1 Introduction
The calculation of the two-site resistance in a resistor network has gained considerable attention for more than 170 years.[1-6] The infinite resistor lattice that is a uniform tiling with resistors in any dimension can systematically be treated by the Laplacian matrix of the difference equations governed by Kirchhoff’s and Ohm’s laws. Everything about a resistor lattice can be described by its Laplacian. The Green’s function can be obtained by inverting the Laplacian matrix, which is related to the two-site resistance on a resistor lattice.
In Ref. [7], Cserti has used Green’s function method to calculate the resistance for several infinite lattice structures of resistors. In Ref. [8], Cserti et al. have presented a general theory based on the Green’s function for calculating two-point resistances in infinite d-dimensional uniform tilings with resistors. Also, the Green’s function theory has been developed to study several types of defects, including a broken resistor (capacitor), a replaced resistor (capacitor), and an extra resistor (capacitor) between two nonconnected lattice sites.[9-12] Based on this method[7-8] considerable works have been performed to determine the resistance and capacitance between two arbitrary lattice sites in infinite lattices of various topologies.[13-17]
However, there are still some infinite resistor lattices, which stayed unresolved in the literature. In this work, we apply the general formulation of two-point resistance established in Ref. [8] to a pentagonal lattice structure of equal resistors. For more details on this formulation, we refer to Ref. [8].
2 Pentagonal Lattice Structure
Let us consider a two-dimensional pentagonal lattice structure of identical resistances R, which is a periodic tiling of the plane by the pentagons, as shown in Fig. 1. There are six kinds of lattice sites in each unit cell denoted by $\alpha = 1, \ldots ,6$. The lattice sites in the pentagonal lattice can be represented by ${\rm{(}}{r_p}{\rm{;}}\alpha {\rm{)}}$ where ${r_p} = {p_1}{a_1} + {p_2}{a_2}$ is the lattice vector, ${a_1}, {a_2}$ are the unit cell vectors (see\quad Fig. 1), and ${p_1},{p_2}$ are arbitrary integers.
Fig. 1 (Color online) The pentagonal lattice structure of resistors network. |
Denote the electric potential at the lattice site ${\rm{(}}{r_p}{\rm{;}}\alpha {\rm{)}}$ by ${V_\alpha }({r_p})$ and the net current entering the lattice at the site ${\rm{(}}{r_p}{\rm{;}}\alpha {\rm{)}}$, from outside the lattice, by ${I_\alpha }({r_p})$. Using Kirchhoff’s and Ohm’s laws, one may write the currents ${I_\alpha }({r_p})$ at site ${\rm{(}}{r_p}{\rm{;}}\alpha {\rm{)}}$ (with$ \alpha = 1, \ldots ,6)$ as the following
$${I_1}(r) =\dfrac{{{V_1}(r) - {V_2}(r)}}{R} + \dfrac{{{V_1}(r) - {V_5}(r-a_1-{a}_2)}}{R} + \dfrac{{{V_1}(r) - {V_6}(r - {a_2})}}{R}, $$
$${I_2}(r) =\dfrac{{{V_2}(r) - {V_1}(r)}}{R} + \dfrac{{{V_2}(r) - {V_3}(r)}}{R} + \dfrac{{{V_2}(r) - {V_4}(r - {{a_1}})}}{R}{\rm{ + }}\dfrac{{{V_2}({r}) - {V_6}({r})}}{R}, $$
$${I_3}(r) = \frac{{{V_3}(r) - {V_2}(r)}}{R} + \frac{{{V_3}(r) - {V_4}(r)}}{R}{\rm{ + }}\frac{{{V_3}(r) - {V_5}(r - {a_2})}}{R}, $$
$${I_4}({r}) = \frac{{{V_4}({r}) - {V_3}({r})}}{R}{\rm{ + }}\frac{{{V_4}({r}) - {V_2}({r} + {{a}_1})}}{R} + \frac{{{V_4}({r}) - {V_5}({r})}}{R}, $$
$${I_5}({r}) = \frac{{{V_5}({r}) - {V_1}({r} + {{a}_1} + {{a}_2})}}{R}{\rm{ + }}\frac{{{V_5}({r}) - {V_3}({r} + {{a}_2})}}{R} + \frac{{{V_5}({r}) - {V_4}({r})}}{R} + \frac{{{V_5}({r}) - {V_6}({r})}}{R}, $$
$${I_6}({r}) = \frac{{{V_6}({r}) - {V_1}({r} + {{a}_2})}}{R} + \frac{{{V_6}({r}) - {V_2}({r})}}{R} + \frac{{{V_6}({r}) - {V_5}({r})}}{R}.$$
The electric potential and current at lattice site $(r_p; \alpha)$ are usually given by their Fourier transforms as the following:
$${V_\alpha }({r_p}) = \frac{{{A_c}}}{{{{(2\pi )}^2}}}\int_{1BZ} {{{\rm d}^2}k} \;V(k)\, {\rm e}^{i k k \cdot k r_p}\,, \quad {I_\alpha } ({r_p}) = \frac{{{A_c}}}{{{{(2\pi )}^2}}}\int_{1BZ} {{{\rm d}^2}k} \;I(k)\, {\rm e}^{i k \cdot r_p}\,, $$
Substituting Eq. (7) into Eqs. (1)-(6), one can write the set equations (1)-(6) in matrix form as
$$ L(k)V(k) = - R I(k), $$
where $L(k)$ is the Fourier transform of the Laplacian matrix of the pentagonal lattice
$$ L(k) = \left( {\begin{array}{*{20}{c}} { - 3}&1&0&0& {\rm e}^{ -i k \cdot ( a_1 + a_2) } & {\rm e}^{ - i k \cdot a_2 }\\ 1&{ - \,4}&1 & {e}^{ -i k \cdot a_1} &0&1\\ 0&1&{ - 3}&1& {e}^{ - i k \cdot a_2} & 0\\ 0 & {\rm e}^{ i k \cdot a_1} &1&{ - 3}&1&0\\ {\rm e}^{i k \cdot ( a_1 + a_2) } &0& {\rm e}^{i k \cdot a_2} & 1&{ - \,4}&1\\ {\rm e}^{ i k \cdot a_2} &1&0&0&1&{ - 3} \end{array}} \right), $$
and $V(k)\;{\rm{and}}\;I(k)$ are vectors
$$ V(k) = \left( {\begin{array}{*{20}{c}} {{V_1}(k)}\\ \vdots \\ {{V_6}(k)} \end{array}} \right),\;\;\;\;\;I(k) = \left( {\begin{array}{*{20}{c}} {{I_1}(k)}\\ \vdots \\ {{I_6}(k)} \end{array}} \right). $$
Changing the variables $k \cdot {a_i}$ to ${\theta _i} (i = 1,2)$, Eq. (9) becomes
$$ L({\theta _1},{\theta _2}) \!=\! \left( {\begin{array}{*{20}{c}} { - 3}&1&0&0&{{{\rm e}^{ - {\rm i}({\theta _1} + {\theta _2})}}}&{{{\rm e}^{ - {\rm i}{\theta _2}}}}\\ 1&{ - \,4}&1&{{{\rm e}^{ - {\rm i}{\theta _1}}}}&0&1\\ 0&1&{ - 3}&1&{{{\rm e}^{ - {\rm i}{\theta _2}}}}&0\\ 0&{{{\rm e}^{{\rm i}{\theta _1}}}}&1&{ - 3}&1&0\\ {{{\rm e}^{{\rm i}({\theta _1} + {\theta _2})}}}&0&{{{\rm e}^{{\rm i}{\theta _2}}}}&1&{ - \,4}&1\\ {{{\rm e}^{{\rm i}{\theta _2}}}}&1&0&0&1&{ - 3} \end{array}}\right).$$
The lattice Green’s function matrix G can be obtained from the definition
$$ L({\theta _1},{\theta _2})G({\theta _1},{\theta _2}) = - {1}. $$
Solving Eq. (12), the elements of the Green’s function matrix ${G_{{\alpha_1}{\alpha _2}}}({\theta _1},{\theta _2})$ are given by
$$ {G_{11}} = {G_{66}} = {[\det L]^{ - 1}}\left[ {192 - 28\cos {\theta _1} - 28\cos {\theta _2} - 2\cos ({\theta _1} + {\theta _2}) - 6\cos ({\theta _1} - {\theta _2})} \right],\\ {G_{33}} = {G_{44}} = {[\det L]^{ - 1}}\left[ {192 - 28\cos {\theta _1} - 28\cos {\theta _2} - 2\cos ({\theta _1} - {\theta _2}) - 6\cos ({\theta _1} + {\theta _2})} \right],\\ {G_{22}} = {G_{55}} = {[\det L]^{ - 1}}\left[ {160 - 16\cos {\theta _1} - 16\cos {\theta _2}} \right],\\ {G_{12}} = G_{21}^* = {[\det L]^{ - 1}}[ {75 + {{\rm e}^{{\rm i}{\theta _1}}} + 9{{\rm e}^{ - {\rm i}{\theta _1}}} + 33{{\rm e}^{ - {\rm i}{\theta _2}}} - 3{{\rm e}^{{\rm i}{\theta _2}}} - {{\rm e}^{ - 2{\rm i}{\theta _2}}} + 3{{\rm e}^{{\rm i}({\theta _1} - {\theta _2})}} + 11{{\rm e}^{ - {\rm i}({\theta _1} + {\theta _2})}}} ],\\ {G_{13}} = G_{31}^* = {[\det L]^{ - 1}}[ {42 + 42{{\rm e}^{ - {\rm i}{\theta _1}}} + 18{{\rm e}^{ - {\rm i}{\theta _2}}} + 3{{\rm e}^{{\rm i}{\theta _2}}} + 3{{\rm e}^{ - {\rm i}({\theta _1} - {\theta _2})}} + 18{{\rm e}^{ - {\rm i}({\theta _1} + {\theta _2})}} + {{\rm e}^{{\rm i}({\theta _1} - {\theta _2})}} + {{\rm e}^{ - {\rm i}(2{\theta _1} + {\theta _2})}}} ],\\ {G_{14}} = G_{41}^* = {[\det L]^{ - 1}}[ {18 + 42{{\rm e}^{ - {\rm i}{\theta _1}}} + 3{{\rm e}^{ - 2{\rm i}{\theta _1}}} + 18{{\rm e}^{ - {\rm i}{\theta _2}}} + {{\rm e}^{{\rm i}{\theta _2}}} + {{\rm e}^{ - 2{\rm i}{\theta _2}}} + 42{{\rm e}^{ - {\rm i}({\theta _1} + {\theta _2})}} + 3{{\rm e}^{ - {\rm i}(2{\theta _1} + {\theta _2})}}} ],\\ {G_{15}} = G_{51}^* = {[\det L]^{ - 1}}[ {11 + 9{{\rm e}^{ - {\rm i}{\theta _1}}} + 33{{\rm e}^{ - {\rm i}{\theta _2}}} + 3{{\rm e}^{ - 2{\rm i}{\theta _2}}} + 75{{\rm e}^{ - {\rm i}({\theta _1} + {\theta _2})}} - {{\rm e}^{ - {\rm i}({\theta _1} - {\theta _2})}} - 3{{\rm e}^{ - {\rm i}(2{\theta _1} + {\theta _2})}} + {{\rm e}^{ - {\rm i}({\theta _1} + 2{\theta _2})}}} ],\\ {G_{16}} = G_{61}^* = {[\det L]^{ - 1}}[ {24 + 5{{\rm e}^{ - {\rm i}{\theta _1}}} + 86{{\rm e}^{ - {\rm i}{\theta _2}}} - {{\rm e}^{{\rm i}{\theta _2}}} - 4{{\rm e}^{ - 2{\rm i}{\theta _2}}} + 24{{\rm e}^{ - {\rm i}({\theta _1} + {\theta _2})}} - 4{{\rm e}^{{\rm i}({\theta _1} - {\theta _2})}} - {{\rm e}^{ - {\rm i}(2{\theta _1} + {\theta _2})}} - {{\rm e}^{{\rm i}({\theta _1} - 2{\theta _2})}}} ],\\ {G_{23}} = G_{32}^* = {[\det L]^{ - 1}}[ {75 + 33{{\rm e}^{{\rm i}{\theta _1}}} - 3{{\rm e}^{ - {\rm i}{\theta _1}}} - {{\rm e}^{ - 2{\rm i}{\theta _1}}} + 9{{\rm e}^{ - {\rm i}{\theta _2}}} + {{\rm e}^{{\rm i}{\theta _2}}} + 11{{\rm e}^{{\rm i}({\theta _1} - {\theta _2})}} + 3{{\rm e}^{{\rm i}({\theta _1} + {\theta _2})}}} ],\\ {G_{24}} = G_{42}^* = {[\det L]^{ - 1}}[ {33 - {{\rm e}^{{\rm i}{\theta _1}}} + 75{{\rm e}^{ - {\rm i}{\theta _1}}} - 3{{\rm e}^{ - 2{\rm i}{\theta _2}}} + 3{{\rm e}^{{\rm i}{\theta _2}}} + 11{{\rm e}^{ - {\rm i}{\theta _2}}} + {{\rm e}^{ - {\rm i}({\theta _1} - {\theta _2})}} + 9{{\rm e}^{ - {\rm i}({\theta _1} + {\theta _2})}}} ],\\ {G_{25}} = G_{52}^* = 32{[\det L]^{ - 1}}[ {1 + {{\rm e}^{ - {\rm i}{\theta _1}}} + {{\rm e}^{ - {\rm i}{\theta _2}}} + {{\rm e}^{ - {\rm i}({\theta _1} + {\theta _2})}}}],\\ {G_{26}} = G_{62}^* = {[\det L]^{ - 1}}[ {75 + 9{{\rm e}^{ - {\rm i}{\theta _1}}} + {{\rm e}^{ - {\rm i}{\theta _1}}} + 33{{\rm e}^{ - {\rm i}{\theta _2}}} - 2{{\rm e}^{{\rm i}{\theta _2}}} - {{\rm e}^{ - 2{\rm i}{\theta _2}}} + 11{{\rm e}^{ - {\rm i}({\theta _1} + {\theta _2})}} + 3{{\rm e}^{{\rm i}({\theta _1} - {\theta _2})}}} ],\\ {G_{34}} = G_{43}^* = {[\det L]^{ - 1}}[ {86 + 24{{\rm e}^{ - {\rm i}{\theta _1}}} - 4{{\rm e}^{{\rm i}{\theta _1}}} - {{\rm e}^{ - 2{\rm i}{\theta _1}}} + 23{{\rm e}^{ - {\rm i}{\theta _2}}} - 3{{\rm e}^{{\rm i}{\theta _2}}} - {{\rm e}^{ - 2{\rm i}{\theta _2}}} + 5{{\rm e}^{ - {\rm i}({\theta _1} + {\theta _2})}} - {{\rm e}^{{\rm i}({\theta _1} + {\theta _2})}}} ],\\ {G_{35}} = G_{53}^* = {[\det L]^{ - 1}}[ {33 + 11{{\rm e}^{ - {\rm i}{\theta _1}}} + 3{{\rm e}^{{\rm i}{\theta _1}}} + 75{{\rm e}^{ - {\rm i}{\theta _2}}} - {{\rm e}^{{\rm i}{\theta _2}}} - 3{{\rm e}^{ - 2{\rm i}{\theta _2}}} + 9{{\rm e}^{ - {\rm i}({\theta _1} + {\theta _2})}} + {{\rm e}^{{\rm i}({\theta _1} - {\theta _2})}}} ],\\ {G_{36}} = G_{63}^* = {[\det L]^{ - 1}}[ {42 + 3{{\rm e}^{ - {\rm i}{\theta _1}}} + 18{{\rm e}^{{\rm i}{\theta _1}}} + 42{{\rm e}^{ - {\rm i}{\theta _2}}} + 3{{\rm e}^{ - {\rm i}({\theta _1} + {\theta _2})}} + {{\rm e}^{{\rm i}({\theta _1} + {\theta _2})}} + 18{{\rm e}^{{\rm i}({\theta _1} - {\theta _2})}} + {{\rm e}^{{\rm i}({\theta _1} - 2{\theta _2})}}} ],\\ {G_{45}} = G_{54}^* = {[\det L]^{ - 1}}[ {75 + {{\rm e}^{ - {\rm i}{\theta _1}}} + 9{{\rm e}^{{\rm i}{\theta _1}}} + 33{{\rm e}^{ - {\rm i}{\theta _2}}} - 3{{\rm e}^{{\rm i}{\theta _2}}} - {{\rm e}^{ - 2{\rm i}{\theta _2}}} + 3{{\rm e}^{ - {\rm i}({\theta _1} + {\theta _2})}} + 11{{\rm e}^{{\rm i}({\theta _1} - {\theta _2})}}} ],\\ {G_{46}} = G_{64}^* = {[\det L]^{ - 1}}[ {42 + 42{{\rm e}^{{\rm i}{\theta _1}}} + 18{{\rm e}^{ - {\rm i}{\theta _2}}} + 3{{\rm e}^{{\rm i}{\theta _2}}} + {{\rm e}^{ - {\rm i}({\theta _1} + {\theta _2})}} + {{\rm e}^{{\rm i}(2{\theta _1} + {\theta _2})}} + 3{{\rm e}^{ - {\rm i}({\theta _1} + {\theta _2})}} + 18{{\rm e}^{{\rm i}({\theta _1} - {\theta _2})}}} ],\\ {G_{56}} = G_{65}^* = {[\det L]^{ - 1}}[ {75 + 33{{\rm e}^{{\rm i}{\theta _1}}} - 3{{\rm e}^{ - {\rm i}{\theta _1}}} - {{\rm e}^{ - 2{\rm i}{\theta _1}}} + {{\rm e}^{ - {\rm i}{\theta _2}}} + 9{{\rm e}^{{\rm i}{\theta _2}}} + 11{{\rm e}^{{\rm i}({\theta _1} + {\theta _2})}} + 3{{\rm e}^{{\rm i}({\theta _1} - {\theta _2})}}} ], $$
where $\det L = 340 - 144\cos {\theta _1} - 144\cos {\theta _2} - 56\cos {\theta _1}\cos {\theta _2} + 2\cos 2{\theta _1} + 2\cos 2{\theta _2}$ is the determinant of the Laplacian matrix L.
The effective resistance between the origin ${\rm{(}}0,0{\rm{;}}{\alpha _1}{\rm{)}}$ and lattice site ${\rm{(}}{p_1}{\rm{,}}\;{p_2}{\rm{;}}{\alpha _2}{\rm{)}}$ in a two-dimensional pentagonal lattice of identical resistances R can be computed from the general resistance formula for any infinite lattice structure of resistor networks given in Ref. [8] with d=2,
$$ \hspace{-0.6cm}\frac{{{R_{{\alpha _1}{\alpha _2}}}({p_1},{p_2})}}{R}\!\! =\!\! \int\limits_{ - \pi }^\pi {\frac{{{\rm d}{\theta _1}}}{{2\pi }}} \int\limits_{ - \pi }^\pi {\frac{{{\rm d}{\theta _2}}}{{2\pi }}} \\ [ {{G_{{\alpha _1}{\alpha _1}}}({\theta _1},{\theta _2})\! +\! {G_{{\alpha _2}{\alpha _2}}}({\theta _1},{\theta _2})\! - \! {G_{{\alpha _1}{\alpha _2}}}({\theta _1},{\theta _2})} {{\rm e}^{ - {\rm i}({p_1}{\theta _1}\! +\! {p_2}{\theta _2})}} {\! - {G_{{\alpha _2}{\alpha _1}}}({\theta _1},{\theta _2}){{\rm e}^{{\rm i}({p_1}{\theta _1}\! +\! {p_2}{\theta _2})}}}], $$
where ${G_{{\alpha _1}{\alpha _2}}}({\theta _1},{\theta _2})$ (with ${\alpha _1}{\rm{,}}{\alpha _2}{\rm{ = 1,}} \ldots {\rm{,6}}$ ) have been given in Eq. (13).
3 Numerical Results
It is difficult to find the two-site resistance analytically since the integrand in Eq. (14) is a very complicated function. However, the resistances ${R_{{\alpha _1}{\alpha _2}}}(0,0)$ }between sites ${\alpha _1}{\rm{ = 1,}} \ldots {\rm{,6}}$ and ${\alpha _2}{\rm{ = 1,}} \ldots {\rm{,6}}$ that belong to the same unit cell and the resistances ${R_{{\alpha _1}{\alpha _2}}}(1,0)$ between sites ${\alpha _1}{\rm{ = 1,}} \ldots {\rm{,6}}$ and ${\alpha _2}{\rm{ = 1,}} \ldots {\rm{,6}}$ that belong to adjacent unit cells are numerically computed and listed in Tables 1 and 2, respectively.
Table 1 Numerical values of the resistance ${R_{{\alpha _1}{\alpha_2}}}(0,0)$ (in units of R). |
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4 Conclusion
In this paper, we have computed the effective resistance between any pair of lattice sites in an infinite pentagonal lattice network of equal resistors. The calculations are based on the lattice Green’s function approach. This problem could be of pedagogical interest for undergraduate physics students and would provide an application of the Green’s function theory.