Regular Resistor Lattice Networks in Two Dimensions ( Archimedean Lattices )

In this paper, we investigate the two-vertex resistance on four Archimedean lattices. This technique is based on the Green's function corresponding to the Laplacian matrix of the lattice. The Laplacian matrix of the resistor network is determined by applying basic principles (Kirchhoff's and Ohm's laws) in electrical circuit analysis, and Fourier transforms of the electric potential and current. We present some numerical results regarding the resistance between nearby lattice vertices.

The paper is arranged as follows.In section 2 we briefly review the general formulation given in Ref. (Cserti et al., 2011)for obtaining the general formula of the effective resistance between any nodes in any lattice structure of uniform tilings of resistors.In section 3 we apply the formulation to compute the vertex to vertex resistance in the Archimedean lattices that mentioned above and present some numerical results regarding the resistance between certain pairs of vertices.In section 4 we give a short conclusion and discussion.

Review the Formulation
In this section, elements of the methodology of the two point resistance are briefly reviewed (for more details see Cserti et al., 2011).Denote the electric potential and current at vertex { ; } α r by ( ) V α r and ( ) I α r respectively.Applying Kirchhoff's current and Ohm's laws at site { ; } α r , the currents ( ) I α r in the unit cell can be written in the form: where ( , ) L αβ ′ r r is the Laplacian matrix of the network.The potential and current at vertex { ; } α r can be written in terms of their Fourier transforms: where 0 Ω is volume of unit cell and k is the reciprocal lattice vector in first Brilliuon zone.Using equations ( 2) and (3), equation (1) may be written as (4) where ( )  L k is the Fourier transform of the Laplacian matrix ( s by s matrix) and ( ) V r , ( ) I k are the Fourier transforms (column matrices) of the potential and current.The Fourier transform of the Green's function is given by (5) where I is s by s identity matrix.
From equations ( 4) and (5) we have The computation of the two-point resistance is now reduced to solving equation (1) or ( 6) for ( ) V α r with the current distribution at any vertex is given by 1 2 , and ( ) , and 0, otherwise Therefore, combining equations ( 2), ( 6) and ( 8) the electrical potential distribution at arbitrary vertex of the lattice is Substituting potential distribution given above into equation ( 7), the two-node resistance can be written as It has been pointed out in (Cserti et al.,2011) that the lattice structure can be deformed into d-dimensional hypercubic lattice without changing the two point resistance of a resistor network.For hypercubic lattice the unit cell vectors are orthogonal and have the same magnitude.Thus , if one makes the transformations For two dimensional lattices the resistance between the origin { ; } α 0 and vertex 1 2 { , ; }
Substituting equations ( 2) and (3) into ( 13) and ( 14), the Laplacian matrix of the (3 3 .4 2 ) lattice, after changing .( 1 ,2) , can be written as and the lattice Green's function can be obtained from equation ( 5): where is the determinant of the Laplacian matrix.Now the equivalent resistance between any two sites can be calculated from equation (12).As an example, the resistance between the origin { ; 1} α = 0 and the site{ ; 2} β = 0 (see figure 1) is given by The numerical calculation of this integral is R 12 (0,0)=0.400648R.Numerical values for some other resistances are displaced in Table 1.As in previous subsection the Fourier transform of the Laplacian matrix reads and its Green's function can be obtained from equation( 5).Using equation ( 12) one can numerically calculate the resistance between any two sites.As an example, the resistance between nodes{ ; 1} (69 40 cos 20 cos 10 cos cos cos 2 ) (0, 0) 170 72 cos 72 cos 28 cos cos cos 2 cos 2 The numerical value of this integral is 12 (0, 0) 0.403775 R R = .
In Table 2, we list numerical values for some extra resistances.
As in the previous subsections the Fourier transform of the Laplacian matrix of the (3.4.6.4) network can easily be obtained : www.ccsen The resista between th 11

R
The numer (1, 0) The perturbation of uniformly tiled resistor lattices by replacing one resistor by another one in the perfect lattice (Owaidat et al., 2014) or removing one resistor from the perfect lattice (Cserti et al., 2002) can be applied to the above Archimedean lattices of resistors.
Finally, the present work can be extended to study the classical lattice dynamics and the vibrational modes of atoms in the harmonic approximation as another application of the Green's function approach.
Consider a resistor lattice network structure which is a periodic lattice of d-dimensional space vertices in the lattice.Let { ; } α r denote any vertex, thus the unit cell and the lattice site can be specified b 1 .., d l l l are any integers.Assume, without lose of generality, that each resistor has of resistance R.