Electrical resistance between pairs of vertices of a conducting cube and continuum limit for a cubic resistor network

A method is given of determining analytically the driving-point resistance between any pair of vertices of: (i) a homogeneous conducting cube, (ii) a N × N × N cubic lattice of identical resistors. For the latter, the limit N → ∞ is considered and the condition of the ohmic equivalence between these two systems is investigated.


Introduction
We intend to compute the electrical resistance between different pairs of vertices of a homogeneous ohmic cube. To the authorʼs knowledge, such a calculation deceptively simple cannot be found anywhere except the unwieldy Fourier series-based formula for the potential at any point of the cube, obtained by Weiss et al in [1] from which the authors discussed the derivation of the van der Pauw formula. The van der Pauw technique is commonly used in the semiconductor industry for electrical transport measurements on solid materials [2]. It permits for instance to obtain the conductivity of infinitely thin samples of arbitrary shape from measurements of current and voltage difference considering four points along the periphery of the sample. However, the van der Pauw method is valid only for infinitely thin samples so that the effects of sample thickness on the accuracy of results may be significant requiring that the van der Pauw formula be corrected and generalized (see [1,3]). Beyond its theoretical aspects, the calculations laid down in the paper and the exact determination of drivingpoint resistances between any pair of vertices of a cube may contribute to improve this experimental technique.
Apart the known resistance between two distinct points of an infinite ohmic medium (see e.g. [4], page 297), the problem for the conducting cube is usually addressed in the literature as the continuum limit of a cubic resistor network (see e.g. [5,6]). For both cases, the equations to be solved are well-known and related to Poisson-type equations such as encountered in electrostatics. But the difficulty both for the cubic medium and the finite network is to take in account the boundary conditions for deriving the exact analytical solution as simply as possible and readily usable in applications.
The bulk of this paper is devoted to the conducting cube (section 1 and followings) while the continuum limit of the cubic resistor network will be discussed in the 4th and last section. The results reported here will show that, in a largely classical context, there exists a very deep connection between the condensed matter physics and the analytic number theory.

Electrostatic model
where δ is the Dirac distribution on 3  . Hence, in virtue of the Ohmʼs law V J s =and in the absence of other current sources or fields, the potential is solution of the following Poisson equation which must be solved given Neumann boundary conditions which ensure that no current can flow through the faces of the cube, nor electric charges can accumulate on the faces. Hence, analogy with the electrostatics is immediate considering a cube of dielectric constant σ holding two opposite point charges I  .
As it is known (see e.g. [7]), the equation (3) may have an exact closed-form solution but this requires to formulate the latter boundary value problem on the 3-flat torus 3 ) . It amounts thus, by the so-called method of images [8], to periodize the problem in the three coordinate directions (period a a a 2 2 2´) and to consider a charge distribution of type body-centered cubic (bcc) for a crystal lattice, the cube  being thus a (quarter of) unit cell which contains two point sources I 8  taking in account their coordination number.
So doing, we have thus led to solve on 3  , where this time, δ denotes a Dirac distribution on 3  (i.e. a 3D Dirac comb on 3  , of period a a a 2 2 2´) and f r r d 0 in accordance with the Gauss theorem. Roughly speaking, the boundary value problem posed for the cube has been thereby redrafted by periodization for the whole space making it more convenient to solve.  can be expressed by the following integral a  v  n m p  r  k  2  i  e  e  ,  , ,  8   v  a   k   k  kr  3 i .

Indeed, recall that (5) is the exponentially convergent series
or also, by application of the Poisson summation formula, a sum of periodized Gaussians as,  Moreover, one can show that the integral (7) is absolutely convergent on any compact not containing the origin.
it follows well after integration the identity (6). This fundamental solution is given up to an arbitrary constant (chosen equal to zero) and is of mean value 0, both conditions allowing to fix a zero potential reference at any point of the torus in the absence of current. In the context of electrostatics, the Poisson equation (6) shows that G r ( ) may correspond to the electrostatic potential at the point r due to unit point charges at lattice points ak 2 , all of 3-space being negatively and uniformly charged (of density a 1 8 3 - ) for an overall electroneutrality, a such charge distribution being sometimes called a jellium crystal and G r ( ) the jellium potential [10]. As already mentioned, the fundamental solution given by (7) is well-defined anywhere on the torus except at the origin r 0 mod = L ( )where it possesses a singularity. Indeed, since (see (9)), as v tends to zero and it ensues that the integral (7) is well divergent at the origin. Nevertheless, it is quite interesting for the rest of the paper to notice that when r 0 = it is possible to extract from the integral (7) a finite part in the Hadamard sense denoted G O and equal to [11], using here and in the sequel, as it is usual for these modular functions, the following notation For the electrostatic problem, the regularization term a 1 4  pe e + ( ) ( )which removes the singularity from the jellium potential (7) at r 0 = may be viewed as the 3-Coulomb potential, at the distance a e , due to a solitary unit charge located at the origin of 3  , plus the (null) contribution of the uniformly charged 'jelly'. As a result, the finite part G O may be thus interpreted as the electrostatic potential due to the crystal and seen by the charge origin, its self-contribution being removed, result which can be thus linked with the lattice sum arising from the related Poisson equation by summing all 3-Coulomb terms as follows a a n m p k 1 8 the prime on the summation sign indicating omission of the term k 0 = (the notation 1 ( ) a is those introduced by Zucker, see [12,13]).
One can check that (11) and ( which agrees up to the fifth decimal with the Zucker's result tabulated in his outstanding work [12]. We can hope to find a more accurate result by using more refined numerical integration procedure (see for instance [14]).
At first glance, it seems that there is a contradiction between the latter negative result and the sum of positive terms (12) but it is a long-known fact that such an infinite series is purely formal and cannot converge as it is (see e.g. [10,13,15]). We see briefly below that the correct comprehension of (12) whose physical sense is undeniable must be found in analytic number theory using tools of complex analysis.
Let us indicate that the Hadamard finite-part (11) and relatives are already implicitly considered in the finitesize scaling theory for defining in an ad hoc way some Madelung-type constants describing different physical situations (geometry system, interparticle interaction, etc) (see [16,17]).

Series representation
A standard approach (detailed in [10], see also [15]) is indeed to consider the following 3-dimensional zeta function (again, the prime means that any singularities are to be ignored in the sum). First, (14) is realized through the Mellin transform of the generalized 'theta function' and then, applying the Poisson summation to g Q leads to the reflection formula where Γ is the Gamma function. In addition, it is well-known in analytic number theory (see e.g. [18]) that the function (15) has a meromorphic continuation in the entire s-plane beyond the line s 3 = R , except for simple poles at s=0 and s=3.
As a result, taking s=2, u r = and v 0 = , it ensues for our purposes that Z Z r 0 0 r 2; , 1; , )is identical to the integral (7) (up to the factor a 1 8p ). We have thereby established the following series representations for the fundamental solution in both forms,   3 3 these equalities being in principle not everywhere pointwise, but understood through the analytic continuation of relevant zeta functions for any r 3  Î . Nevertheless, it is noteworthy that • the first series (denoted 1 S ) is the formal Fourier series of G we may directly derived by solving (6) considering As it is, 1 S is obviously divergent at the origin (mod Λ), otherwise it is conditionally convergent.
• While the second one (denoted 2 S ) simply corresponds to the electrostatic superposition of 3-Coulomb potentials at point r due to all lattice point charges if r 0 ¹ , otherwise the self-potential of the single origin charge is removed.
So doing, notice from 2 S and (12) that one has here G G a showing, as already emphasized by Crandall and Buhler [10], that the analytic continuation leading to the previous results, by removing the singularity due to the origin charge, allots to the fundamental solution G when divergent the finite-part value (11) equal to the lattice sum (12).

Resistance between the vertices O and A
The fundamental solution (7) or (17) being now available, the potential V at any point r of the torus 3  (and thus, of the cube ) may be exactly obtained from (4) via the triple convolution product i.e. using for instance the integral representation (7) and a translation property for 3 Q , It is easy to verify that (19) satisfies well to the Neumann boundary conditions. Since G is even, notice that holds the reflection formula V V r r r In the light of the regularization procedure discussed previously, the voltage difference felt by the electric current flowing between the points O and A is thus equal to where the constant M depending on the formula used for G, is equally expressed as • or the second one, Considering the related Zucker's lattice sums as tabulated in [12], one thus obtains respectively and we can note that the agreement between all these numerical results is highly satisfactory. This constant M is well-known in electrostatics of ionic crystals as the Madelung constant for the bcc crystal structure built with the cube  (for instance, a CsCl crystal [7,13]). Moreover, M is found positive meaning for consistency that the current must be considered flowing through the cube from A to O. Finally, denoting by R a 1 s = a standard resistance value (this is for instance the very known electric resistance between two opposite faces of the cube), the driving-point resistance (also called, two-point resistance) between the vertices O and A has exactly the value For a silver cube of edge a=1 cm, at 20°C, 6.30 10 7 1 s =´Wm −1 one finds R 2.06 10 OA 6 »´W -.

Other resistances
In a similar way, the driving-point resistance between the vertices O and B on the face diagonal, and between O and C on one edge are obtained by solving the Poisson equation (3) on the torus 3  with the current densities for the first and the other configuration respectively, it is straightforward to derive the following resistances

Cubic resistor network
Consider a N N Ń´cubic lattice formed from equal resistors R(N) with nodes n m p , , n = ( )and integers n m p N , , 0, , 1 = ¼ -(see figure 2). We intend in this section to find the driving-point resistance between vertices of this cubic network, and to discuss when N tends to infinity, the continuum limit and conditions for the ohmic equivalence with the previous conducting cube .

Resistance between two diagonally opposite nodes
Let R OA N ( ) the driving-point resistance between the diagonally opposite nodes 0 0, 0, 0 = ( ) and ) . Assuming a current I flowing from the node 0 to A n and in the absence of other external current sources, the electric potential V n ( ) at any node n can be obtained by applying the Ohm's and Kirchhoff's laws. Hence, the desired resistance is readily given by the ratio This is a classical problem of multilinear algebra which has been recently solved by Wu [19] in an elegant manner using the formalism of dyadic algebra or, in a more abstract form, by Jafarizadeh et al [20] whose calculation is based on stratification of the underlying graph and the Stieltjes transform of the spectral distribution associated with the network. Here, we prefer an alternative approach using the lattice Green's function method appearing in many problems of solid state physics [5,6,21], and which can be likened to a discrete formulation of problems discussed in the previous sections. Indeed, according to the superposition theorem of current distributions, the physical situation is identical to those for an infinite 3-dimensional cubic lattice with the following current distribution, even and and δ is the Kronecker unit impulsion,

= -
). Such a current distribution ensures by the superposition theorem that no current can flow through the sides of the finite square network N 0, , is easily obtained using the discrete Fourier transform and shift properties, knowing that Hence, one finds the well-known result (see e.g.
the prime on the summation indicating as usual that the term k 0 = must be omitted.
Therefore, from (31), the electric potential at any node N 2 or using a trigonometric identity for the latter term in brackets, Notice that the latter voltage difference is positive in agreement with the fact that, above, the current has been chosen flowing into the network from the node 0 to A n . Thus, the expected two-point resistance R OA and some exact values for N 2 6   are listed in table 1. In particular, we actually found the value 5/6 for N=2 which is the well-known solution of the '12 resistors cube' problem.
Further numerical tests show also that N N a( ) , N 2  , thus also N a( ), is a positive increasing sequence.
Moreover, by the Rayleigh's Monotonicity Law [22], the following exact bounds can be obtained for N 2  , Indeed, let us remark that N N a( ) is a normalized two-point resistance between O and A for any N 2  (which amounts equivalently to set R N 1 = ( ) for all N). Then, cutting three adjacent faces of the cubic network except the 12 resistors cell at the common vertex (see figure 4) yields to Taking the limit N  +¥, the series in the RHS of (39) is still not convergent, but we have seen in section 3.1 that such an infinite lattice sum, considered as the analytic continuation evaluation of a relevant zeta function, is related to the Madelung constant (22). As a result, it appears reasonable to conjecture that (36) by an appropriate rearrangement of terms may retain a finite value as N increases, viz.
Surprisingly, this limit value is negative. The change of sign made to a +¥ ( )and originating in the analytic continuation machinery implemented for making convergent (36) as N tends to +¥, means that the current asymptotically must flow, for mathematical consistency, from the node A n to 0 i.e.
. Nevertheless, from a physical point of view, it is fortunate that such an adjustment does not affect the definition of the driving-point resistance R OA N ( ) . Recall that a similar result was found in section 3.1 concerning the conducting cube . Therefore, from (37), one writes,

M
Comparing (42) and (25) shows that the cubic resistor network formed from equal resistor R N N a s = ( ) is asymptotically 'ohmically' equivalent to an homogeneous conducting cube of edge a and conductivity σ.
Since 1 sis a resistance per unit length, the resistance R(N) connecting two nodes is thus the resistance for a segment of length a/N which can be viewed as a fictitious distance separating these two nodes. This is consistent with the fact that each edge of the cubic network has N nodes (thus N 1 resistances) whose fictitious total length is asymptotically a i.e. the actual side of the equivalent cube .

Other resistances
The driving-point resistance between the nodes 0 0, 0, 0 = ( ) and N N N e e 1, 0, 1 1 Hence, analogously to the previous section, it is direct to show that on the one hand,

M M
which show, comparing with (28), that the continuum limit of such a cubic resistor network is exactly the conducting cube .

Perspectives and conclusions
First, all results presented in this paper both for the cubic medium, the finite cubic resistor network and its continuum limit may be easily generalized in any spatial dimensionality n 3  . The case n=2 (i.e. in a 3-dimensional approach, the electrical resistance between two edges of a square prism) is more delicate to study owing to the existence of logarithmic singularities for the potentials and the impossibility to fix a zero potential reference at infinity for a 2D Coulomb potential. Nevertheless, these difficulties may be overcome as already discussed in [7,23]. Secondly, we leave it to the reader to adjust the calculations of sections 2 and 3 when considered a cuboid, for instance a cuboid a a b´. In that case, setting the axial ratio a b a = and defining the Madelung-type constant