilibrium in
background of $f(\Re,T)$ theory by using hydrostatic equilibrium equation.

In search of more realistic stellar configuration the researchers need to be
connected with the macroscopic properties of stars which is determined through
observations of the microphysics. A new method of stellar structure modeling
was born for which standard approaches were made which includes assumptions of
the metric function, energy density, anisotropic parameter, matter content
which allows connection to physics , masses linked to redshift and compactness
of stellar structure model. EoS is one such approach. It basically links the
radial pressure to the density to microphysics linearly. Applying barotropic
EoS basically helps in understanding complicated microphysics at a macroscale
level. We consider EoS in the form $p_{r}=p_{r}(\rho)$ that reduces the
difficulty in finding exact model of the modified field equations by single
generating function [36][37]. The colour-flavour locked EoS [38] describes
strange quark matter. It is also used in examining the surface tension of
stars which restrict the bag constant and tangential pressure included with
the EoS model.

For constructing a model, Negreiros and collaborators have modeled for compact
objects consisting of matter which follows MIT bag modeling (EoS) on the
surface of the compact stars[39][40]. In these paper, we have studied a new
class of anisotropic general solutions to modified Einsteins general
relativity field equations for relativistic compact stellar models by
considering Buchdahl metrics within the framework of $f(\Re,T)$ gravity
theory.And simplified the solutions using MIT bag EoS equations for the
compact stars Her X-1. This paper is structured as follows: in Sec.2, the
basic of MIT bag EoS is presented. In Sec.3, a general mathematical framework
of our $f(\Re,T)$ gravity theory has been shown, in sec.4, anisotropic matter
distributions in $f(\Re,T)$ gravity theory and their solutions of anisotropic
Buchdahl metric are b discussed. In Sec.5, we have been represented the
matching condition for interior space-time in the framework of $f(\Re,T)$
theory by smooth joining with exterior Schwarzchild geometry. In Sec.6, we
have described various physical properties for present model analytically as
well as graphically. In Sec.7, we concluded the detail discussion on the
determined results and their implication on the various cosmological cases.

##  2 MIT bag EOS

In study for compact objects, it is important to know that the principle of
thermodynamic in which variables are connected. It is described that the
microphysics properties of compact stars are linked to the macroscopic
properties by EoS, particularly their radii and masses. Now depending upon the
nature of interaction, the models were categorized by threebroad types: 1.
Non-relativistic potential models, 2. Relativistic theoretical field models
and 3. Relativistic DBHF models [41]. Futhermore, one can classify EoS in two
classes : first, EoS have a pressure which vanishes when density tends to zero
and second, self-bound EoS have a pressure which vanishes at significantly
finite density whereas our MIT bag model EoS can be referred to be self bound
EoS [42].

Now, compact stars are basically considered with strange quark matter which is
the ultimate ground of state of matter. It basically describes that the inward
and outward pressure of the bag helps in stabilizing the system [43] [44]. So,
the MIT bag EoS model is the simplest equation of state to study equilibrium
configuration of such models. To clear the properties of compact stars with
high densities, many considered EoS $p_{r}=p_{r}(\rho)$ as approximated linear
function of energy density [45][46][47]. Mathematically, MIT bag EoS model is
given by

$p_{r}=\frac{1}{3}(\rho-4B)$ (1)

where B is the bag constant. In original MIT bag model, the value of bag
constant was $B=55Mev/fm^{3}$. In recent articles, the range of $B$ are found
to be from $60$ to $90Mev/fm^{3}$ for strange quark stars [48][49][50].

However, a linear relationship guarantees the casuality condition. Although
the relationship between radial pressure and energy density is complex, still
the core of the object is approximately linear and the standard equation for a
stellar structure is given by the following expression

$p_{r}=\chi(\rho-\rho_{s})$ (2)

where $\chi$ is a positive constant. It is clear from the equation that for
$r=R$, the radial pressure vanishes i.e $p_{r}=0$. This happens due to the
energy density at zero pressure i.e $\rho(R)=\rho_{s}$, which is a non-
vanishing quantity. In case, EoS obeys polytropic form then the density
vanishes at the boundary $p_{r}=0$.

##  3 Basic field equation in $f(\Re,T)$ gravity

In this section, we formulate the general $f(\Re,T)$ as in [7] gravity theory.
While obtaining the Einstein’s field equations, the Ricci scalar in Einstein-
Hilbert action is given as

$\displaystyle S_{E}=\frac{1}{16}\int{\Re\sqrt{-g}}d^{4}x$ (3)

Now one can get $f(\Re,T)$ field equation from eqn.3, if we choose $f(\Re,T)$
in place of Ricci scalar $\Re$. Thus, the full action for the modified theory
of gravity is given by

$\displaystyle
S=\frac{1}{16}\int{f(\Re,T)\sqrt{-g}d^{4}x}+\int{L_{M}\sqrt{-g}d^{4}x}$ (4)

where in the first term $f(\Re,T)$ is an arbitrary function of Ricci scalar
$\Re$ with $g$ as the determinant of the metric tensor $g_{\mu\nu}$ and $T$ is
the trace of the stress-energy tensor,$T_{\mu\nu}$. And in the second term,
$L_{M}$ is the matter Lagrangian density which is related to stress-momentum
tensor and given as

$\displaystyle
T_{\mu\nu}=-\frac{2}{\sqrt{-g}}\frac{\delta(\sqrt{-g}L_{M})}{\delta
g^{\mu\nu}}$ (5)

with trace $T=g^{\mu\nu}T_{\mu\nu}$. Assuming that the Lagrangian density
$L_{M}$ only depends on the metric tensor components $g_{\mu\nu}$, we obtain
eqn.(5) as

$\displaystyle T_{\mu\nu}=g_{\mu\nu}L_{M}-\frac{2\partial(L_{M})}{\partial
g^{\mu\nu}}$ (6)

By varying eqn.(4) with respect to $g_{\mu\nu}$, we obtained the field
equation

$\displaystyle(\Re_{\mu\nu}-\bigtriangledown_{\mu}\bigtriangledown_{\nu})f_{\Re}(\Re,T)+g_{\mu\nu}\diamondsuit
f_{\Re}(\Re,T)-\frac{1}{2}f(\Re,T)g_{\mu\nu}$ $\displaystyle=$ $\displaystyle
8\pi T_{\mu\nu}-f_{T}(\Re,T)(T_{\mu\nu}+\varPsi_{\mu\nu})$ (7)

where $f_{\Re}(\Re,T)=\frac{\partial f(\Re,T)}{\partial\Re}$ and
$f_{T}(\Re,T)=\frac{\partial f(\Re,T)}{\partial T}$. The covariant derivative
is denoted by $\bigtriangledown_{\mu}$ and the diamond operator $\diamondsuit$
is the D’Alambert operator which is defined as

$\displaystyle\diamondsuit=\frac{\partial_{\mu}(\sqrt{-g}g^{\mu\nu}\partial_{\nu})}{\sqrt{-g}}$

$\displaystyle\varPsi_{\mu\nu}=g^{\alpha\beta}\frac{\delta
T_{\alpha\beta}}{\delta g^{\mu\nu}}$

Now we performed the covariant derivative of eqn.(7) by which we obtained the
energy-momentum tensor as

$\displaystyle\bigtriangledown^{\mu}T_{\mu\nu}=\frac{f_{T}(\Re,T)}{8\pi-
f_{T}(\Re,T)}[(T_{\mu\nu}+\varPsi_{\mu\nu})\bigtriangledown^{\mu}lnf_{T}(\Re,T)+\varPsi_{\mu\nu}\bigtriangledown^{\mu}-\frac{1}{2}g^{\mu\nu}\bigtriangledown^{\mu}T]$
(8)

Clearly from eqn.(8) we can see that the stress-energy momentum tensor
$T_{\mu\nu}$ in $f(\Re,T)$ theory does not follow the conservation law as in
Einstein general relativity (GR) due to the presence of some non-minimal
matter geometry coupling in its formulation. By using eqn.(6), the tensor
$\varPsi_{\mu\nu}$ is given by

$\displaystyle\varPsi_{\mu\nu}=-2T_{\mu\nu}+g_{\mu\nu}L_{M}-2g^{\alpha\beta}\frac{\partial^{2}L_{M}}{\partial
g^{\mu\nu}\partial g^{\alpha\beta}}$ (9)

To find the field equations, we consider the interior of star to be filled
with a perfect fluid source along with the energy-momentum tensor of the form

$\displaystyle
T_{\mu\nu}=(\rho+p_{t})u_{\mu}u_{\nu}-p_{t}g_{\mu\nu}+(p_{r}-p_{t})g_{\mu\nu}$
(10)

provided the four-velocity $u_{\nu}$ satisfied $u_{\mu}u^{\mu}=-1$ and
$u_{\nu}\bigtriangledown^{\mu}u_{\nu}=0$, $p_{r}$ and $p_{t}$ is the radial
and tangential pressure respectively . Taking the matter Lagrangian as
$L_{M}=-p$ where $p=\frac{1}{3}(p_{r}+2p_{t})$ [51, 52], we obtain eqn.(9) as

$\displaystyle\varPsi_{\mu\nu}=-2T_{\mu\nu}-pg_{\mu\nu}$ (11)

In this paper, we assumed the function $f(\Re,T)$as $f(\Re,T)=\Re+2f(T)$ where
$f(T)$ is the function of trace $(T)$ of the stress-energy tensor of matter.
Here, we choose $f(T)=\xi T$ to determine the modified theory of gravity,
where $\xi$ is a coupling constant.
Using the expression in eqn.(7), we obtain

$\displaystyle G_{\mu\nu}=8\pi T_{\mu\nu}+\xi
Tg_{\mu\nu}+2\xi(T_{\mu\nu}+pg_{\mu\nu})$ (12)

When $f(\Re,T)\equiv\Re$, then eqn.(7) reduces to Einstein field equations. By
putting $f(\Re,T)=\Re+2\xi T$ and eqn.(11) in eqn.(8) we get

$\displaystyle\bigtriangledown^{\mu}T_{\mu\nu}=-\frac{\xi}{2(4\pi+\xi)}[g_{\mu\nu}\bigtriangledown^{\mu}T+2\bigtriangledown^{\mu}(pg_{\mu\nu})]$
(13)

Now, we can write eqn.13 in form of

$\displaystyle\bigtriangledown^{\mu}T^{eff}_{\mu\nu}=0$

So the above equation represents the conservation of the energy-momentum
tensor $T^{eff}_{\mu\nu}$ for effective matter distribution. Thus, we can
easily obtain the conservation equation in Einstein’s gravity by putting
$\xi=0$. For studying cosmological and astrophysical phenomenon, $f(\Re,T)$
has been widely considered because of its advantages in explaining such space
cases.

##  4 Field equation in $f(\Re,T)$ modified theory

We assumed the interior spacetime of the spherically symmetric stellar
structure by the metric coordinates $(t,r,\theta,\phi)$

$ds^{2}=e^{\nu(r)}dt^{2}-e^{\lambda(r)}dr^{2}-r^{2}(d\theta^{2}+sin^{2}d\phi^{2})$
(14)

where $\nu$ and $\lambda$ are the metric potentials and are function of the
radial coordinate, $r$.

Now, by using eqns 10, 12 and 14, we have the Einstein field equations for
$f(\Re,T)$ gravity theory as

$\displaystyle
8\pi\rho^{eff}=e^{-\lambda}\big{(}\frac{\lambda^{\prime}}{r}-\frac{1}{r^{2}}\big{)}+\frac{1}{r^{2}}=8\pi\rho+\frac{\xi}{3}(9\rho-
p_{r}-2p_{t})$ (15) $\displaystyle 8\pi
p_{r}^{eff}=e^{-\lambda}\big{(}\frac{\nu^{\prime}}{r}+\frac{1}{r^{2}}\big{)}-\frac{1}{r^{2}}=8\pi
p_{r}-\frac{\xi}{3}(3\rho-7p_{r}-2p_{t})$ (16) $\displaystyle 8\pi
p_{t}^{eff}=\frac{e^{-\lambda}}{2}\big{(}\nu^{\prime\prime}+\frac{1}{2}\nu^{\prime^{2}}+\frac{\nu^{\prime}-\lambda^{\prime}}{r}-\frac{1}{2}\nu^{\prime}\lambda^{\prime}\big{)}=8\pi
p_{t}-\frac{\xi}{3}(3\rho-p_{r}-8p_{t})$ (17)

where the prime($\prime$)denote the differentiation with respect,$r$.

To solve eqns 15, 16 and 17, we used Buchdahl solution which was generated
through a assuming spherically symmetric fluid spheres of Einstein’s equation.
The most widely studied metric ansatz is Buchdahl metric

$\displaystyle e^{\lambda}=\frac{K(1+Cr^{2})}{K+Cr^{2}},K=\frac{7}{4}$ (18)

Where $K$ and $C$ are two parameters that are characterized by the geometry of
the star. For complete solution of these equations, we introduce
transformations

$\displaystyle e^{\nu}=Y^{2}$ (19)

For this anisotropic matter distribution in modified theory, it is necessary
that the anisotropic fluid must satisfy conservation equation as

$\displaystyle
p_{r}^{\prime}+\frac{\nu^{\prime}}{2}(\rho+p_{r})-\frac{2}{r}(p_{r}-p_{t})=\frac{\xi(3\rho^{\prime}-p_{r}^{\prime}-2p_{t}^{\prime})}{6(4\pi+\xi)}$
(20)

In another way, the energy density $(\rho)$, radial pressure $(p_{r})$ and
tangential pressure $(p_{t})$ for anisotropic stellar model in $f(\Re,T)$
theory are given as

$\displaystyle\rho=\frac{8\pi\rho^{eff}}{(8\pi+4\xi)}+\frac{8\pi(3\rho^{eff}+p_{r}^{eff}+2p_{t}^{eff})\xi}{3(8\pi+2\xi)(8\pi+4\xi)}$
(21) $\displaystyle p_{r}=\frac{8\pi
p_{r}^{eff}}{(8\pi+2\xi)}+\frac{8\pi(3\rho^{eff}-p_{r}^{eff}-2p_{t}^{eff})\xi}{3(8\pi+2\xi)(8\pi+4\xi)}$
(22) $\displaystyle p_{t}=\frac{8\pi
p_{t}^{eff}}{(8\pi+2\xi)}+\frac{8\pi(3\rho^{eff}-p_{r}^{eff}-2p_{t}^{eff})\xi}{3(8\pi+2\xi)(8\pi+4\xi)}$
(23)

Where $\xi noteq-4\pi$ and $-2\pi$. For solving eqns 21 \- 23, we have assumed
the equation of state(EoS) in the interior of the stellar structure to be
governed by MIT bag model. For anisotropic fluid, we have considered the
relation between the energy density and the radial pressure of the fluid by

$\displaystyle\rho=3p_{r}+4B$ (24)

So, from eqn 24, we derive the well known MIT bag model EoS as

$\displaystyle p_{r}=\frac{1}{3}(\rho-4B)$ (25)

where $B$ is the bag constant approximation between the range of the values
$57MeV/fm^{3}$ to $94MeV/fm^{3}$. With introduction to MIT bag model, eqns 15,
16 and 17 can be written as

$\displaystyle\rho^{eff}=\frac{3(3+Cr^{2})}{56\pi(1+Cr^{2})^{2}}$ (26)
$\displaystyle p_{r}^{eff}=\frac{(3+Cr^{2})}{56\pi(1+Cr^{2})^{2}}-D1$ (27)
$\displaystyle
p_{t}^{eff}=\frac{(3+Cr^{2})}{56\pi(1+Cr^{2})^{2}}-D1+\frac{15}{28}\frac{\beta
Cr^{2}}{(1+Cr^{2})^{3}}$ (28)

where $D1=\frac{4B}{3C}$ and $\beta=\frac{3}{4}$.

##  5 Matching boundary conditions

As to ensure a well behaved stellar interior configurations for all compact
structure which are mainly bounded objects, smooth geometry at the surface
$r=R$ of the configuration need to be joined the inner space-time with the
exterior space-time. The well-known Schwarzschild metric which is taken as
exterior solution, can be written as [53]

$ds^{2}=\big{(}1-\frac{2m}{r}\big{)}dt^{2}-\big{(}1-\frac{2m}{r}\big{)}^{-1}dr^{2}-r^{2}(d\theta^{2}+sin^{2}\theta
d\phi)$

at the boundary $r=R$. With definition of continuity, the metric co-efficient
$e^{\nu}$ and $e^{\lambda}$ across the surface boundaries $(r=R)$ and the
radial pressure vanishes at the boundary, with equations,

$1-\frac{2m}{r}=e^{\nu(R)}=e^{-\lambda(R)}$ $p_{r}(r=R)=0$

On using the boundary conditions in the above equations, we obtained the value
of arbitrary constants as

$\displaystyle
D1=\frac{(3+Cr^{2})(\pi+3\xi)}{28\pi(8\pi+3\xi)(1+Cr^{2})^{2}}-\frac{15\beta
Cr^{2}\xi}{336\pi(8\pi+3\xi)(1+Cr^{2})^{3}}$ (29)

For solution of $Y$,

$\displaystyle
N1=\frac{log(Cr^{2}+1)}{6}-\frac{5log(4Cr^{2}+7)}{48}+\frac{7\pi(3log(4Cr^{2}+7)-4Cr^{2})D1}{8}+\frac{3log(4Cr^{2}+7)}{16}$
$\displaystyle CD=exp(N1)$ $\displaystyle
CN=\sqrt{\frac{7+4Cr^{2}}{7(1+Cr^{2})}}$ $\displaystyle N2=\frac{CN}{CD}$
$\displaystyle
N3=\frac{log(CR^{2}+1)}{6}-\frac{5log(4CR^{2}+7)}{48}+\frac{7\pi(3log(4CR^{2}+7)-4CR^{2})D1}{8}+\frac{3log(4CR^{2}+7)}{16}$
$\displaystyle N4=exp(N3)$

Therefore,

$\displaystyle Y=N4.N2$ (30)

##  6 Physical properties in $f(\Re,T)$ gravity

Here, all the physical requirements of the anisotropic stellar model were
examined analytically as well as graphically to provide validation and
stabilization of the system in the framework of $f(\Re,T)$ gravity theory.

###  6.1 Metric functions, energy density and pressures

In this paper, we have considered the metric potentials $e^{\lambda}$ and
$e^{\nu}$ as in Buchdahl assumptions. In previous section 5, the behavior of
metric potential matched inner geometry to the exterior space-time in smoothly
manner at $r=R$ to get value of constants. In regard to study the behavior of
the main salient features of the model i.e density $\rho$, radial $p_{r}$ and
tangential $p_{t}$ pressure respectively, the physical parameters of any
compact object describing stellar structure should be monotonically decreasing
towards the surface and should be maximum at the center. And the radial
pressure should vanishes at the boundary whereas the energy density and
tangential pressure are non-zero at the boundary.

We have shown the graphical presentation of the metric functions in fig.1,
density, and pressures ($p_{r}$ and $p_{t}$) in fig.2 for different values of
$\xi$.

Figure 1: Above graphs represent metric potentials against radial coordinate
($r/R$) of compact star Her X-1 for different values of $\xi$.

Figure 2: Above graphs represent energy density, radial pressure and
tangential pressure respectively against radial coordinate ($r/R$) of compact
star Her X-1 for different values of $\xi$.

###  6.2 Anisotropy factor and ratio of radial and tangential pressure

In context of anisotrophy, the radial pressure is different from tangential
pressure i.e, ${p_{r}}neq{p_{t}}$. So, the effective anisotropy factor is
given as $\Delta=p_{t}-p_{r}$, this improves the stability and balancing
mechanism and also increasing the value of the redshift. Depending on the
signs of anisotropy, one can define the equilibrium mechanism contribution of
it, i.e, where its positive, $\Delta>0\implies p_{t}>p_{r}$ or negative,
$\Delta<0\implies p_{t}<p_{r}$. In first case, the system experiences a
repulsive force which helps in counter balancing the gravitational gradient
whereas in second case, the anisotropy force helps compress the object with
gravitational force. And if the nuclear force is unable to overcome the
gravitational force then the object forms a blackhole with unusual behaviors
which means the attractive force because of anisotropy damages the balance and
stability of the stellar model. It clearly confirms that the singularity
depends on hydrostatic force exerted by the matter inside the star. The ratio
of pressures i.e $\frac{p_{r}}{p_{t}}$ also confirms outward behavior of the
anisotrophy factor.

We have graphically shown the behavior of anisotropy factor and ratio of
pressures in fig.3 for different values of $\xi$.

Figure 3: Above graphs represent anisotrophy factor and the ratio
$\frac{p_{r}}{p_{t}}$ respectively against radial coordinate ($r/R$) of
compact star Her X-1 for different values of $\xi$.

###  6.3 Energy conditions

In this section, we verify the energy conditions for our choice
$f(\Re,T)=\Re+2\xi T$ field equations. The authenticity of the energy
condition is investigated by [54] constraints namely, null energy condition
(NEC), weak energy condition (WEC), strong energy condition(SEC), dominant
energy condition(DEC) and lastly trace energy condition(TEC), which are
mathematically given as

$\displaystyle NEC:\rho(r)\geq 0$ $\displaystyle WEC:\rho(r)+p_{i}(r)\geq 0$
$\displaystyle SEC:\rho+p_{r}+2p_{t}\geq 0$ $\displaystyle
DEC:\rho(r)-p_{i}(r)\geq 0$ $\displaystyle TEC:\rho(r)-p_{r}(r)-2p_{t}(r)\geq
0$

where $i$=radial $r$, transverse $t$. When an observer crosses the null
diagram, NEC depicts that the quantity of matter density is positive , when an
observer transverses a time-like diagram, WEC indicates that the matter
density is always positive whereas when tested by the relating observers, SEC
depicts that for the observer the trace of the tidal tensor is always positive
and DEC indicates that the mass-energy never flows faster than light and
finally, TEC is basically the extension of SEC and DEC which indicates that
the energy density is positive [55]. Graphical behaviur of these energy
conditions for different values of $\xi$ is provided in fig.4 that indicates
the energy conditions are satisfying throughout the star surface for different
values of $\xi$.

Figure 4: Above graphs represent different energy conditions against radial
coordinate ($r/R$) of compact star Her X-1 for different values of $\xi$

.

###  6.4 Compactness relation

For stellar structure, the value of mass function vanishes at the center of
the star and is maximum at the boundary. In case of Buchdahl [56], the mass-
radius ratio relation should be less than $\frac{8}{9}$. Mathematically, the
mass function of the compact stellar structure is defined as

$m(r)=\frac{3Cr^{3}}{14(1+Cr^{2})}$

The compactness factor has an upper bound of $\frac{4}{9}$. For our present
model compactness is obtained as

$u(r)=\frac{m(r)}{r}$ (31)

Compactness factor are classified in different categories like if for normal
star $u(r)=10^{-5}$ where as for white dwarfs it is $u(r)=10^{-3}$. In case of
neutronstar, $u(r)\in(10^{-1},\frac{1}{4})$, for ultra compact star
$u(r)\in(\frac{1}{4},\frac{1}{2})$ and for Black hole $u(r)=\frac{1}{2}$ [38].
Graphical representation of compactness is given in fig. 5 for different
values of $\xi$.

Figure 5: Above graph represent compactness factor against radial coordinate
of compact star Her X-1.

###  6.5 Gravitational and surface redshift

From definition of the gravitational redshift,
$Z=\frac{\lambda_{0}-\lambda_{e}}{\lambda_{e}}$ where $\lambda_{0}$ is the
observed wavelength at a distance $r$ and $\lambda_{e}$ is the emitted
wavelength from the surface of the compact object [57]. So, the redshift is
defined as

$Z=\sqrt{e^{-\nu(R)}}-1$ (32)

The surface redshift depends on the mass and radius of stellar object. With
increase in mass, radius increases which increases surface gravity. Thus, the
surface redshift is given by

$Z_{s}=\frac{1}{\sqrt{1-2u}}-1$ (33)

Graphical representation of the gravitational redshift is given in fig. 6 for
different values of $\xi$.

Figure 6: Above graphs represent gravitational redshift and surface redshift
against radial coordinate of compact star Her X-1 for different values of
$\xi$.

###  6.6 Causality condition

According to Herrera [58], physical acceptability of a model can be studied
from the speed of sound. For stable stellar structure, square of radial
$(v^{2}_{r})$ and tangential $(v^{2}_{t})$ should be less than 1 [59]. It
basically tells that the sound waves do not travel at arbitrary speeds and
this know as causality condition. Regardless of the material content of the
star the only difference is that in case of anisotropic, the propagation is in
two directions of the sphere i.e, radial and transverse directions where as in
case of isotropic the speed of sound decreases subliminally. So, the causality
condition given by $0\leq v_{r}\leq 1$ and $0\leq v_{t}\leq 1$.
Mathematically, these velocities define as

$\displaystyle v^{2}_{r}=\frac{dp_{r}}{d\rho}$ (34) $\displaystyle
v^{2}_{t}=\frac{dp_{t}}{d\rho}$ (35)

For $\xi>0$, the radial propagation is greater than tangential propagation.
Similarly, for $\xi<0$, both the propagation are decreasing in nature.
Graphically, we represented both causality conditions in fig. 7, which
confirms the stablity of the system for all $\xi$.

Figure 7: Above graph represent causality condition against radial coordinate
of compact star Her X-1 for different values of $\xi$.

###  6.7 Herrera-cracking concept

Another method for checking the stability condition is by Herrera-Cracking
condition, which basically determines the stability of the system by taking
variation in distribution of speed of sound into account. The differences
between both velocities should be less than 1 i.e

$|v^{2}_{t}-v^{2}_{r}|\leq 1$

The graphical representation in fig. 8 shows that the obtained stellar system
satisfies the stability condition.

Figure 8: Above graph represent stability factor against radial coordinate of
compact star Her X-1 for different values of $\xi$.

###  6.8 Adiabatic index

For a relativistic anisotropic, the stellar configuration of a compact body
with spherically symmetric configuration is examined by the adiabatic index as
it defines the intensity of equation of state at a defined density [60][61].
Many authors [62][63][64] provided different methods to check the stability of
the compact objects by following the work of Chandrasekhar [65]. In work of
Heintzmann and Hillebrandt [66] for the proposed spherically symmetric model
the stability of the configuration holds when $\Gamma>4/3$, where $\Gamma$ is
the adiabatic index given as

$\displaystyle\Gamma=\frac{\rho+p_{r}}{p_{r}}\frac{dp_{r}}{d\rho}$
$\displaystyle=\frac{\rho+p_{r}}{p_{r}}v^{2}_{r}$ (36)

The graphical presentation of adiabatic index is given in fig. 9 which shows
that the stability of $\Gamma$ and is greater than $4/3$ for all values of
$\xi$.

Figure 9: Above graph represent adiabatic index against radial coordinate of
compact star Her X-1 for different values of $\xi$.

###  6.9 EOS parameter

Here, we define and check the nature of radial and transverse equation of
state parameters. Expressions defining equation of state parameters are given
as follows

$\displaystyle W_{r}=\frac{p_{r}}{\rho}$ (37) $\displaystyle
W_{t}=\frac{p_{t}}{\rho}$ (38)

where $p_{r}$ and $p_{t}$ indicates the radial and tangential pressures
whereas $\rho$ indicates the density of the anisotropic fluid. The graphical
presentation of the EOS parameters is given in fig. 10, which showed the
behaviour of both parameters to be monotonically decreasing for different
values of $\xi$.

Figure 10: Above graphs represent equation of state(EOS) parameters against
radial coordinate of compact star Her X-1 for different values of $\xi$.

###  6.10 TOV equation

In this section, we have undergone checking the equilibrium condition of the
stellar model by Tolman-Oppenheimmer-Volkoff(TOV) [67][68], using different
forces acting on our present system in the framework of $f(\Re,T)$ theory
gravity. As we know that the Einstein field equations for $f(\Re,T)$ gravity
theory which leads to the energy conservation for our stellar model as
$\bigtriangledown^{\mu}T_{\mu\nu}=0$. So, the conservation equation of the
energy-momentum tensor which implies the modified form of the TOV equation in
context of $f(\Re,T)$ gravity theory for our system as follows

$-\frac{dp_{r}}{dr}-\frac{1}{2}\nu^{\prime}(\rho+p_{r})+\frac{2}{r}(p_{t}-p_{r})+\frac{\chi(3\rho^{\prime}+p_{r}^{\prime}+2p_{t}^{\prime})}{3(8\pi+2\chi)}=0$
(39)

Where as the equilibrium equation has four forces namely: the hydrostatic
force $F_{h}$, the gravitational force $F_{g}$, the anisotropic force $F_{a}$
and finally the force related to modified gravity $F_{m}$. However, the
explicit form of these forces are given by

$\displaystyle F_{h}=\frac{-dp_{r}}{dr}$ (40) $\displaystyle
F_{g}=\frac{\nu^{\prime}}{r}(\rho+p_{r})$ (41) $\displaystyle
F_{a}=\frac{2}{r}(p_{t}-p_{r})$ (42) $\displaystyle
F_{m}=\frac{\chi(3\rho^{\prime}+p_{r}^{\prime}+2p_{t}^{\prime})}{3(8\pi+2\chi)}$
(43)

For $\xi=0$, the conservation equation in Einstein-Maxwell gravity. The
modified TOV equation describes that the sum of the different forces in our
stellar model should be zero i.e

$F_{g}+F_{h}+F_{a}+F_{m}=0$ (44)

The behaviour of these forces is graphically represented in fig. 11 for
different values of $\xi$.

Figure 11: Above graphs represent the behavior of gravitational force $F_{g}$, hydrostatic force $F_{h}$, anisotropic force $F_{a}$ and force due to modified theory $F_{m}$ against radial coordinate of compact star Her X-1 for different values of $\xi$. Figure 12: Above graphs represent the behavior of bag constant against different values of $\xi$ of compact star Her X-1. Table 1: Values of the model parameters of central density, surface density, central radial pressure, adiabatic index, surface redshift, compactness and bag constant for different values of $\xi$ for the compact star Her X-1 by assuming $M=0.85M_{\odot}$ and $R=8.1km$.  $\xi$ | $\rho_{c}$ | $\rho_{s}$ | $p_{r_{c}}$ | $\Gamma_{r=0}$ | $Z_{s}(R)$ | $U(R)$ | $B$
---|---|---|---|---|---|---|---
$0$ | $0.051157$ | $0.00737$ | $0.014596$ | $1.5106154$ | $0.203423$ | $0.154750$ | $0.002457$
$0.2$ | $0.052289$ | $0.007546$ | $0.014583$ | $1.489027$ | $0.219592$ | $0.164218$ | $0.002402$
$0.4$ | $0.053479$ | $0.00773$ | $0.014553$ | $1.475531$ | $0.235761$ | $0.173686$ | $0.002342$
$0.6$ | $0.054731$ | $0.007923$ | $0.014502$ | $1.461064$ | $0.251930$ | $0.183154$ | $0.002282$
$0.8$ | $0.05605$ | $0.008126$ | $0.014428$ | $1.445505$ | $0.268099$ | $0.192622$ | $0.002218$
$1$ | $0.057444$ | $0.00834$ | $0.014326$ | $1.428709$ | $0.284268$ | $0.202090$ | $0.002151$

##  7 Discussion and conclusion

In this exposition, we have investigated a new class of general solutions for
the spherically symmetric anisotropic stellar structure. To our best
knowledge, the investigation is based on the simplified form of $f(\Re,T)$
function. We have employed the physically motivated Buchdahl ansatz[56] for
the metric potentials. Our system is governed by simplified phenomenological
MIT bag model equation of state, i.e $p_{r}=\frac{1}{3}(\rho-4B)$. We have
choose $f(\Re,T)=\Re+2\xi T$ basing on it we put our model to rigorous tests
for checking its regularity, causality and stability which emphasizes on the
role of $\xi$, coupling constant. It has been shown that our system conserves
energy-momentum tensor in the framework of $f(\Re,T)$ gravity theory. Through
the boundary conditions, we fixed the free constants which are arises due to
integration of the field equations. In connection to features, we bring out
the contributions from the modified theory we have explored several physical
aspects based on our findings for various values of $\xi$, coupling constant.
All these have related very interesting advocacy in favour of physical
acceptance of the model. We have illustrated the observations graphically for
Her X-1 as the representative of he compact stars.

Here, we have presented characteristics of the metric potentials by $e^{\nu}$
and $e^{\lambda}$ in fig.1, which depicts that our stellar configuration is
free of geometrical singularity. In fig.2, we have shown the variation of the
energy density($\rho$), radial pressure ($p_{r}$) and tangential pressure
($p_{t}$) respectively. It is clearly visible that all these three parameters
are maximum at the center and monotonically decreasing towards the surface so
that one can achieve minimum result on the surface. Fig.3 depicts the
anisotropy feature of the system that shows minimum value of anisotropy i.e
zero at the center and maximum at the boundary in the present modified
$f(\Re,T)$ gravity theory where in general relativity maximum anisotropy at
the boundary carries the property of the anisotropic strange stars.

Also in order to evaluate the physical liability of this stellar model in the
framework of $f(\Re,T)$ gravity theory, we have examined the energy
conditions, in fig.4 which tells that our stellar structure is consistent with
all the energy conditions. Then in fig.5, we presented the variation in
compactness factor and in fig.6, we shown the gravitational and surface
redshift of the compact star Her X-1. The variation in the casuality
conditions and Herrera-Cracking concept are studied and presented against
radial coordinate $r/R$ in figs.7 and 8, respectively. Further in fig.9, we
presented the variation of the adiabatic index against radial coordinate $r/R$
and concluded that the value of $\Gamma$ is greater than $4/3$ which confirms
that our model is stable against the radial pulsation. The variation of EoS
parameters as in fig.10, depicts that the anisotropic fluid distribution is
real and naturaly non-exotic.

Lastly to show the stability of the stellar model in terms of the equilibrium
of forces we have examined the modified TOV equation as in fig.11 in the
background of $f(\Re,T)$ theory of gravity. The variation of all four forces
namely gravitational force, hydrostatic force, anisotropic force and force due
to modified gravity has been shown for different values of $\xi$. We see that
the gravitational force ($F_{g}$) is always negative and hydrostatic force
($F_{h}$) is always positive, the anisotropic force ($F_{a}$) is almost
negligible compared to other forces so that it coincides along X-axis. Whereas
the modified force ($F_{m}$) is in between the hydrostatic force and
anisotropic force. That concludes that $F_{g}$ is balanced by both $F_{h}$ and
$F_{m}$ for our stellar structure. In fig.12, we have presented the varialtion
of bag constant against coupling constant ($\xi$) which shows that the values
of the bag constant are less than the value of $\xi$ i.e
$\xi=0,0.2,0.4,0.6,0.8,1$. In table 1, we obtained central density, surface
density, central radial pressure, adiabatic index, surface redshift,
compactness and bag constant for different values of $\xi$ for the compact
star Her X-1 by assuming $M=0.85M_{\odot}$ and $R=8.1km$.

Einstein’s theory has been successfully tested mainly in the regime of weak
gravity through solar system and laboratory experiments. But still this theory
faces strong stringent constraints in the region of strong gravity specially
near the region of black hole, highly densed compact stars and expanding
Universe. It has been noticed that because of $f(\Re,T)$ theory, the maximal
mass limits rise higher than their standard values in general relativity for
different parametric values of $\xi$. Hence, such stellar configurations in
the background of modified theory gravity can also be used in explaining
massive stellar systems like massive pulsars, super-Chandrasekhar stars and
magnetstars,etc which are not clearly explained by general Einstein’s theory.
Now, to summarize the paper, we have obtained a well-behaved model for
anisotropic compact stars in presence of $f(\Re,T)$ modified theory of gravity
and its results have been analyzed both analytically and graphically. We have
shown the solution of stellar system depends on the MIT bag model EoS which is
already used by several researchers for modeling compact stars. Finally, it is
important to say that by taking $\xi\implies 0$, results of GR theory in four
dimensions are recovered.

## Acknowledgment

The Authors would like to express their sincere gratitude towards Department
of Mathematics, Central University of Jharkhand, Ranchi, India for the
necessary support and encouragement while writing this paper and finalizing.

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# Polarization of charged spherical dielectric core-shell

A. Marin  University of Bucharest, Faculty of Physics, Măgurele, PO BOX MG11,
077125, Romania  O. Ianc  University of Bucharest, Faculty of Physics,
Măgurele, PO BOX MG11, 077125, Romania  T. O. Cheche  University of Bucharest,
Faculty of Physics, Măgurele, PO BOX MG11, 077125, Romania  Corresponding
author email<EMAIL_ADDRESS>
###### Abstract

Two point charges are placed in a spherical dielectric core-shell embedded in
a dielectric environment, one of the charges being located in the core and the
other in the shell. The core, shell, and environment are characterized by
different dielectric constants. Analytical solutions for the electrostatic
energy of the system and the polarization charge density at the core-shell and
shell-environment interfaces are obtained by a classical approach.

Keywords: dielectric, electrostatic energy, polarization charge, core-shell

##  1 Introduction

Understanding the electro-optical properties of materials are of crucial
importance in engineering components and devices which operate by the light-
material interaction. The quantum treatments of optical properties [1, 2, 3,
4] and spin Hall conductivity[5, 6, 7] in semiconductor structures, quantum
approaches of the dynamics of photosynthetic system [8, 9], the quantized
versus the classical picture of crystal lattice oscillations [10, 11] all are
based on the electrical properties of matter. Application of conservation laws
is another powerful tool in modelling the classical [14, 13, 12, 15, 16] or
quantum motion. The electrostatic properties of dielectric materials are of
interest for both academic learning and experimental research. Analytical
solution for the polarization charge on a dielectric sphere due to the
presence of a point charge is presented in the literature [17, 18].

In this work we consider a core-shell dielectric sphere embedded in a
dielectric environment with two point charges located one in the core and the
other in the shell.The core has the relative dielectric constant
$\varepsilon_{1}$ and radius $r_{0}$, the shell has the relative dielectric
constant $\varepsilon_{2}$ and core+shell radius is $R$, and the dielectric
environment has the relative dielectric constant $\varepsilon_{3}$. The goal
of the work is two-fold, namely: i) calculus of the work classically necessary
to assemble the charge distribution of two charges; ii) calculus of the
polarization charge generated at the dielectric interfaces. The first task is
motivated by the occurrence of the electron-hole pair after the light
irradiation of type II core-shell semiconductor nanostructures, the so-called
quantum dots (QDs), with the electron located in the shell and the hole
located in the core in the fundamental quantum state. The second task once
completed offers an intuitive picture on the charge induced by polarization
and a result which can be compared with that obtained by a quantum density
functional theory calculations [18]. The structure of the article is as
follows. In Sec. 2 the potential generated by the two point charges is
obtained for the dielectric core-shell. Section 3 provides the expression of
the electrostatic energy of the system of the two point charges in dielectric
in both Brus [19] and core-shell geometry. In Sec. 4, the surface charge
induced by polarization by the two point charges in the and dielectric core-
shell is obtained and graphically represented. Section 5 is dedicated to
conclusions.

##  2 The potential

This section presents a derivation for the potentials generated by each of the
charges in the system. The point charge $q$ located in $\mathbf{s}_{i}$ at
distance $s_{i}=|\mathbf{s}_{i}|$ to the center of core generates the charge
density $\rho(\mathbf{r})=q\delta\left(\mathbf{r}-\mathbf{s}_{i}\right)$;
$i=1,2$ for core, shell, respectively. Consequently, the Poisson equation for
isotropic dielectric constant in each region $\Delta\phi=\rho/\varepsilon_{i}$
becomes a Laplace equation excepting the points occupied by the charges. To
describe the potential of the point charge we use expansion [17]:

$\frac{1}{|\mathbf{r}-\mathbf{s}|}=\sum_{n=0}^{\infty}\frac{r_{<}^{n}}{r_{>}^{n+1}}P_{n}(\cos\theta)$
(1)

and take into account the azimuthal symmetry of the problem in writing the
solution of Laplace equation.

Figure 1: Spherical core-shell dielectric. Cross-section through the center of
the sphere. $\varepsilon_{1}$, $\varepsilon_{2}$, and $\varepsilon_{3}$ are
relative dielectric constants.

We can write the potential created by charge $q_{1}$ located at
$\mathbf{s}_{1}$ on $z$ axis in $\mathbf{r}$ at distance $r=|\mathbf{r}|$ from
origin and angle $\theta_{1}$ with respect to $z$ axis as follows:

$\displaystyle\phi\left(s_{1},r,\theta_{1}\right)$
$\displaystyle=\begin{cases}&\sum_{n=0}^{\infty}\left[\frac{q_{1}}{4\pi\varepsilon_{0}\varepsilon_{1}s_{1}}\left(\frac{r}{s_{1}}\right)^{n}+A_{1n}r^{n}\right]P_{n}\left(\cos\theta_{1}\right),0<r<s_{1}\\\
&\sum_{n=0}^{\infty}\left[\frac{q_{1}}{4\pi\varepsilon_{0}\varepsilon_{1}r}\left(\frac{s_{1}}{r}\right)^{n}+A_{1n}r^{n}\right]P_{n}\left(\cos\theta_{1}\right),s_{1}<r<r_{0}\end{cases}$
(2a) $\displaystyle\phi\left(s_{1},r,\theta_{1}\right)=$
$\displaystyle\sum_{n=0}^{\infty}\left[B_{1n}r^{n}+C_{1n}r^{-n-1}\right]P_{n}\left(\cos\theta_{1}\right),r_{0}<r<R$
(2b) $\displaystyle\phi\left(s_{1},r,\theta_{1}\right)$
$\displaystyle=\sum_{n=0}^{\infty}D_{1n}r^{-n-1}P_{n}\left(\cos\theta_{1}\right),r>R$
(2c)

with $A_{1n},B_{1n},C_{1n},D_{1n}$ constants. In eqs. (2a) the first term of
the sum is generated by the point charge and all the others in eqs. (2) are
polarization charges induced by $q_{1}$. Boundary conditions, taking into
account the orthogonality of the Legendre polynomials, are written as follows:

$\displaystyle\left.\frac{\partial\phi}{\partial\theta_{1}}\right|_{r_{0_{-}}}$
$\displaystyle=\left.\frac{\partial\phi}{\partial\theta_{1}}\right|_{r_{0_{+}}},$
(3a) $\displaystyle\left.\varepsilon_{1}\frac{\partial\phi}{\partial
r}\right|_{r_{0_{-}}}$
$\displaystyle=\left.\varepsilon_{2}\frac{\partial\phi}{\partial
r}\right|_{r_{0_{+}}},$ (3b)
$\displaystyle\left.\frac{\partial\phi}{\partial\theta_{1}}\right|_{R_{-}}$
$\displaystyle=\left.\frac{\partial\phi}{\partial\theta_{1}}\right|_{R_{+}},$
(3c) $\displaystyle\left.\varepsilon_{2}\frac{\partial\phi}{\partial
r}\right|_{R_{-}}$
$\displaystyle=\left.\varepsilon_{3}\frac{\partial\phi}{\partial
r}\right|_{R_{+}}.$ (3d)

From the system of equations (3) in solving eqs. (2) one obtains

$\displaystyle\phi\left(s_{1},r,\theta_{1}\right)=\frac{q_{1}}{4\pi\varepsilon_{0}\varepsilon_{1}}\frac{1}{\left|\mathbf{r}-\mathbf{s}_{1}\right|}+\sum_{n=0}^{\infty}\underbrace{\sigma_{1n}^{A}\frac{s_{1}^{n}}{R^{2n+1}}}_{A_{1n}}r^{n}P_{n}\left(\cos\theta_{1}\right),\quad
0<r<r_{0}$ (4a)
$\displaystyle\phi\left(s_{1},r,\theta_{1}\right)=\sum_{n=0}^{\infty}[\underbrace{\sigma_{1n}^{B}\frac{s_{1}^{n}}{R^{2n+1}}}_{B_{1n}}r^{n}+\underbrace{\sigma_{1n}^{C}s_{1}^{n}}_{C_{1n}}\frac{1}{r^{n+1}}]P_{n}\left(\cos\theta_{1}\right),r_{0}<r<R$
(4b)
$\displaystyle\phi\left(s_{1},r,\theta_{1}\right)=\sum_{n=0}^{\infty}\underbrace{\sigma_{1n}^{D}s_{1}^{n}}_{D_{1n}}\frac{1}{r^{n+1}}P_{n}\left(\cos\theta_{1}\right),\quad
r>R$ (4c)

with the constants $A_{1n},B_{1n},C_{1n},D_{1n}$ given in Appendix.

Next, we write the potential created by charge $q_{2}$ located in
$\mathbf{s}_{2}$ on (a new) $z$ axis in the field point $\mathbf{r}$ located
at distance $r$ from the origin in the shell and angle $\theta_{2}$ with
respect to $z$ axis as follows:

$\displaystyle\phi\left(s_{2},r,\theta_{2}\right)=\sum_{n=0}^{\infty}A_{2n}r^{n}P_{n}\left(\cos\theta_{2}\right),\quad
0<r<r_{0}$ (5a)
$\displaystyle\begin{split}&\phi\left(s_{2},r,\theta_{2}\right)\\\ &\hskip
19.91684pt=\begin{cases}&\sum_{n=0}^{\infty}\left[\frac{q_{2}}{4\pi\varepsilon_{0}\varepsilon_{2}s_{2}}\left(\frac{r}{s_{2}}\right)^{n}+B_{2n}r^{n}+C_{2n}r^{-n-1}\right]P_{n}\left(\cos\theta_{2}\right),r_{0}<r<s_{2}\\\
&\sum_{n=0}^{\infty}\left[\frac{q_{2}}{4\pi\varepsilon_{0}\varepsilon_{2}r}\left(\frac{s_{2}}{r}\right)^{n}+B_{2n}r^{n}+C_{2n}r^{-n-1}\right]P_{n}\left(\cos\theta_{2}\right),s_{2}<r<R\end{cases}\end{split}$
(5b)
$\displaystyle\phi\left(s_{2},r,\theta_{2}\right)=\sum_{n=0}^{\infty}D_{2n}r^{-n-1}P_{n}\left(\cos\theta_{2}\right),\quad
r>R$ (5c)

with $A_{2n},B_{2n},C_{2n},D_{2n}$ constants. In eqs. (5b) the first term of
the sum is generated by the point charge $q_{2}$ and all the others in eqs.
(5) are polarization charges induced by $q_{2}$. Boundary conditions, taking
into account the orthogonality of the Legendre polynomials give:

$\displaystyle\left.\frac{\partial\phi}{\partial\theta_{2}}\right|_{r_{0_{-}}}=\left.\frac{\partial\phi}{\partial\theta_{2}}\right|_{r_{0_{+}}},$
(6a) $\displaystyle\left.\varepsilon_{1}\frac{\partial\phi}{\partial
r}\right|_{r_{0_{-}}}=\left.\varepsilon_{2}\frac{\partial\phi}{\partial
r}\right|_{r_{0_{+}}},$ (6b)
$\displaystyle\left.\frac{\partial\phi}{\partial\theta_{2}}\right|_{R_{-}}=\left.\frac{\partial\phi}{\partial\theta_{2}}\right|_{R_{+}},$
(6c) $\displaystyle\left.\varepsilon_{2}\frac{\partial\phi}{\partial
r}\right|_{R_{-}}=\left.\varepsilon_{3}\frac{\partial\phi}{\partial
r}\right|_{R_{+}}.$ (6d)

From the system of equations (6) in solving eqs. (5) one obtains:

$\displaystyle\phi\left(s_{2},r,\theta_{2}\right)=\sum_{n=0}^{\infty}\underbrace{\sigma_{2n}^{A}\left(s_{2}\right)s_{2}^{-n-1}}_{A_{2n}}r^{n}P_{n}\left(\cos\theta_{2}\right),0<r<r_{0}$
(7a)
$\displaystyle\begin{split}&\phi\left(s_{2},r,\theta_{2}\right)=\frac{q_{2}}{4\pi\varepsilon_{0}\varepsilon_{2}}\frac{1}{\left|\mathbf{r}-\mathbf{s}_{2}\right|}\\\
&\hskip
28.45274pt+\sum_{n=0}^{\infty}[\underbrace{\sigma_{2n}^{B}\left(s_{2}\right)s_{2}^{-n-1}}_{B_{2n}}r^{n}+\underbrace{\sigma_{2n}^{C}\left(s_{2}\right)s_{2}^{n}}_{C_{2n}}r^{-n-1}]P_{n}\left(\cos\theta_{2}\right),r_{0}<r<R\end{split}$
(7b)
$\displaystyle\phi\left(s_{2},r,\theta_{2}\right)=\sum_{n=0}^{\infty}\underbrace{\sigma_{2n}^{D}\left(s_{2}\right)s_{2}^{-n-1}}_{D_{2n}}r^{-n-1}P_{n}\left(\cos\theta_{2}\right),r>R$
(7c)

with the constants $A_{2n},B_{2n},C_{2n},D_{2n}$ given in the Appendix. In
Appendix, the expressions obtained for the potentials of the two charges in
eqs. (4) and (7) are checked by comparison with those from the literature for
charge in a dielectric sphere embedded in a dielectric matrix.

##  3 The electrostatic energy

In this section we calculate the electrostatic energy $W$ of a system
consisting of dielectrics of spherical geometry and two point charges
incorporated in. By definition $W$ is given by the work classically necessary
to assemble the two charges in the system.

###  3.1 Brus’ problem

Following the model of Brus [19], we first calculate the work necessary to
assemble two charges in a dielectric sphere of relative dielectric constant
$\varepsilon_{1}$ and radius $r_{0}$ embedded in a dielectric of relative
dielectric constant $\varepsilon_{2}$ as shown in Fig. 2.

Figure 2: Brus’ problem: Point charges $q_{1}$, $q_{2}$ located inside the
dielectric sphere of radius $r_{0}$. For computational reasons charges
$q_{1}$, $q_{2}$ are imagined as being located on the surface of small spheres
of radius $\delta$.

First, we compute the work necessary to create one charge $q$ inside the
sphere. For this we express the infinitesimal potential generated by an
infinitesimal charge $dq$ put on the surace of a sphere of small (atomic size)
radius $\delta$ located inside the sphere:

Figure 3: Infinitesimal charge placed on the surface of sphere of small radius
$\delta$.

$d\phi\left(s^{\prime},r,\theta\right)=\frac{dq}{4\pi\varepsilon_{0}\varepsilon_{1}}\frac{1}{\left|\mathbf{r}-\mathbf{s}^{\prime}\right|}+\sum_{n=0}^{\infty}\underbrace{d\sigma_{1n}^{A}\frac{\left(s^{\prime}\right)^{n}}{r_{0}^{2n+1}}}_{dA_{1n}}r^{n}P_{n}\left(\cos\theta^{\prime}\right),0<r<r_{0}$
(8)

where
$d\sigma_{1n}^{A}=\frac{dq(\varepsilon-1)(n+1)}{\varepsilon_{1}(1+n+\varepsilon
n)}$, and $\varepsilon=\varepsilon_{1}/\varepsilon_{2}$ (see Appendix, eq.A.3
for the particular case $\varepsilon=1$). For $\delta$ of atomic size one has
$s\approx s^{\prime}\gg\delta$, the $z$ axis passes through $O^{\prime}$ (see
Fig. 3), and the integration over the small sphere surface in (eq. 8) results
in:

$\phi(s,r,\theta)=\frac{q}{4\pi\varepsilon_{0}\varepsilon_{1}}\frac{1}{|\mathbf{r}-\mathbf{s}|}+\sum_{n=0}^{\infty}\underbrace{\sigma_{1n}^{A}\frac{s^{n}}{r_{0}^{2n+1}}}_{A_{1n}}r^{n}P_{n}(\cos\theta),0<r<r_{0}$
(9)

where
$\sigma_{1n}^{A}=\frac{q(\varepsilon-1)(n+1)}{\varepsilon_{1}(1+n+\varepsilon
n)}$ and $\theta$ is the angle between $\mathbf{r}$ and $\mathbf{s}$. Then the
work necessary to create the total charge $q$ is the work necessary to bring
it from infinity to the small sphere surface centered at $\mathbf{s}$ (by a
sequential infinitesimal process), that is:

$\displaystyle\tilde{W}_{q}=\int_{0}^{q}dq\phi\left(s,s+\delta,\delta\gamma\right)\stackrel{{\scriptstyle\operatorname{small}\delta}}{{\longrightarrow}}\frac{q^{2}}{8\pi\varepsilon_{0}\varepsilon_{1}}\frac{1}{\delta}+\sum_{n=0}^{\infty}\int_{0}^{q}dq\sigma_{n}^{A}\frac{s^{2n}}{r_{0}^{2n+1}}\underbrace{P_{n}(\cos\delta\gamma)}_{\begin{subarray}{c}\delta\gamma\rightarrow
0\text{ when }s\gg\delta\end{subarray}}$ (10)
$\displaystyle=\underbrace{\frac{q^{2}}{8\pi\varepsilon_{0}\varepsilon_{1}}\frac{1}{\delta}}_{W_{\text{Born
}}}+\frac{q^{2}}{8\pi\varepsilon_{0}}\underbrace{\sum_{n=0}^{\infty}\frac{(\varepsilon-1)(n+1)}{\varepsilon_{1}(1+n+\varepsilon
n)}\frac{s^{2n}}{r_{0}^{2n+1}}}_{W_{q}}$

where $\delta$ is radius of the small sphere on which the infinitesimal charge
$dq$ is added, and $\delta\gamma$ the angle between $\mathbf{s}\left(z\right.$
axis) and $\mathbf{s}+\boldsymbol{\delta}$.

For the Brus’ problem (see Fig. 2), the work necessary to bring the charge
$q_{j}$ from infinity to $\mathbf{s}_{j}$ (which is located inside the
dielectric sphere of radius $r_{0}$ of relative dielectric constant
$\varepsilon_{1}$) in the field generated by the charge $q_{i}$ (also located
inside the dielectric sphere of radius $r_{0}$) is calculated by using the
potential from eq. (9). Thus we have:

$\displaystyle
W_{ij}=\int_{0}^{q_{j}}dq_{j}\phi\left(s_{i},s_{j},\theta_{ij}\right)$ (11)
$\displaystyle=$
$\displaystyle\frac{q_{i}q_{j}}{4\pi\varepsilon_{0}}\left[\frac{1}{\varepsilon_{1}\left|s_{i}-s_{j}\right|}+\sum_{n=0}^{\infty}\frac{(\varepsilon-1)(n+1)}{\varepsilon_{1}(1+n+\varepsilon
n)}\frac{s_{i}^{n}s_{j}^{n}}{r_{0}^{2n+1}}P_{n}(\cos\theta_{ij})\right]$

with $\theta_{ij}=\gamma$ the angle between $\mathbf{s}_{i}$ and
$\mathbf{s}_{j}$ and $i\neq j$. Then, the work classically necessary to
assemble the charge distribution (excluding the Born term) is

$\displaystyle W_{\text{diel }}=$ $\displaystyle
W_{q_{1}}+W_{q_{2}}+\frac{1}{2}\left(W_{12}+W_{21}\right)=\underbrace{\frac{q_{1}q_{2}}{4\pi\varepsilon_{0}}\frac{1}{\varepsilon_{1}\left|\mathbf{s}_{1}-s_{2}\right|}}_{W_{\text{charge
}}}+$ (12)
$\displaystyle\underbrace{\frac{1}{4\pi\varepsilon_{0}}\sum_{n=0}^{\infty}\frac{(\varepsilon-1)(n+1)}{\varepsilon_{1}(1+n+\varepsilon
n)r_{0}^{2n+1}}\left[\frac{q_{1}^{2}s_{1}^{2n}+q_{2}^{2}s_{2}^{2n}}{2}+q_{1}q_{2}s_{1}^{n}s_{2}^{n}P_{n}(\cos\gamma)\right]}_{W_{\text{pol
}}}$

###  3.2 Core-shell problem

Next, we calculate the polarization energy for the charge distribution in
spherical core-shell sketched in Fig. 1.

The energy $W_{q_{1}}$ necessary to assemble the charge $q_{1}$ at
$\mathbf{s}_{1}$ and the energy $W_{q_{2}}$ necessary to assemble the charge
$q_{2}$ at $\mathbf{s}_{2}$ are calculated by using the potential from eq.
(4a) and respectively eq. (7a), with the recipe used to obtain eq. (10). Thus,
without the Born term, we have

$\displaystyle
W_{q_{1}}={\frac{q_{1}^{2}}{8\pi\varepsilon_{0}}\sum_{n=0}^{\infty}\frac{1}{\widetilde{\varepsilon}_{1n}^{A}}\frac{s_{1}^{2n}}{R^{2n+1}}}$
(13)

$\displaystyle
W_{q_{2}}=\frac{q_{2}^{2}}{8\pi\varepsilon_{0}}\frac{1}{s_{2}}\sum_{n=0}^{\infty}\left[\frac{1}{\tilde{\varepsilon}_{2n}^{B}\left(s_{2}\right)}+\frac{1}{\tilde{\varepsilon}_{2n}^{C}\left(s_{2}\right)}\right]$
(14)

The work necessary to bring the charge $q_{2}$ from infinity to
$\mathbf{s}_{2}$ in the field generated by the charge $q_{1}$ located at
$\mathbf{s}_{1}$ is calculated by using the potential from eq. (4b) with the
recipe used to obtain eq. (11). Thus we have:

$\displaystyle W_{12}=\int_{0}^{q_{2}}dq\phi\left(s_{1},s_{2},\gamma\right)$
(15)
$\displaystyle=\frac{q_{1}q_{2}}{4\pi\varepsilon_{0}}\sum_{n=0}^{\infty}\left(\frac{1}{\tilde{\varepsilon}_{1n}^{B}}\frac{s_{1}^{n}s_{2}^{n}}{R^{2n+1}}+\frac{s_{1}^{n}}{\tilde{\varepsilon}_{ln}^{C}}\frac{1}{s_{2}^{n+1}}\right)P_{n}\left(\cos\gamma\right)$

The work necessary to bring the charge $q_{1}$ from infinity to
$\mathbf{s}_{1}$ in the field generated by the charge $q_{2}$ located at
$\mathbf{s}_{2}$ is calculated by using the potential from eq. (7a) with the
recipe used to obtain eq. (11). Thus we have:

$\displaystyle
W_{21}=\int_{0}^{q1}dq\phi\left(s_{2},s_{1},\gamma\right)=\frac{q_{2}q_{1}}{4\pi\varepsilon_{0}}\sum_{n=0}^{\infty}\frac{s_{2}^{-n-1}s_{1}^{n}}{\tilde{\varepsilon}_{2n}^{A}\left(s_{2}\right)}P_{n}\left(\cos\gamma\right)$
(16)

Similarly to the procedure used for eq. (12) one can obtain the electrostatic
energy $W_{diel}$ of the core-shell system.

##  4 The polarization charge

In this section we compute the surface charge induced by polarization at the
core-shell and shell-environment interfaces for the structure sketched in Fig.
1. By using the superposition principle, the electric potential in the
structure can be obtained by adding the potentials generated by the two
charges. Thus, by using the expressions obtained in eqs. (4) and (7) and the
notations from Fig. 4, the potential at the position vector $\mathbf{r}$ is

Figure 4: The angular spherical coordinates used in the application of
Legendre addition theorem. A similar choice of coordinates is employed for
$q_{1}$ (not represented here for clarity reasons).

$\displaystyle\phi\left(s_{1},s_{2},r,\theta_{1},\theta_{2},\varphi_{1},\varphi_{2},\theta,\varphi\right)=\phi_{1}\left(s_{1},r,\gamma_{1}\right)+\phi_{2}\left(s_{2},r,\gamma_{2}\right)$
(17)
$\displaystyle=\sum_{n=0}^{\infty}f_{1n}\left(s_{1},r\right)P_{n}\left(\cos\gamma_{1}\right)+\sum_{n=0}^{\infty}f_{2n}\left(s_{2},r\right)P_{n}(\cos\gamma_{2}).$

To calculate the charge induced by polarization we need the field that,
accordingly to the piecewise potential expressions from eqs. (4) and (7), is
defined as follows:

$\displaystyle\mathbf{E_{1L}(r)}$
$\displaystyle=-\boldsymbol{\nabla}\phi(\mathbf{r}),\quad s_{1}<r\leq r_{0}$
(18a) $\displaystyle\mathbf{E_{12}(r)}$
$\displaystyle=-\boldsymbol{\nabla}\phi(\mathbf{r}),\quad r_{0}<r\leq R$ (18b)
$\displaystyle\mathbf{E_{13}(r)}$
$\displaystyle=-\boldsymbol{\nabla}\phi(\mathbf{r}),\quad r>R$ (18c)
$\displaystyle\mathbf{E_{21}(r)}$
$\displaystyle=-\boldsymbol{\nabla}\phi(\mathbf{r}),\quad r\leq r_{0}$ (18d)
$\displaystyle\mathbf{E_{2S}(r)}$
$\displaystyle=-\boldsymbol{\nabla}\phi(\mathbf{r}),\quad r_{0}\leq r\leq
s_{2}$ (18e) $\displaystyle\mathbf{E_{2L}(r)}$
$\displaystyle=-\boldsymbol{\nabla}\phi(\mathbf{r}),\quad s_{2}<r\leq R$ (18f)
$\displaystyle\mathbf{E_{23}(r)}$
$\displaystyle=-\boldsymbol{\nabla}\phi(\mathbf{r}),\quad r>R$ (18g)

The gradient, in spherical coordinates, can be obtained by writing the
potential generated by $q_{j}$ as a function of variables $r,\theta,\varphi$
by using the Legendre addition theorem:

$\displaystyle
P_{n}\left(\cos\gamma\right)=P_{n}\left(\cos\theta\right)P_{n}\left(\cos\theta_{j}\right)$
(19)
$\displaystyle+2\sum_{m=1}^{m=n}\frac{(n-m)!}{(n+m)!}P_{n}^{m}\left(\cos\theta\right)P_{n}^{m}\left(\cos\theta_{j}\right)\cos[m(\varphi-\varphi_{j})],$

with $j=1,2$. The polarization (dipole moment volume density) is defined as
$\mathbf{P}=\varepsilon_{0}\chi_{e}\mathbf{E}$, with
$\chi_{e}=\varepsilon/\varepsilon_{0}-1$ the dielectric susceptibility. Since
inside each dielectric $\boldsymbol{\nabla}{\cdot}\mathbf{E}=0$, except at the
point charges, the polarization-surface charge density,
$\sigma_{pol}=-\boldsymbol{\nabla}\cdot{\mathbf{P}}$, is also zero except both
the point charges and the interfaces, where a jump in the $\chi_{e}$ value
occurs. Then the polarization-surface charge density at the two interfaces

$\displaystyle\sigma_{pol}^{12}=-(\mathbf{P_{2}}-\mathbf{P_{1}})\cdot\mathbf{n},$
(20a)
$\displaystyle\sigma_{pol}^{23}=-(\mathbf{P_{3}}-\mathbf{P_{2}})\cdot\mathbf{n},$
(20b)

with $\mathbf{n}$ the radial unit vector oriented in the direction of
increasing radius, is written as:

$\displaystyle\left[\chi_{e_{2}}(\mathbf{E_{12}}|_{r=r_{0{+}}}+\mathbf{E_{2S}}|_{r=r_{0{+}}})-\chi_{e_{1}}(\mathbf{E_{1L}}|_{r=r_{0{-}}}+\mathbf{E_{21}}|_{r=r_{0{-}}})\right]\cdot\mathbf{n}=-\frac{\sigma_{pol}^{12}}{\varepsilon_{0}},$
(21a)
$\displaystyle\left[\chi_{e_{3}}(\mathbf{E_{13}}|_{r=R_{+}}+\mathbf{E_{23}}|_{r=R_{+}})-\chi_{e_{2}}(\mathbf{E_{12}}|_{r=R_{-}}+\mathbf{E_{2L}}|_{r=R_{-}})\right]\cdot\mathbf{n}=-\frac{\sigma_{pol}^{23}}{\varepsilon_{0}}.$
(21b)

When computing the dot product, we are left only with the radial component of
$\mathbf{E}$, so applying eqs. (18) gives:

$\displaystyle-\chi_{e_{2}}\left(\frac{\partial\phi_{12}}{\partial
r}|_{r=r_{0_{+}}}+\frac{\partial\phi_{2S}}{\partial
r}|_{r=r_{0_{+}}}\right)+\chi_{e_{1}}\left(\frac{\partial\phi_{1L}}{\partial
r}|_{r=r_{0_{-}}}+\frac{\partial\phi_{21}}{\partial
r}|_{r=r_{0_{-}}}\right)=-\frac{\sigma_{pol}^{12}}{\varepsilon_{0}},$ (22a)
$\displaystyle-\chi_{e_{3}}\left(\frac{\partial\phi_{13}}{\partial
r}|_{r={R_{+}}}+\frac{\partial\phi_{23}}{\partial
r}|_{r={R_{+}}}\right)+\chi_{e_{2}}\left(\frac{\partial\phi_{12}}{\partial
r}|_{r={R_{-}}}+\frac{\partial\phi_{2L}}{\partial
r}|_{r={R_{-}}}\right)=-\frac{\sigma^{23}_{pol}}{\varepsilon_{0}},$ (22b)

One can check that the system behaves symmetrically when reflected on $Oxy$
plane. Indeed, by applying the known formula
$P_{n}^{m}(-x)=(-1)^{n+m}P_{n}^{m}(x)$ to eq. (19), it can be easily deduced
that the reflection $\theta\to\pi-\theta,\,\theta_{j}\to\pi-\theta_{j}$ leaves
$P_{n}(\cos\gamma)$ unchanged. In Fig. 5 we represented the polarization
charge density for several values and positions of the point charges obtained
from eqs. (22a, 22b). Other cases are shown in Appendix. The code used for our
numerical simulations is available here.

Figure 5: Polarization charge density for
$r_{0}=1,R=2,s_{1}=0.5,\,s2=1.5,\,\varepsilon_{1}=2,\,\varepsilon_{2}=3,\,\varepsilon_{3}=4$,
$\theta_{1}=\theta_{2}=0$ and: a) $q_{1}=1,\,q_{2}=-1$; b)
$q_{1}=-1,\,q_{2}=1$; c) $q_{1}=0,\,q_{2}=1$; d) $q_{1}=1,\,q_{2}=1$ (SI units
and rationalized to $\varepsilon_{0}$ dielectric constants).

##  5 Conclusions

For the system formed by two point charges placed in a spherical dielectric
core-shell embedded in a dielectric environment, one of the charges being
located in the core and the other in the shell, we obtained the the position-
dependent potential, the electrostatic energy, and the polarization charge
located at the core-shell and shell-environment interfaces. To obtain the
analytical solutions we used a Legendre polynomial series expansion of the
potential and appropriate boundary conditions. Some calculation checks are
implemented. We checked the expression for the potential for particular cases
with [21], the expressions for the potential energy with [19] and the surface
polarization charge density formulae by reflecting on $xy$ plane. Both
classical expressions of electrostatic energy and polarization charge can be
used for the comparison of classical results with results obtained by using
microscopic approaches as those used in refs., e.g., [18, 22, 23]. The
electrostatic energy we obtained can be quantized and used for an increased
accuracy description of the electronic energy structure and optical spectra
simulation in core-shell QDs[2].

## Appendix

Next, the details required to obtain the complete solutions are provided as
follows.
The constants $A_{1n},B_{1n},C_{1n},D_{1n}$ that occur in eqs. (4) are as
follows:

$\displaystyle
A_{1n}=\frac{q_{1}}{4\pi\varepsilon_{0}\widetilde{\varepsilon}_{1n}^{A}}\frac{s_{1}^{n}}{R^{2n+1}}=\sigma_{1n}^{A}\frac{s_{1}^{n}}{R^{2n+1}},$
(A.1a)
$\displaystyle\begin{split}&\frac{1}{\widetilde{\varepsilon}_{1n}^{A}}=(1+n)\\\
&\hskip
11.38092pt\times\frac{\left[\varepsilon_{1}+n\left(\varepsilon_{1}+\varepsilon_{2}\right)\right]\left(\varepsilon_{2}-\varepsilon_{3}\right)+\left(R/r_{0}\right)^{1+2n}\left(\varepsilon_{1}-\varepsilon_{2}\right)\left[\varepsilon_{3}+n\left(\varepsilon_{2}+\varepsilon_{3}\right)\right]}{Dn(n)\varepsilon_{1}},\end{split}$
(A.1b) $\displaystyle
B_{1n}=\frac{q_{1}}{4\pi\varepsilon_{0}\widetilde{\varepsilon}_{1n}^{B}}\frac{s_{1}^{n}}{R^{2n+1}}=\sigma_{1n}^{B}\frac{s_{1}^{n}}{R^{2n+1}},$
(A.1c)
$\displaystyle\frac{1}{\widetilde{\varepsilon}_{1n}^{B}}=\frac{(1+2n)(1+n)\left(\varepsilon_{2}-\varepsilon_{3}\right)}{Dn(n)},$
(A.1d) $\displaystyle
C_{1n}=\frac{q_{1}s_{1}^{n}}{4\pi\varepsilon_{0}\widetilde{\varepsilon}_{1n}^{C}}=\sigma_{1n}^{C}s_{1}^{n},$
(A.1e)
$\displaystyle\frac{1}{\widetilde{\varepsilon}_{1n}^{C}}=\frac{(1+2n)\left[\varepsilon_{3}+n\left(\varepsilon_{2}+\varepsilon_{3}\right)\right]}{Dn(n)},$
(A.1f) $\displaystyle
D_{1n}=\frac{q_{1}s_{1}^{n}}{4\pi\varepsilon_{0}\widetilde{\varepsilon}_{1n}^{D}}=\sigma_{1n}^{D}s_{1}^{n},$
(A.1g)
$\displaystyle\frac{1}{\widetilde{\varepsilon}_{1n}^{D}}=\frac{(1+2n)^{2}\varepsilon_{2}}{Dn(n)},$
(A.1h)
$\displaystyle\begin{split}&Dn(n)=n(1+n)\left(\varepsilon_{1}-\varepsilon_{2}\right)\left(\varepsilon_{2}-\varepsilon_{3}\right)\left(r_{0}/R\right)^{1+2n}\\\
&\hskip
34.14322pt+\left[\varepsilon_{2}+n\left(\varepsilon_{1}+\varepsilon_{2}\right)\right]\left[\varepsilon_{3}+n\left(\varepsilon_{2}+\varepsilon_{3}\right)\right].\end{split}$
(A.1i)

The constants $A_{2n},B_{2n},C_{2n},D_{2n}$ that occur in eqs. 7 are as
follows:

$\displaystyle
A_{2n}=\frac{q_{2}}{4\pi\varepsilon_{0}\widetilde{\varepsilon}_{2n}^{A}\left(s_{2}\right)}s_{2}^{-n-1}=\sigma_{2n}^{A}\left(s_{2}\right)s_{2}^{-n-1},$
(A.2a)
$\displaystyle\frac{1}{\widetilde{\varepsilon}_{2n}^{A}}=\frac{(1+2n)\left[\varepsilon_{3}+n\left(\varepsilon_{2}+\varepsilon_{3}\right)+\left(\varepsilon_{2}-\varepsilon_{3}\right)(1+n)\left(s_{2}^{1+2n}/R\right)^{1+2n}\right]}{Dn(n)},$
(A.2b) $\displaystyle
B_{2n}=\frac{q_{2}}{4\pi\varepsilon_{0}\widetilde{\varepsilon}_{2n}^{B}\left(s_{2}\right)}s_{2}^{-n-1}=\sigma_{2n}^{B}\left(s_{2}\right)s_{2}^{-n-1},$
(A.2c)
$\displaystyle\begin{split}&\frac{1}{\widetilde{\varepsilon}_{2n}^{B}}=\left(\varepsilon_{2}-\varepsilon_{3}\right)(1+n)\\\
&\hskip
19.91684pt\times\frac{{\left(\varepsilon_{2}-\varepsilon_{1}\right)n\left(r_{0}/R\right)^{1+2n}+\left[\varepsilon_{2}+\left(\varepsilon_{1}+\varepsilon_{2}\right)n\right]\left(s_{2}/R\right)^{1+2n}}}{\varepsilon_{2}Dn(n)},\end{split}$
(A.2d) $\displaystyle
C_{2n}=\frac{q_{2}}{4\pi\varepsilon_{0}\widetilde{\varepsilon}_{2n}^{C}\left(s_{2}\right)}s_{2}^{-n-1}r_{0}^{1+2n}=\sigma_{2n}^{C}\left(s_{2}\right)s_{2}^{-n-1}r_{0}^{1+2n},$
(A.2e)
$\displaystyle\frac{1}{\widetilde{\varepsilon}_{2n}^{C}}=\frac{\left(\varepsilon_{1}-\varepsilon_{2}\right)\left\\{-n\left[\varepsilon_{3}+\left(\varepsilon_{2}+\varepsilon_{3}\right)n\right]+n(1+n)\left(\varepsilon_{3}-\varepsilon_{2}\right)\left(s_{2}/R\right)^{1+2n}\right\\}}{\varepsilon_{2}Dn(n)},$
(A.2f) $\displaystyle
D_{2n}=\frac{q_{2}}{4\pi\varepsilon_{0}\widetilde{\varepsilon}_{2n}^{D}\left(s_{2}\right)}s_{2}^{-n-1}r_{0}^{1+2n}=\sigma_{2n}^{D}\left(s_{2}\right)s_{2}^{-n-1}r_{0}^{1+2n},$
(A.2g)
$\displaystyle\frac{1}{\widetilde{\varepsilon}_{2n}^{D}}=\frac{(1+2n)\left\\{-n\left(\varepsilon_{1}-\varepsilon_{2}\right)+\left[\varepsilon_{2}+\left(\varepsilon_{1}+\varepsilon_{2}\right)n\right]\left(s_{2}/r_{0}\right)^{1+2n}\right\\}}{Dn(n)}.$
(A.2h)

The expressions obtained for the potentials of the two charges in eqs. (4) and
(7) are checked by comparison with those from ref.[21] for one point charge in
a spherical dielectric embedded in a dielectric matrix as follows.
If in eq. (A.1) one sets $\varepsilon_{2}=\varepsilon_{3}=1$, we obtain:

$\displaystyle
A_{1n}=\frac{(n+1)q_{1}\left(\varepsilon_{1}-1\right)r_{0}^{-2n-1}s_{1}^{n}}{4\pi\varepsilon_{0}\varepsilon_{1}\left(n\varepsilon_{1}+n+1\right)},$
(A.3a) $\displaystyle B_{1n}=0,$ (A.3b) $\displaystyle
C_{1n}=\frac{(2n+1)q_{1}s_{1}^{n}}{4\pi\varepsilon_{0}\left(n\varepsilon_{1}+n+1\right)},$
(A.3c) $\displaystyle
D_{1n}=\frac{(2n+1)q_{1}s_{1}^{n}}{4\pi\varepsilon_{0}\left(n\varepsilon_{1}+n+1\right)}.$
(A.3d)

which are similar to the expressions from ref.[21], p. 83.
If we set $\varepsilon_{1}=\varepsilon_{2}=\varepsilon_{3}$, the space becomes
a homogeneous dielectric and eqs. (A.1) transform to:

$\displaystyle A_{1n}=0,$ (A.4a) $\displaystyle B_{1n}=0,$ (A.4b)
$\displaystyle
C_{1n}=\frac{q_{1}s_{1}^{n}}{4\pi\varepsilon_{0}\varepsilon_{1}},$ (A.4c)
$\displaystyle
D_{1n}=\frac{q_{1}s_{1}^{n}}{4\pi\varepsilon_{0}\varepsilon_{1}}.$ (A.4d)

which introduced in eqs. (4) express the potential of a point charge placed at
distance $s_{1}$ to the origin of system of reference.
If in eqs. (A.2) one sets $\varepsilon_{2}=\varepsilon_{1}=1$, we obtain:

$\displaystyle
A_{2n}=\frac{1}{4\pi\varepsilon_{0}}\frac{(2n+1)q_{2}s_{2}^{-n-1}}{\varepsilon_{1}n+n+1},$
(A.5a) $\displaystyle B_{2n}=0,$ (A.5b) $\displaystyle
C_{2n}=-\frac{1}{4\pi\varepsilon_{0}}\frac{(\varepsilon_{1}-1)nq_{2}r_{0}^{2n+1}s_{2}^{-n-1}}{\varepsilon_{1}n+n+1},$
(A.5c) $\displaystyle
D_{2n}=\frac{1}{4\pi\varepsilon_{0}}q_{2}s_{2}^{n}-\frac{1}{4\pi\varepsilon_{0}}\frac{(\varepsilon_{1}-1)nq_{2}r_{0}^{2n+1}s_{2}^{-n-1}}{\varepsilon_{1}n+n+1}.$
(A.5d)

which are similar to the expressions from ref. [21], p. 86.
The setting $\varepsilon_{3}=\varepsilon_{2}=\varepsilon_{1}$ in eqs. (A.2)
implies:

$\displaystyle
A_{2n}=\frac{1}{4\pi\varepsilon_{0}}\frac{q_{2}s_{2}^{-n-1}}{\varepsilon_{1}},$
(A.6a) $\displaystyle B_{2n}=0,$ (A.6b) $\displaystyle C_{2n}=0,$ (A.6c)
$\displaystyle
D_{2n}=\frac{1}{4\pi\varepsilon_{0}}\frac{q_{2}s_{2}^{n}}{\varepsilon_{1}}.$
(A.6d)

which introduced in eqs. (7) express the potential of a point charge placed at
distance $s_{2}$ to the origin of system of reference.

In Fig. A1 we represented the polarization charge density for additional
values and positions of the point charges obtained from eqs. (22a, 22b).

Figure A1: Polarization charge density for
$r_{0}=1,R=2,s_{1}=0.5,s2=1.5,\varepsilon_{1}=2,\varepsilon_{2}=3,\varepsilon_{3}=4$
and: a) $\theta_{2}=\pi/2,\,q_{1}=-1,\,q_{2}=1$; b)
$\theta_{2}=\pi/2,\,q_{1}=1,\,q_{2}=1$; c)
$\theta_{2}=3\pi/4,\,q_{1}=-1,\,q_{2}=1$; d)
$\theta_{2}=3\pi/4,\,q_{1}=1,\,q_{2}=1$ (SI units and rationalized to
$\varepsilon_{0}$ dielectric constants).

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