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Dielectric function for a material containing hyperspherical inclusions to O(c2 ) II. Method of images By T. C. C h o y1 , A r i s Alexopoulos1 a n d M. F. T h o r p e2 1

Department of Physics, Monash University, Clayton, Victoria 3168, Australia 2 Department of Physics and Astronomy, Michigan State University, East Lansing, MI 48824, USA Received 16 October 1997; accepted 3 April 1998

We have developed the method of images for the solution of two hyperspheres in d dimensions in a uniform field in order to study the O(c2 ) coefficient κ in the dielectric constant of the medium. The method employs extensions of image theory for classical charges and dipoles for dielectric spheres, which produce novel line charge and line dipole densities. Numerical results for κ with arbitrary integer d are consistent with our previous paper I (this issue), and demonstrate that the image method shows superior convergence properties. Although computationally less convenient, the procedure is, in principle, calculable term-by-term and leads to useful analytic results in the weak scattering limit. This work also produces a simple algebraic approximation that is useful everywhere. Keywords: dielectric inclusions; multipole expansions; images; hyperspheres; two inclusions; concentration of inclusions

1. Introduction The method of images is a well-known technique for the solution of problems in classical electrostatics (Jackson 1975; Landau et al . 1984) and electromagnetic theory, including antennae (Slater 1942). The latter, as an application of images in dynamical cases, is particularly interesting and offers useful insight for the analysis of fundamental forces, like the Van der Waals attraction of atoms and molecules near surfaces (den Hertog & Choy 1995). The theory of images was initiated by Lord Kelvin (1848), who must have been the first to observe that the potential due to a charge Q outside a perfect conducting sphere in three dimensions is mathematically equivalent to that due to two point charges. One of these is at the image point dK = a2 /R with charge QK = −aQ/R, where a is the radius of the sphere and R the distance of the charge from the centre. The other point charge equal to −QK , assuming the sphere is uncharged, is located at the centre of the sphere. This result is in fact valid for any dimension d, as is evident by examining the proofs in standard texts (Jackson 1975; Landau et al . 1984, see also our § 2). Point charges solve the case of a point charge outside a dielectric disc in two dimensions only (Binns & Lawrenson 1973). Less well known is the case of a dielectric sphere and the first results for three dimensions are hidden in an old classic text of Carl Neumann (1883). Generalization Proc. R. Soc. Lond. A (1998) 454, 1993–2013 Printed in Great Britain

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c 1998 The Royal Society

TEX Paper

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T. C. Choy and others ε0 QK Q

ε1

R

*

*

b1

a

ρ (z) Figure 1. The Neumann image for a point charge Q outside a d-dimensional sphere. The origin is at the centre of the hypersphere, and the point image Qk and the line image ρ(z) are shown.

of Kelvin images for a dielectric sphere is rather non-trivial. A modern discussion of this subject, relevant to our study of the bispherical system, can be found in Lindell (1993) and Bussemer (1994), whose results are crucial to this paper. Suffice it to say here, the image of a point charge outside a dielectric sphere is a point charge at the image point dK , plus a line charge ρ(z) from dK to the centre O, with a power law distribution except in two dimensions. We will call this the Neumann image solution, following Lindell (1993). Thus in general the image of dipoles outside a dielectric sphere will lead to line charges and line dipoles, a highly complicated affair. From this viewpoint, in the unique case of two dimensions, Neumann’s theory reduces to Kelvin’s theory. Especially intriguing is the fact that when only the parallel dipole configurations are considered, the d = 2 case is really quite simple by comparison to d > 2, even for the perfect conductor, see § 4. In the present paper, we will continue the study of our previous paper I (Choy et al . 1998) by exploiting some of these image results and their non-trivial extensions to the bispherical system in an arbitrary d-dimensional space. We will demonstrate that the method of images furnishes a complementary approach to the multipole moment expansion solution of paper I and that it simultaneously offers better convergence and some new physical insight, albeit at the expense of greater algebraic complexity. Moreover, this theory, we believe, will serve as the basis in some future work for the study of the resummation of the slowly convergent series of paper I. This paper was initially motivated by the recent work of Djordjevi´c et al . (1996), who applied an image theory of dipoles (Binns & Lawrenson 1973) to study the bispherical system in two dimensions. That the method of Djordjevi´c et al . (1996) cannot be extended readily to arbitrary integer dimensions follows from our previous remarks about the complexity of line dipole images in general for d > 2. In § 2 we explore the image of dipoles outside a d-dimensional sphere. In § 3 we discuss higher-order dipole images for any dimension d and dielectric constant  up to the third reflection image p3 . We will formulate, but not solve, both the perpendicular and parallel configurations in general. Unfortunately, this formulation, though elegant, does not seem to be amenable to an iterative solution beyond p4 , except in the perfectly conducting limit. In this limit, which we treat in § 4, we have been able to calculate all images up to the seventeenth reflection image p17 . Our numerical results here are in accord with paper I, thus demonstrating better convergence properties using the image method. We will show that our method for the perfect conductor can be extended indefinitely, limited only by available computing Proc. R. Soc. Lond. A (1998)

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power, while the general dielectric case appears intractable. In § 5, we show results using a weak scattering approximation, which is good everywhere and particularly so in the limit of holes, see § 3 of paper I. In § 6, we further simplify the integral that appears in the weak scattering limit, to obtain a very simple algebraic approximation for κ that is useful for all d and for all ratios α = 1 /0 , where 1 is the dielectric constant of the inclusion, and 0 is the dielectric constant of the host. For convenience, we consider in this paper, as was done in § 3 of paper I, symmetrical bispheres. These results with graphs are compared to § 3 and to the multipole expansion results of paper I. In § 7 we conclude with some discussion of higher-dimensional images, other limits and a few intriguing questions which arise from this work. Our notations will adhere closely to that of paper I.

2. Image theory in d dimensions In this section we discuss the extension of the Neumann image theory due to point charges in d dimensions. Subsequently, we use this theory to derive images for dipoles, which forms the basis for this work. (a) Neumann images of a point charge We begin our discussions by summarizing the results (for details see Appendix A) for the d-dimensional extensions of the Neumann image theory. This is an important first step, as the line charge image theory of Neumann was originally derived only in three dimensions. A priori, there are no reasons to expect that in higher dimensions these line charges do not transform into hypersurfaces of higher dimensions than one. If this were the case, then the mathematical complexity of the present paper would have been unmanageable. Fortunately, Appendix A shows that for a point charge Q outside a d-dimensional dielectric sphere (see figure 1) whose permittivity is 1 , there exists an image point charge at the image point dK given by  d−2  a α−1 , (2.1) QK = −Q α+1 R where α = 1 /0 is the relative permittivity between the sphere and the medium. In addition, there exists a line image from the centre O to the image point dK , which we rename as b1 = dK = a2 /R, given by the charge density  d−3  (d−3−α)/(α+1) α−1 Q a z (d − 2) . (2.2) ρ(z) = 2 a R (α + 1) b1 These are nice results, which appear as minor modifications of the three-dimensional Neumann image theory. We note the following. 1. The image charge QK diminishes with dimensionality d since a < R for nonoverlapping spheres. 2. The image line charge ρ(z) is a one-dimensional line density only, that reduces to a point charge −QK at the origin for d = 2, and the result becomes that of Binns & Lawrenson (1973). Proc. R. Soc. Lond. A (1998)

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R

p1

p 02

ρ 2 (z) Figure 2. First image dipole p⊥0 and line dipole ρ⊥ 2 2 (z) as derived from the Neumann theory for the perpendicular configuration. The origin is at the centre of the hypersphere.

Q2

p1

ρ 02

* p 02 k0

k0

Figure 3. First image dipole p2 and line charge ρ2 (z) and point charge Q2 as derived from the Neumann theory for the parallel configuration. The origin is at the centre of the hypersphere.

3. The image charge has a power-law density distribution which depends both on d and on α while the polarizability factor, (α − 1)/(α + 1) ((1.2) in paper I), is that of the two-dimensional sphere. 4. In the perfectly conducting limit α → ∞, the line charge density ρ(z) vanishes for all d except for the survival of a point charge at the origin, whose strength is −QK , as required by charge neutrality. In other words, the line charge reduces to a delta function ρ(z) = −QK δ(z), in accordance with Kelvin image theory. 5. For d = 3, the line charge density ρ(z) is a constant for holes, when α → 0. 6. For all d, the line charge density ρ(z) is also a constant at the single value α = d − 3. We will leave the reader to ponder these observations and their implications. Here we are primarily interested in using the d-dimensional generalization of the Neumann image theory to study dipole images as in Djordjevi´c et al . (1996), which will be applied to the bispherical system in any dimension d. (b) First perpendicular dipole image We will use the above results to derive the image of a point dipole of strength p1 outside a d-dimensional sphere, as shown in figures 2 and 3. As in paper I, we must consider separately the perpendicular (figure 2) and the parallel (figure 3) configurations. The complementary character of this work to that in paper I will manifest itself when we show that here it is the perpendicular configuration that is mathematically the simpler, which is the converse of paper I. Again, postponing details to Proc. R. Soc. Lond. A (1998)

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Appendix B, the image for a point dipole p⊥ 1 in d dimensions and perpendicular to the line between the spheres R (see figure 2) is an image point dipole p⊥0 at the 2 image point b1 given by  d  a ⊥ α−1 p⊥0 = −p . (2.3) 2 1 α+1 R ⊥ The point dipole p⊥ 1 also creates a line dipole image ρ2 (z) for 0 < z < b1 , given by  d−1  (d−2)/(α+1) p⊥ α−1 a z 1 ⊥ (d − 2) . (2.4) ρ2 (z) = 2 a R (α + 1) b1 ⊥ These results constitute the first perpendicular image set (p⊥0 2 , ρ2 ). Note that we write the image line charge as a vector ρ sometimes, where the direction of the vector is along the line itself. For d = 2 the line dipole distribution (2.4) disappears completely (Binns & Lawrenson 1973). Integrating over this density and adding it to the image (2.3) above, we get the total dipole moment for the first image p⊥ 2 given by   d  α−1 ⊥ ⊥ a p2 = −p1 . (2.5) R α + (d − 1)

Here we see the recovery of the d-dimensional polarizability factor β, missing in the theory so far. Note also that the orientation of the dipole p⊥0 and the line dipole 2 density ρ⊥ 2 (z) are opposite in sign, which is a consequence of charge neutrality. We summarize the above results by writing the perpendicular image distribution as given by ⊥0 ⊥ p⊥ 2 (z) = p2 δ(z − b1 ) + ρ2 (z),

(2.6)

whose integral recovers (2.5). We leave the study of higher-order images to the next section. Here we will mention that although the result for the total dipole moment (2.5) appears to be simple, the detailed distribution (2.6) is required explicitly for the next generation. In fact the straightforward guess following the work of Djordjevi´c et al . (1996) for the total perpendicular dipole moment p⊥ 3 of the form   d  a α−1 ⊥ ⊥ , (2.7) p3 = −p2 R − b1 α + (d − 1) is not correct (see § 3). Still (2.7) is a useful formula to keep in mind when we study approximations later. Let us now look at the parallel configuration for the first image. (c) First parallel dipole image The complexity of the bispherical problem first appears in this case. In Appendix C we have derived the following result using the generalized d-dimensional Neumann k theory. The parallel dipole p1 first generates a point dipole at the image point b1 given by  d  a k0 k α−1 p2 = p1 , (2.8) α+1 R Proc. R. Soc. Lond. A (1998)

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similar to, but opposite in sign to, (2.3). The dipole p1 here does not create a line dipole density as in the perpendicular case, but a line charge density from 0 < z < b1 , of the form  (d−3−α)/(α+1) k  d−2 p1 a α(α − 1) z k0 (d − 2) . (2.9) ρ2 (z) = 2 a R (α + 1)3 b1 In addition, we also have an image charge at the image point b1 given by  d−1 k p α(α − 1) a . Q2 = − 1 a (α + 1)2 R k0

(2.10) k

We summarize these results by writing the first parallel image set as (p2 , ρ2 ) where the charge density is k

k0

ρ2 (z) = ρ2 + Q2 δ(z − b1 ).

(2.11)

While deriving these results, charge neutrality furnishes a useful check, as is easily k seen by integrating (2.11). the net dipole moment p2 from the charge R Calculating k density (2.11) via (d − 2) zρ2 (z) dV and then adding it to (2.8), we have the total parallel dipole moment:    d α−1 k k a . (2.12) (d − 1) p2 = p1 R α + (d − 1) Before proceeding, we note that a useful check of this result is that the net angularly averaged dipole moment p¯2 should vanish, see § 3 of paper I and the remarks following (3.7) there. This is the notorious divergent term which must vanish for the theory to be meaningful, as discussed in paper I. It is easy to show that this is indeed the case, for upon performing the angular integrations first over the total k dipole moments (2.5) and (2.12) and noting that |p⊥ 1 | = |p1 | = p1 , k

p¯2 = p2 + (d − 1)p⊥ 2 = 0.

(2.13)

⊥ Although this total averaged dipole moment is zero, the image sets (p⊥0 2 , ρ2 ) and do not vanish from the scene completely. The theory depends on the totality of higher-order images generated from these sets, which we discuss in the following section.

k0 k (p2 , ρ2 )

3. Higher-order images Having secured the basics, we now proceed to calculate the higher-order images. We k0 k ⊥ are naturally thinking about the next image sets (p⊥0 3 , ρ3 ) and (p3 , ρ3 ), and how they will transform. A moment’s reflection will show that in the perpendicular case the mapping goes from point dipoles and line dipoles to point dipoles and line dipoles and so on. The parallel case is similar; the mapping goes from a point dipole with a line charge to another point dipole with line charge and so on. In all cases the density distributions develop ever-increasing complexities. In the parallel case, there is an additional step required for computing the total dipole moment from the resultant charge densities, increasing the work, but otherwise straightforward (provided these densities are available). Proc. R. Soc. Lond. A (1998)

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(a) Perpendicular higher-order images A careful examination of the higher-order images shows that the total dipole moment of the nth image is given by d  Z bn−2 α−1 a ⊥ = − p (z) dz, (3.1) p⊥ n n−1 α+d−1 0 R−z where the key quantity here is the dipole density distribution which we write as ⊥0 ⊥ p⊥ n (z) = pn δ(z − bn−1 ) + ρn (z).

Here the point dipoles

(3.2)

p⊥0 n

are given by the recursion relation (cf. (2.7))   d a ⊥0 ⊥0 α − 1 pn+1 = −pn , α+1 R − bn−1

(3.3)

and bn are the image positions, represented by the usual continued fraction (Djordjevi´c et al . 1996) bn =

a2 , R − bn−1

(3.4)

with the limits b0 = 0 and b1 = a2 /R. The dipole density ρ⊥ n (z) is formally given by the following integral recursion relation:  d−1  (d−2)/(α+1) p⊥0 α−1 a z n ⊥ (d − 2) ρn+1 (z) = 2 a R − bn−1 (α + 1) bn    d  Z bn−1 a α−1 a2 ⊥ 0 − ρn (z ) δ − z dz 0 α+1 R − z0 R − z0 0  d−1 Z bn−1 ⊥ 0 ρn (z ) α − 1 a − (d − 2) a (α + 1)2 R − z 0 0     (d−2)/(α+1) z(R − z 0 ) a2 × θ − z dz 0 . (3.5) a2 R − z0 Unfortunately, we know of no closed form solution of this system of difference integral equations, apart from resorting to direct step-by-step iteration. The starting point must of course be the expression (2.6). By this procedure we have computed the third image distribution ρ⊥ 3 (z) whose details we will not exhibit further here, except to note that it is not a convenient expression. Here we briefly mention that (3.5) has an algebraic kernel, though it is not of the difference type. It appears that the first few iterations are manageable, but the prospects diminish beyond ρ⊥ 4 (z). The system of equations (3.5) deserves further study and could be useful for improving the convergence properties of the multipole expansion series in paper I. At this point we merely quote the result for the total perpendicular dipole moment p⊥ 3 calculated by this method (cf. (2.7)):  d (α − 1)2 ⊥ ⊥ a p3 = p1 R (α + 1)(α + d − 1)  Z b1  (d−2)/(α+1)  d  d   dz d−2 z a a − . (3.6) × R − b1 α+1 b1 R − z b1 0 Proc. R. Soc. Lond. A (1998)

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As the prefactor to the integral term vanishes in the limit d → 2, (3.6) clearly reproduces earlier results (Djordjevi´c et al . 1996). Note that in the limit of the perfect conductor (§ 4), this term also vanishes, leading to a considerable simplification. The same cannot be said for holes, however, as the integral survives, but is nevertheless of a higher order in β. In § 5, we will see that this justifies a weak-scattering approximation. (b) Parallel higher-order images Again by careful examination of the results obtained for this case in § 2 c, we now k0 see that at the nth generation, there exists a point dipole pn at the image point k bn−1 and a line charge ρn (z) from the centre O to that point. The point dipole at the next generation is easy to write down:   d a k0 k0 α − 1 pn+1 = pn . (3.7) α+1 R − bn−1 k

The structure of the line charge at the next order ρn+1 (z) is the sum of several parts. We will write down these contributions and explain their origins: k

k0

k

k

ρn+1 (z) = Qn+1 δ(z − bn−1 ) + ρn+1 (z) + ρ˘n+1 (z) + ρ˜n+1 (z).

(3.8)

Now the first two terms arise in a similar manner to § 2 c and they are given by the point charge (cf. (2.10))  d−1 k0 pn α(α − 1) a (3.9) Qn+1 = − a (α + 1)2 R − bn−1 and by the charge density (cf. (2.9)) d−2  (d−3−α)/(α+1) k0  pn α(α − 1) z a k0 (d − 2) . ρn+1 (z) = 2 a R − bn−1 (α + 1)3 bn

(3.10)

k

Next the line charge of the nth generation ρn (z) maps onto a sequence of point charges, which forms a line charge density given by (cf. (2.1))    d−2  Z bn−1 a α−1 a2 k k 0 ρ˘n+1 (z) = − ρn (z ) δ − z dz 0 , (3.11) 0 0 α + 1 R − z R − z 0 and another line charge due to (2.2), namely  d−3 Z bn−1 k 0 ρn (z ) α − 1 a k ρ˜n+1 (z) = (d − 2) a (α + 1)2 R − z 0 0   (d−3−α)/(α+1)  z(R − z 0 ) a2 × θ − z dz 0 . (3.12) a2 R − z0 Once again charge neutrality is an important check and this can be obtained by integrating (3.10)–(3.12). Note that in the integral recursion relations (3.11) and k (3.12), it is the full density of the previous generation ρn (z 0 ) given by (3.8) that must be known before the partial densities can be calculated for the next generation. Herein lies the complication of the parallel system. Furthermore, havingRsomehow obtained k these densities, we must also compute the dipole moment (d − 2) zρn+1 (z) dV that Proc. R. Soc. Lond. A (1998)

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κ 10 8 d =1

6

d =4 d =3

4 d =2 d =2 d =3 d =4 -3

2 d =1 -2

-1

1

2

3

log 10 (α)

Figure 4. Graph of κ versus log10 (α) for d = 1, 2, 3, 4. The dashed curve is the result using the full third order result from equation (3.16), and the solid lines are the multipole results from paper I (figure 4). Here α = i /0 is the ratio of the dielectric constant of the inclusions to that of the host material. k

is to be added to (3.7) to find the total dipole moment pn+1 . This increases the work but is otherwise straightforward. Just like (3.5), the system of equations (3.8)–(3.12) does not seem to be amenable to a closed-form solution. Direct iteration is tedious k but can be carried out by hand up to ρ3 (z), starting from (2.11), with little prospects k k for going beyond ρ4 (z). Again we do not exhibit the details of ρ3 (z), which is even ⊥ more messy than ρ3 (z). Here, as before, we will merely quote the result for the total k parallel dipole moment p3 calculated via the above procedure (cf. (2.12):  d  d (d − 1)(α − 1)2 a k k a p3 = p1 R R − b1 (α + 1)(α + d − 1)  d−1 α(α − 1)2 k 2 a − p1 (d − 2) R (α + 1)3 (α + d − 1) d−1  (d−3−α)/(α+1) Z b dz a z × R−z b1 b1 0   d−1  d−1  2 α(α − 1) a d−2 k a . (3.13) + p1 R R − b1 (α + 1)2 α + d − 1 At this stage, we mention a few observations that might be useful for future work. Certainly, the key to further progress primarily lies in the solution of the pair of integral relations (3.5) and (3.12). The main hurdle appears not so much with the integrals, as in the region of integration; note the θ function in the integrands. As a k result of this, closed-form expressions for the distributions ρ⊥ 3 (z) and ρ3 (z) already involve non-elementary functions. Hence at the next generation, it appears to involve indefinite integrals over these functions. Thus the prospect of going beyond this level diminishes. Apart from the usual perfect conductor and d = 2 limits which are familiar by now, another useful observation is that (3.5) and (3.12) can be expanded as a power series in ξ = a/R. Indeed, we will make contact with the results of paper I in this way. By averaging over angles and using a Clausius–Mossotti-like formula, as Proc. R. Soc. Lond. A (1998)

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in § 3 of paper I, the coefficient κ up to the total averaged image p¯3 is now given by   2 2 α−1 κ = dβ + d(d − 1)β α+1  Z 1 (d−2)/(1+α)    Z 1/2 x 2 α−1 sd−1 ds + (d − 2) dx . (3.14) × 2 )d (1 − s α + 1 (1 − s2 x)d 0 0 This result admits a power-series expansion in s, under the integrals, as discussed above. Although it involves a double integral, we can rewrite (3.14) as an infinite series by introducing the Fd (β) function (see paper I and Djordjevi´c et al . (1996), whose F (β) = βF2 (β) here). This is defined by κ = dβ 2 + β 3 Fd (β), where the series Fd (β) = d(d − 1)

 ∞  d+2s  X 1 d+s−1 s=0

2

s

α+d−1 , α(s + 1) + d + s − 1

(3.15)

(3.16)

which can be viewed as a partial resummation of (3.8) in paper I to all terms in β 3 and is shown in figure 4. Leaving numerical details to later, we conclude this section by stating the leading-order values for small β: F1 (0) = 0, F2 (0) = 23 = 0.666 . . . , F3 (0) = 18 (20 − 9 ln 3) = 1.264 . . . and F4 (0) = 44 27 = 1.629 . . . . This is an exact result for d = 1 and in agreement with Djordjevi´c et al . (1996) for d = 2. For d = 3, the result quoted by Jeffrey (1973), F3 (0) = 87 80 = 1.087 . . . , only involved a part of the third-order dipole term, which can be obtained by taking the first two terms of the series (3.16) in the limit α → 1. We must also remember that β = β1 = (α − 1)/(α + d − 1) has a dimensional dependence. Comparison of this series also shows agreement with (3.9) of paper I up to the appropriate order in β.

4. Perfect conductor In our study of the image method so far, we have seen that in order of difficulty, the d = 2 general dielectric stands out as the easiest, while the next simplest case is the perfect conductor for all d. Indeed, although § 2 shows that Kelvin’s image theory is applicable both to a dielectric for d = 2 and for a perfect conductor in any d, a careful study of § 3 shows that for d > 2 the perfect conductor is still complicated by the parallel case. This is because the point charges generated by the parallel configuration cannot be neglected, except for two dimensions. Nevertheless, the bookkeeping for these point charges is still manageable and this is the case we will treat in this section. The perpendicular case is by now clear, as it leads only to point dipoles in each generation (see (3.3)). Here we will use the opportunity to discuss the point-dipoles-only approximation. (a) Point-dipoles-only approximation An examination of § 3 a and § 3 b shows that a large part of the result is contained k0 in the point dipoles pn and p⊥0 n , and we know that in two dimensions there are no other contributions. Therefore we start by keeping only all point dipoles; an Proc. R. Soc. Lond. A (1998)

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approximation that is exact in two dimensions. The function Fd (1) now takes the form (Djordjevi´c et al . 1996) Z ∞ X 2d(d − 1) d−1 dR R p¯2s+1 Fd (1) ≈ ωp E0 a2d s=1 =

∞ X

fs ,

(4.1)

s=1

where fs is given by the integral Z 1/2 fs = 2d(d − 1) 0

[1 +

x2sd−d−1 dx. k 2s−k 2k d k=1 (−1) ( k )x ]

Ps

(4.2)

This integral is rather interesting. For integer s and d, the integrand is a rational function of polynomials that can be integrated in closed form by reduction formulae (Gradshteyn & Ryzhik 1980). For example, for d = 2, we have   √ √ √ √ 4 2 5 5 f2 = 5 + √ {ln( 5 + 3) − ln( 5 − 3) + ln( 5 − 2 ) − ln( 5 + 2 )} 5 5 = 0.0556728 . . . , (4.3) in agreement with Djordjevi´c et al . (1996). The latter, however, used a digamma function representation which does not seem to be generalizable to d > 2. These integrals can be reduced using Mathematica 3.0 and by summing (4.1) to s = 19 terms; we obtained the value for d = 2 as F2 (1) = 0.7449145 . . . , agreeing with the value F2 (1) = 0.7449896 . . . up to the fourth digit, as quoted by Djordjevi´c et al . (1996), who summed the series (4.1) to over 100 terms using digamma functions. In table 1, we show the values of (4.1) summed to eight terms for d = 2, 3, 4. Note that F3 (1) = 0.895007 . . . , which is to be compared with the values in the last three rows of table 1, in particular, the known value of Jeffrey (1973) (see paper I), i.e. F3 (1) = 1.51 . . . . We now incorporate the full image series to improve upon this result. (b) Total dipole moments p4 and p5 At this point we will investigate the contributions from images beyond the point dipoles of § 3 a for the perfect conductor. We are only concerned here with the parallel dipole moments, as the perpendicular case is just the point dipole sequence p⊥0 n . For k this subsection the superscript k will be suppressed, and we will denote pn = pn for brevity. Here we need to keep track of the point charges in each generation, which are depicted in figure 5. From them, together with p0n , we calculate the total dipole moment at each generation. The computation of the charges is straightforward but lengthy. We give the results for the preaveraged total dipole moment pn for n = 2, 3, 4, 5. We begin by rewriting p2 in (2.12) as p2 = p1 ω1d (d − 1),

(4.4)

where ωn = bn /a. The next generation gives us p3 as p3 = p1 ω1d [(d − 1)ω2d + (d − 2)ω1−1 (ω2d−1 − ω1d−1 )], Proc. R. Soc. Lond. A (1998)

(4.5)

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T. C. Choy and others Table 1. Perfect conductor: image method

(The value of Fd (1) is calculated for d = 2, 3, 4 for the perfect conductor (α → ∞) using the image method. The approximate series (4.1) is evaluated up to s = 8 terms, and converges much faster than the multipole series in paper I. We also show the result of incorporating the full dipole images up to p¯17 with much improved results. The known results from Djordjevi´c et al . (1996) for d = 2 and from Jeffrey (1973) for d = 3 are shown for comparison. The result for d = 1 is 0. It can be seen that the approximate series (4.1) is only useful for d = 2, where it is exact.) s

d=2

d=3

d=4

1 2 3 4 5 6 7 8

0.666666 0.722340 0.735509 0.740152 0.742196 0.743232 0.743813 0.744164

0.842707 0.884733 0.891834 0.893781 0.894482 0.894783 0.894929 0.895007

0.814815 0.839237 0.842187 0.842816 0.843002 0.843069 0.843098 0.843111

to p¯3 to p¯4 to p¯5 to p¯6 to p¯7 to p¯8 to p¯9 to p¯10 to p¯11 to p¯12 to p¯13 to p¯14 to p¯15 to p¯16 to p¯17

0.666666 0.666666 0.722340 0.722340 0.735509 0.735509 0.740152 0.740152 0.742195 0.742195 0.743231 0.743231 0.743812 0.743812 0.744163

1.166666 1.291666 1.394103 1.427576 1.455031 1.468273 1.479289 1.485755 1.491220 1.494825 1.497916 1.500118 1.502030 1.505672 1.506934

1.370370 1.550925 1.652564 1.691469 1.716301 1.729623 1.738861 1.744690 1.748973 1.751943 1.754221 1.755901 1.757231 1.758257 1.759089

known

0.744989 . . .

1.51 . . .

and it appears that there might be a pattern emerging at p4 given by 2(d−1)

p4 = p1 ω1d [(d − 1)ω2d ω3d + (d − 2)ω1−2 {ω2d−1 ω3d−1 − 2ω1d−1 ω2d−1 + ω1

}].

(4.6)

Unfortunately, this does not prove to be the case, for when we have finally computed p5 the expression turns out to be  p5 = p1 ω1d (d − 1)ω2d ω3d ω4d + (d − 2)ω1−3 {ω2d−1 ω3d−1 ω4d−1 (1 − ω12 ) 2(d−1)

− 2ω1d ω2d−2 ω3d−1 − ω1d−1 ω2

2(d−1)

+ 2ω1

3(d−1)

ω2d−1 + ω12d−1 ω2d−2 − ω1

}]. (4.7)

This implies that higher-order dipole moments will increase in complexity. Indeed this is the case; however, the scheme presented is well defined and we have computed these higher-order contributions in analytical form using Mathematica 3.0 code. Proc. R. Soc. Lond. A (1998)

Dielectric function to O(c2 ) for hyperspheres p1

*

O

Q

(3) 3

*

(5) 5

*

Q2

-Q 2

*

*

p 02

Q

(2) 3

*

Q (1) 3

Q

*

*

p 03

O

Q

2005

(4)

Q5

*

Q

(3) 5

*

(1) 4

O

(2)

Q4

*

p 04

(2)

Q

(3) 4

*

Q

(4) 4

* O

(1)

Q5

Q5

*

* p 05

O

k

Figure 5. Sequence of point-charge images for each generation up to p5 . The superscript j in Qjn refers to the image point bj−1 while n refers to the generation pn . Table 2. Weak-scattering approximation (The value of Fd (β) in the weak-scattering approximation (5.2) to the leading order in β.) d

Fd (β)

2 3 4 5

0.666666 1.264061 1.629629 1.769689

The results for Fd (1) calculated up to p¯17 are shown in the bottom part of table 1, and they show the importance of contributions from the point charges. By examining the results of tables 1 and 2, it can be seen that convergence is slow but on the other hand much faster than the multipole expansion method we studied in paper I.

5. Weak-scattering approximation We now return to the integral expression (3.14) and study the behaviour for what we refer to as the weak-scattering limit, i.e. α → 1. Now since the factor βd α−1 = , (5.1) α+1 2 + β(d − 2) it follows that to O(β 3 ) in κ we have the simple result Z 1/2 sd−1 2 ds. Fd (0) = d (d − 1) (1 − s2 )d 0

(5.2)

The integral (5.2) is elementary and we have tabulated the values of Fd (β) in table 2. Using (3.15) and (5.2), we can write the coefficient κ of the c2 term in the dielectric Proc. R. Soc. Lond. A (1998)

2006

T. C. Choy and others κ d =1

6

d =4 d =3

4 d =2 2 d =2 d =1 d =3 d =4 -3

-2

-1

1

2

3

log 10 ( α )

Figure 6. Graph of κ versus log10 (α) for d = 1, 2, 3, 4. The dashed curve is the weak-scattering approximation as discussed in the text, and the solid lines are the multipole results from figure 4 of paper I.

constant as κ = dβ 2 + β 3 d2 (d − 1)

Z 0

1/2

sd−1 ds. (1 − s2 )d

(5.3)

We can also easily expand the integrand in (5.2); the first two terms upon integration 3 yield F3 (0) = 87 80 in agreement with the β terms of Jeffrey (1973). Note that the 3 latter does not contain all the β terms, as opposed to our image method here. This is an important observation and may be useful for the resummation of the multipole series in paper I. Finally, for holes where α → 0, the series (3.16) can be easily 7 summed to give the value of κ = 12 = 0.583333 . . . , in good agreement with Jeffrey (1973) and table 2 in paper I which gives the value κ = 0.588277 . . . . The weakscattering approximation (5.2) yields κ = 0.591992 . . . which is in agreement to two significant figures. That this approximation is less good for the perfect conductor limit can also be seen here, since (5.2) predicts a value of κ = 4.264061 . . . to be compared with the known value κ = 4.51 . . . of Jeffrey (1973) and our paper I. In figure 6 we have plotted these results as a function of log10 (α), where the weak scattering approximation is compared with the multipole expansion series in paper I. The poor convergence of the latter at O(ξ 21 ) for d = 4 is detected here since the exact solid curve should lie above the dashed curve.

6. A useful algebraic approximation For some purposes it may be useful to have an analytic approximation for the pair term that is good to a few per cent for all dimensions d and for all values of α. This can be achieved by noting that the integral (5.2) that occurs in the weak-scattering limit can be approximately evaluated to give Z 1/2 sd−1 ds ≈ d(d − 1)( 23 )d+1 , (6.1) Fd (β) = d2 (d − 1) (1 − s2 )d 0 Proc. R. Soc. Lond. A (1998)

Dielectric function to O(c2 ) for hyperspheres

2007

1.5 Fd

1

0.5

10

20

30

d Figure 7. Graph of Fd versus the dimension d. The dashed line is the algebraic approximation (6.1) to the weak-scattering result (5.2), shown as a solid line.

which is plotted in figure 7. The agreement is reasonable at all d and with errors of less than 8% for d = 1 up to 30, as shown in figure 7. For very large d, an asymptotic evaluation of the integral in (6.1) is larger by a factor 32 , but this works less well in the range of d shown in figure 7. Using this result and (3.15), we have κ = dβ 2 + β 3 d(d − 1)( 23 )d+1 .

(6.2)

Note that the second term is somewhat smaller than the first term in (6.2), but nevertheless we find it convenient to display the results for κ rather than for Fd in the graphs in both this paper and paper I. The algebraic approximation (6.2) is reasonable for all d and for all dielectric constants of host and inclusion, and leads to a dielectric constant  = 0 [1 + cdβ + c2 {dβ 2 + β 3 d(d − 1)( 23 )d+1 }].

(6.3)

This approximation leads to the exact results for κ both as d → 1 and as d → ∞, where β = (1 − 0 )/(1 + (d − 1)0 ). When d = 1, we have the exact expansion  = 0 [1 + cβ + c2 β 2 + . . . ]  2     1 1 2 +c 1− + ... , = 0 1 + c 1 − α α

(6.4)

which can be obtained by noting that in one dimension κ = β 2 is an exact result (the conductances which are equivalent to the dielectric constants add in parallel to −1 give α/0 = (1 − cβ)−1 ), or equivalently α−1 = (1 − c)−1 0 + c1 ). In the limit that d → ∞, we have κ = 0 and we recover the virtual crystal approximation  = 0 [1 + cdβ] = 0 [1 + c(α − 1)] = (1 − c)0 + c1 .

(6.5)

This is equivalent to saying that the various pieces of the dielectric add up in parallel in one dimension and in series in infinite dimensions. In between, the situation is very complex, but the behaviour for a general d goes smoothly between these two limits. Proc. R. Soc. Lond. A (1998)

2008

T. C. Choy and others

κ d =1

6

d =4 d =3

4 d =2 2

d =2

d =1

d =3 d =4 -3

-2

-1

1

2

3

log 10 (α )

Figure 8. Graph of κ versus log10 (α) for d = 1, 2, 3, 4. The dashed line is the algebraic approximation discussed in this section, and the solid line is the result from the multipole expansion shown in figure 4 of paper I. The exact perfect conductor asymptote for d = 4 is as yet unknown.

7. Conclusions In this paper, we have presented an alternative study of the dielectric function to O(c2 ) using the method of images. By an extension of the Neumann image theory to d dimensions in § 2 and to higher-order image dipoles in § 3, we have put the theory of images for the bispherical system into a unified framework. Numerical studies of this paper are consistent with that of paper I, putting the multipole series expansion method in perspective. We have found that the method of images for the dielectric bispherical system is rather difficult to implement in general and that k0 k ⊥ analytical calculations beyond the fourth image sets (p⊥0 4 , ρ4 ) and (p4 , ρ4 ) appear to be intractable. Nevertheless, the perfect conductor limit in § 5 appears to be amenable k0 k ⊥ to extensions beyond the fifth image sets (p⊥0 5 , ρ5 ) and (p5 , ρ5 ). Our calculations for p5 , which is the third non-trivial term for the image series in this case, already converged to a value for Fd (1) that is superior to the 20-term multipole series in paper I. Two approximation methods are also investigated in this paper. In § 4 we studied the point-dipoles-only approximation, known to be exact for d = 2. There we showed that for perfect conductors this approximation contains only about 70% of the result for d = 3. We have also investigated the weak-scattering limit (5.2). We have given a simple algebraic approximation for the pair term in the dielectric function, κ, that is exact in the limits d = 1 and d → ∞, and shows the dimensional dependence of κ explicitly. This should be useful in constructing effective medium theories that attempt to include the pair terms. This work was initiated during a visit to Monash University in the Autumn of 1996 by M.F.T., who thanks Monash University for its hospitality. T.C.C. thanks Michigan State University for its hospitality during a visit in the Spring of 1997, when this work was completed. This work was partly supported by NSF grant CHE 9633798. Proc. R. Soc. Lond. A (1998)

Dielectric function to O(c2 ) for hyperspheres

2009

r' r

θ Q

O

a

Figure 9. The electrostatic potential in d-dimensional space.

Appendix A. Neumann image theory in d dimensions The starting point in our discussions here will be the electrostatic potential in a d-dimensional space. In the geometrical configuration of figure 9, this is given by the expression, remembering that p = (d − 2), 1 1 = 02 . |r0 |p (r − 2ar cos θ + a2 )p/2

(A 1)

Upon using the generating function for the Gegenbauer polynomials, Cnp , defined by ∞ X 1 = tn Cnp (x), (1 − 2xt + t2 )p n=0

(A 2)

(A 1) can be written for the case r > a as

∞  n 1 X a 1 = p Cnp/2 (cos θ), r n=0 r (r02 − 2ar cos θ + a2 )p/2

(A 3)

or for the case of r < a as

∞  n 1 X r 1 = p Cnp/2 (cos θ). a n=0 a (r02 − 2ar cos θ + a2 )p/2

(A 4)

Thereafter, following Stratton (1941), we have for the potential inside a dielectric sphere in d dimensions due to a point charge −

φ (r, θ) =

∞ X

an rn Cnp/2 (cos θ),

(A 5)

n=0

or for the potential outside φ+ (r, θ) =

∞

X Q 1 bn C p/2 (cos θ). p + 0 r2 n=0 rn+p n

(A 6)

Upon using the standard boundary conditions φ+ (r, θ) = φ− , 0 Proc. R. Soc. Lond. A (1998)

∂φ+ ∂φ− = 1 ∂r ∂r

(A 7)

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T. C. Choy and others

at the surface of the sphere r = r1 and noting that ∞  n Q X r Q = Cnp (cos θ), 0 r2p ζ p n=0 ζ

(A 8)

where ζ is the distance of the charge from the centre of the sphere O, we easily obtain an = bn =

Q

2n + p , 1 n + 0 (n + p)

(A 9)

Qnr12n+p 0 − 1 . n+p 0 ζ 1 n + 0 (n + p)

(A 10)

ζ n+p

With the definition 1 /0 = α and ζ = d, as well as reverting to r1 = a, these potentials become ∞ Q X α(2n + p) rn p/2 φ (r, θ) = C (cos θ), 1 n=0 (α + 1)n + p dn+p n −

(A 11)

and φ+ (r, θ) = −Q

∞ X

(α − 1)n a2n+p 1 C p/2 (cos θ). n+p r n+p n (α + 1)n + p d n=0

(A 12)

We can now use Neumann’s trick (Bussemer 1994), which makes use of the identity Z ∞ 1 = dy e−[(α+1)n+p]y (α + 1)n + p 0 Z ∞ 1 = dξ e−nξ e−pξ/(α+1) . (A 13) α+1 0 Upon inserting this in (A 12),  n+p Z ∞ X Q(α − 1) ∞ n dK + −pξ/(α+1) dξ e Cnp/2 (cos θ), φ (r, θ) = − (d/2)−1 (α + 1) 0 r (dd ) K n=0 (A 14) where dK = a2 /d is the image point, and after suitable manipulations with the use of a partial integration (Bussemer 1994), the potential now becomes  p Q(α − 1) dK φ+ (r, θ) = − (α + 1) a p  0 (d−2)/(α+1)−1   Z dK 1 d−2 1 z 0 1 , (A 15) × − dz r2 d K |rz0 =dK |p dK α + 1 0 thereby proving (2.1) and (2.2). We will conclude this derivation by pointing out that the limit d → 2 or p → 0 is in fact pathological. By suitable manipulations the electrostatic potential becomes a logarithmic function and the Gegenbauer polynomials can be shown to reduce to Chebyshev polynomials. However, all the results of §§ 3–5 carry through by analytic continuation as d → 2, thus the limit p → 0 does not need further special attention. Proc. R. Soc. Lond. A (1998)

Dielectric function to O(c2 ) for hyperspheres

2011

Q

*

Q'

δ'

δ

Q'

*

-Q

ρ 2 (z) ⊥ Figure 10. Derivation of the image set (p⊥0 2 , ρ2 ).

Appendix B. Point perpendicular dipole images We derive the first perpendicular dipole images (2.3) and (2.4). We remind ourselves that from the electrostatic potential in d dimensions as in Appendix A, the definition of a point dipole is given by p⊥ 1 = Q(d − 2)δ. Then from the geometry of figure 10 we have δ 0 /δ = b/R. Using the result (2.1) for the image point charge, the point dipole p⊥0 2 is derived:  d−2  a α−1 ⊥0 p2 = −Q (d − 2)δ 0 α+1 R  d−2  b1 a α−1 (B 1) = −Q (d − 2)δ , α+1 R R thus proving (2.3). In the same way as the above, we can consider the two line image charges as forming a line dipole. The density of this line dipole follows from the results of Appendix A:  d−3  (d−3−α)/(α+1)   Q a α−1 z z ⊥ (d − 2)δ (d − 2) ρ2 (z) = a R (α + 1)2 b1 R  d−1  (d−2)/(α+1) α−1 p⊥ a z 1 (d − 2) . (B 2) = a R (α + 1)2 b1 Thus we have proved (2.4). We note that in the perpendicular configuration, the images are particularly simple and charge neutrality is obvious from the geometrical structure. This is not the case for the parallel configuration which we will now examine.

Appendix C. Point parallel dipole images A cursory examination of the geometrical configuration for the parallel case using our knowledge of Neumann images shows that there are several parts to consider. We begin by noting that from the binomial expansion of the electrostatic potential, d−2    1 δ 1 , (C 1) ≈ d−2 1 − (d − 2) R+δ R R Proc. R. Soc. Lond. A (1998)

2012

T. C. Choy and others

Appendix A shows that we have a residual image point charge at the image point given by  d−2  a δ α−1 (d − 2) Q0 = −Q α+1 R R  d−1 k p α−1 a , (C 2) =− 1 a α+1 R with the last line following from the definition of the point dipole. Now the dipole length, a2 a2 a2 − ≈δ , R R+δ R so that in analogy with Appendix B, the point dipole image is given by  d−2  a α−1 k0 (d − 2)δ 0 p2 = Q α+1 R  d−2  b1 a α−1 =Q (d − 2)δ , α+1 R R δ0 =

(C 3)

(C 4)

so that the point dipole image (2.8) follows. Now we look at the line charges. By examining (2.2), we find that the R dependence goes as R((d−2)α)/(α+1) , whereupon by the binomial expansion  ((d−2)α)/(α+1)   1 α δ 1 . (C 5) ≈ ((d−2)α)/(α+1) 1 − (d − 2) R+δ α+1R R Inserting this result into (2.1), we find that there exists a net line charge density as given by (2.9). Finally, in view of the charge density (2.2), there is now an infinitesimal region of non-overlap of opposite charges given by the integral  (d−3−α)/(α+1) Z b−δ(a/R)2  d−3 α−1 Q a dz z 00 b (d − 2) Q = 2 a R (α + 1) b b b  d−1 p1 a α−1 . (C 6) = a R (α + 1)2 Combining (C 2) and (C 6), we are now left with a point charge as given by (2.10). We have now completed the proofs for all the main results used in § 2.

References Binns, K. J. & Lawrenson, P. J. 1973 Electric and magnetic field problems, pp. 48–53. Oxford: Pergamon. Bussemer, P. 1994 Am. J. Phys. 62, 657–658. den Hertog, B. C. & Choy, T. C. 1995 J. Phys.: Condens. Matter 7, 19–28. Choy, T. C., Alexopoulos, A. & Thorpe, M. F. 1998 Proc. R. Soc. Lond. A 454, 1973–1992. (Preceding paper.) Djordjevi´c, B. R., Hetherington, J. H. & Thorpe, M. F. 1996 Phys. Rev. B 53, 14 862–14 871. Proc. R. Soc. Lond. A (1998)

Dielectric function to O(c2 ) for hyperspheres

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Gradshteyn, I. S. & Ryzhik, I. M. 1980 Table of integrals, series and products, pp. 56–70. New York: Academic. Jackson, J. D. 1975 Classical electrodynamics, 2nd edn, pp. 54–60. New York: Wiley. Jeffrey, D. J. 1973 Proc. R. Soc. Lond. A 335, 355–367. Kelvin, Lord 1848 (see Thomson, W. 1884 Reprint of paper on electrostatics and magnetism. 2nd edn, pp. 52–85. London: Macmillan.) Landau, L. D., Lifshitz, E. M. & Pitaevskii, L. P. 1984 Electrodynamics of continuous media, pp. 8–10. London: Pergamon. Lindell, I. V. 1993 Am. J. Phys. 61, 39–44. Neumann, C. 1883 Hydrodynamische Untersuchungen nebst einem Anhang u ¨ber die Probleme der Elektrostatik und der magnetischˆen Induktion, pp. 279–282. Leipzig: Teubner. Slater, J. C. 1942 Microwave transmission, 1st edn, pp. 269–277, 281–288. New York: McGrawHill. Stratton, J. A. 1941 Electromagnetic theory, pp. 204–207. New York: McGraw-Hill.

Proc. R. Soc. Lond. A (1998)