Exact Evaluation of the Resistance in an Infinite Face-Centered Cubic Network

Abstract

The equivalent resistance between the origin and the lattice site (2n,0,0), in an infinite Face Centered Cubic network consisting from identical resistors each of resistance R, has been expressed in terms of the complete elliptic integral of the first kind, and  .The asymptotic behavior is investigated, and some numerical values for the equivalent resistance are presented.

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

“…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.…”

“…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 -…”

“…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…”

“…[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.…”

Remarks on the Lattice Green Function for the Anisotropic Face Centered Cubic Lattice

“…Nowadays, GF is one of the most important concepts in many branches of physics, as many quantities in solid state physics can be expressed in terms of the LGF. In the following there are some examples: statistical model of ferromagnetism such as the Ising model [5], the Heisenberg model [6], spherical model [7], random walk theory [8,9], diffusion [10], band structure [11], and in recent * corresponding author; e-mail: drjasad@yahoo.com years it becomes an important tool in analyzing infinite electric networks [12][13][14][15][16][17][18][19][20][21][22][23][24][25][26].…”