On the uniform tiling with electrical resistors

Transcription

1 On the uniform tiling with electrical resistors M. Q. Owaiat Department of Physics, Al-Hussein Bin Talal University, Ma an,7, Joran Abstract We calculate the effective resistance between two arbitrary lattice points on infinite strip of the triangular lattice laer network) in one imension, an on infinite moifie square an Union Jack lattices in two imensions, an on infinite ecorate simple cubic an base-centere cubic lattices in three imensions by using the general lattice Green's function metho. Keywors: lattice Green s function; moifie square ;Union Jack lattice; base-centere lattice. PACS numbers: b, 0.0.Vv, 4.0.Cv

2 . Introuction The etermination of the effective resistance between any two noes on an infinite lattice network of ientical resistors is a problem of consierable interest in electric circuit theory. The problem has been stuie extensively by many authors using several methos[-6]. ecently, Cserti [6] has presente an elegant metho to calculate the resistance between two arbitrary noes for infinite lattice networks. This approach is base on the lattice Green s function of the Laplacian operator of the ifference equations governe by the Ohm s an Kirchhoff's laws. Cserti et al. [7] have applie the Green s function metho to the problem of a perturbe network that is obtaine by removing one bon from the perfect lattice. Base on this metho, other perturbations are consiere such as replacing one resistor in the perfect lattice with another one[8], introucing single interstitial resistor[9] an two extra interstitial resistors [0] between two arbitrary noes in the perfect lattice. The Green s function metho is also a useful tool to calculate the capacitance of a perfect an a perturbe infinite capacitor networks[-]. Another interesting problem in electric circuit theory is to compute the two-point resistance in a finite lattice network. ecently, Wu[4] has presente a general formulation for computing two-point resistances in a network in terms of the eigenvalues an eigenfunctions of the Laplacian matrix. Jafarizaeh et al.[5] propose an algorithm for the calculation of the effective resistance between two arbitrary noes in an arbitrary istance-regular networks. Cserti et al. [6] have generalize the Green s function metho [6] for a resistor network that is a uniform tiling of -imensional space with electrical resistors. They presente several examples as applications to this metho. Owaiat[7] has use the general Green s function metho [6] to calculate the two-point resistance on the facecentere cubic lattice. In this paper the same technique [6] is applie to other lattices: infinite full laer, moifie square, Union Jack, ecorate simple cubic an base-centere cubic lattices of

3 ientical resistors. The paper is arrange as follows. In section, a review of general formalism for computing the two point resistance on an any infinite lattice network is given 7. In section, several lattice structures in one, two an three imensions with more one type of lattice point with some analytical an numerical results for resistance are presente.. A conclusion is given in section 4. The results for the resistance on a ecorate simple cubic lattice in terms of the resistances on an usual simple cubic lattice are liste in Appenix.. eview the general formulation The formulation of calculating the resistance between two arbitrary sites in an infinite uniform lattice network of resistors is briefly reviewe see, for more etails ef.6). Consier a regular - imensional lattice comprise of repeate unit cells, each contains v verticessites) labele by,,..., v,specifie by lattice siter n a, n i i i wherea, a,..., a are the unit cell vectors an n, n,..., n are integers. If{ r, } enotes any lattice site, then I rn ) an V rn ) enote the current an potential at site{ r n, }, respectively. It is suppose that the current can enter the { rn, } site from a source outsie the lattice. Accoring to Ohm's an Kirchhoff's laws the equations for the currents I r n) for each lattice site in one unit cell can be written as n L rm rn ) V rm ) I r n).) m, where r r ) is a v by v matrix calle Laplacian matrix. The Laplacian matrix.in L m n matrix notation one writes Eq..) as L rm rn ) V rm ) I r n).) m The iscrete Fourier transforms of the current an potential are efine as I -ikr k) I r )e, r V -ikr k) V r )e,,,...,v.)

4 where k is the wave vector in the reciprocal lattice an is limite to the first Brillouin zone[8-]. The general expressions for the inverse Fourier transform are given / a / a 0 ikr I ) [ I )],..., I )e ) / a / a r k k k.5) / a / a 0 ikr V ) [ V )],..., V )e ) / a / a r k k k.6) where is the imensionality an 0 is the volume of the unit cell. Using the above equations, Eq..) can be written as L k) V k) I k ).7) where Lk ) is the Fourier transform of Laplacian matrix an given by -ikr L k) L r )e.8) The lattice Green s function Gk )which is v by v matrix) corresponing to Fourier transform of Laplacian matrix Lk ) is efine as G k) L k ).9) Hence, Eq..7) becomes V k ) = G k) Q k ).0) To compute the resistance r, r ) between two noes { r, } an { r, }, we m n connect { r, } an { r, } to the two terminals of an external battery an measure the m n current I going through the battery while no other noes are connecte to external sources. Let the potentials at the two noes be, respectively, V r m) an V r ). Then, the esire resistance is the ratio m n n 4

5 V rm) V rn) rm, r n).) I The computation of the two-point resistance r, r ) is now reuce to solving.) or.6)for V r m) an V r ) with the current given by n k rk, rm, rk, rn, m I r ) I.) Therefore, combining Eqs..6),.0)an.) one can obtain the following integral expression for the electrical potential in any site of the lattice network: / a / a 0 -ikrm -i krn ) ikrk V k ),..., G )e G )e e ) / a / a r k k k.) Substituting potential istribution given Eq..) into Eq..),the noe-noe resistance can be written as follows: / a / a 0 -i k rn - rm ) i k rn - rm ) m, n),..., G ) G ) G )e G )e ) / a / a r r k k k k k n.4) The resistance oes not change if the lattice structure is eforme an the topology of the lattice structure is preserve. Therefore, any resistor network lattice can be eforme into a - imensional hypercube where the volume of the unit cell is 0 a... a a. If one writes rn rm r a.. a an changes the variables, kai i i,,..., ) in equation.4) then the final result for the resistance between any infinite uniform lattice structure can be written as,.., ).. G,.. ) G,.. ) i.. ) i.. ) G,.. ) e G,.. ) e -.5) 5

6 . Applications.. One- imensional lattice Here we consier a strip of the triangular lattice of resistors as an example of a one - imensional resistor network with next-nearest- neighbor resistors as shown in Fig.. We assume all eges represent a resistance. Here one can choose a unit cell contains two lattice points labele by an. Accoring to Ohm's an Kirchhoff's laws the currents in the unit cell at a lattice point x a are given by I x) 4 V x) V x a) V x a) V x) V x a).) I x) 4 V x) V x a) V x a) V x) V x a) The Laplacian matrix is two by two an can be written as 4 x,0 x, a x, a x,0 x, a L x) x,0 x, a 4 x,0 x, a.) x, a an after changing the variable ka, the Fourier transformation reas i cos 4 e L ) -i.) e cos 4 The lattice Green s function can be evaluate reaily using Eq..9).The result which we want is i 4 cos e G ) -i Det L )) e 4 cos.4) where Det L )) 4 8cos 4cos ) / is the eterminant of matrix L ). The resistance between the origin an site )can be calculate from equation.5): -i i ) G ) G ) G ) e G ) e.5) 6

7 There are four types of resistance: cos ) cos ) ) ).6a) 7 cos ) cos ) ) 4 cos cos cos ).6b) 7 cos ) cos ) ) 4 cos cos cos ).6c) 7 cos ) cos ) Using above equations, some analytical resultsin units of ) are liste below: 0) 0) ) ), 5 5 ) ), ), ) ) 5.. Two imensional lattices In this subsection, we consier two imensional lattices with more than one type of site the square, triangular an honeycomb lattices have been stuie in efs.[5,6] ; the centere square, Kagomé, ice, ecorate square an 4.8.8bathroom tile) lattices in ef.[6])....moifie square lattice The moifie square lattice of the resistor network is shown in Fig., where a unit cell contains s 4 sites numbere by,,,4,an the coorination number is 6.Using Ohm's an Kirchhoff's laws the currents at the lattice site { r, } with,,,4) are given by 7

8 I r) 6 V r) V r) V r a ) V r a ) V r a ) V r) V r a ) 4 4 I r) 6 V r) V r) V r a ) V r) V r a ) V r a ) V r a ) 4 4 I r) 6 V r) V r a ) V r a ) V r) V r a ) V r) V r a ) 4 4 I4 r) 6 V4 r) V r) V r a ) V r a) V r a ) V r) V r a).7) Hence, after changing the variables the Fourier transform of the Laplacian matrix is given by i i i i 6 e e e e i i i i e 6 e e e L, ).8) i i i i e e e 6 e i i i i e e e e 6 an the Green function can be obtaine from Eq..9): G, ) A C D E * C B E F E F C B * * * Det L, )) D E A C * *.9) where Det cos cos L, )) 56 cos cos ) / is the eterminant of matrix L, ), A 76 6cos 6cos 6cos ), i i B 76 6cos 6cos 6cos ) C 48 48e 6e cos 6cos, D e e e E 48 48e 6e cos 6cos, i i i ) i i , F e e e i i i ) Using the result.5) for =, one can fin the resistance between any two sites. For example, the resistance between lattice point an that belong to the same unit cell, we get 4 cos cos cos cos 0,0) 4.0) 4 cos cos cos cos ) 8

9 Performing the integrations analytically, we fin 0,0) ) / 4. From the symmetry of the lattice, one can fin 0,0) 0,0) 4 0,0) 4 0,0) ) / 4. Moreover, Some other analytical results in units of ) are given below: 0,0) 40,0), 4,0), 4,0), 8 8 4,) Union Jack lattice The Union Jack lattice is square tiling with each square ivie into for triangles from the center point as shown in Fig.. Each unit cell contains s 4 sites an coorination numbers are 4 an 8. The Ohm's an Kirchhoff's laws applie to the lattice site { r, }with,,,4) rea: I r) 8 V r) V r) V r a ) V r) V r a ) V r a ) V r a a ) V r) V r a ) 4 4 I r) 4 V r) V r) V r a ) V r) V r a ) I r) 8 V r) V r) V r a ) V r a ) V r a a ) V r) V r a ) V r) V r a ) 4 4 I r) 4 V r) V r) V r a ) V r) V r a ) 4 4.) The Fourier transform the Fourier transform of the Laplacian matrix is given by 8 4 0, ) i j L,where 8 j e j, ).) 0 4 9

10 Now calculating the Green s function from equation.9) one can use the general expression.5) to fin the resistance between arbitrary lattice points on the Union Jack resistor network. Some numerical results in terms of ) are given below: 0,0) 0,0) 0,0) 0,0) 0.65, 4 4 0,0) 0.77, 4,0) 0.76,,0) , 0,0) 0.565,,0) 0.589,,) Lattices in three imensions The usual three-imensional simple, boy-centere an face-centere cubic lattices are stuie in [5,6], [6] an [7], respectively. In this section, we calculate the resistance on other lattices. In particular on the ecorate cubic an base-centere lattices....the ecorate cubic lattice The so-calle ecorate cubic lattice is a simple cubic lattice with one site inserte on each ege as shown in Fig. 4. The unit cell contains s 4 sites labele by,,,4 an the egree or coorination number of all sites is or 6. Using Ohm's an Kirchhoff's law's the equation for the currents for each lattice site in one unit cell can be written as I r) 6 V r) V r) V r a ) V r) V r a ) V r) V r a ) I r) V r) V r) V r a ) I r) V r) V r) V r a ) I r) V r) V r) V r a ).) After changing the variables ka,,), the Fourier transform of the Laplacian matrix is given by i i i 0

11 6 0 0 i,, ) j L where 0 0 j e, j,,..4) 0 0 an the corresponing lattice Green's function : G,, ) A Det L 4,, )) B 4 C.5) where Det,, )) 8cos cos cos L, A B6 cos cos, C6 cos cos. 6 cos cos, One can note that the eterminant of the Laplacian L, ) is eight times the enominator of the integran in the resistance formula for the simple cubic lattice [5,6], so that the resistance on the ecorate simple cubic lattice,, ) can be expresse in terms of the resistances,, ) sc on the simple cubic lattice. The resistance between the origin an the site,, ) on the simple cubic lattice is given by cos ) ).6) cos cos cos,, ) sc The results for the resistance,, ) on a ecorate simple cubic lattice in terms of the resistances,, ) sc ) on an usual simple cubic lattice are given in Appenix. Some results for resistances are presente below:

12 0,0,0) 0,0,0) 40,0,0), 0, 0, 0).975,, 0, 0).0864, 4,, 0) 0.790,,,) 0.867,, 0, 0) Base-centere cubic lattice The base-centere cubic lattice is a three imensional lattice. Besies the resistors on the eges of the cube there are resistors between the centers of its two bases to the corners of these bases as shown in Fig.5. Each unit cell consists of two lattice points. Applying the Ohm's an Kirchhoff's law's one can write the current at each lattice point in one unit cell as I r) V r) V ra ) V ra ) 4 V r) V r) V r a ) i i i V r a ) V r a a ) I r) 4 V r) V r) V r a ) V r a ) V r a a ).7) Therefore, the Fourier transform of the Laplacian matrix is given by i i 0 cos cos cos e ) e ) L,, ).8) i i e ) e ) 4 an the corresponing Green s function : G,, ) i i 4 e ) e ) L) i i ) ) 0 cos cos cos Det e e.9)

13 where Det ) 6 cos cos 4 coscos 8 cos L. Now, the resistance between any two site can be calculate from Eq..5). Below we present some numerical results in units of 0, 0, 0) = 0.4,, 0, 0) = 0.8,, 0, 0) = 0.467,,, 0) = 0.465,,,) = 0.79,, 0, 0) = Conclusions In this work, we have calculate the resistance between any two noes on the infinite laer lattice, the moifie square, Union Jack, ecorate simple cubic an basecentere cubic lattices using the general Green s function metho[6]. For the ecorate simple cubic lattice we have expresse the resistances between two arbitrary sites in terms of the resistances on a simple cubic lattice of resistors. We have presente some analytical an numerical results near the lattices origins.

14 eferences.b. van er Pol an H. Bremmer, Operational Calculus Base on the Two-Sie Laplace Integral, n en. Cambrige University Press, Englan, 955).. P. G. Doyle an J. L. Snell, anom Walks an Electric Networks, The Carus Mathematical Monograph, Series The Mathematical Association of America, USA,984)... E. Aitchison, Am. J. Phys. 964) G. Venezian, Am. J. Phys ) D. Atkinson an F. J. van Steenwijk, Am. J. Phys ) J. Cserti, Am. J. Phys ) J. Cserti, G. Davi an A. Piroth, Am. J. Phys ) M.Q. Owaiat,.S Hijjawi an J. M. Khalifeh, Mo. Phys. Lett. B 900) M.Q. Owaiat,.S Hijjawi an J. M. Khalifeh,J. Phys. A, Math. Theor. 400) M.Q. Owaiat,.S Hijjawi an J. M. Khalifeh, Int.J.Theor. Phys. 50)5 59. J. H. Asa, A. Sakaji,. S. Hijjawi an J. M. Khalifeh,Int. J. Mo. Phys. B 94), ).. M.Q. Owaiat, Joran Journal of Physics. 5 ), -80).. M.Q. Owaiat,.S Hijjawi, J. H. Asa an J. M. Khalifeh, Mo. Phys. Lett. B Accepte). 4. F. Y. Wu, J. Phys. A: Math. Gen ) M.A.Jafarizaeh,. Sufiani, an S. Jafarizaeh, J. Phys. A, Math. Theor )

15 6. J ozsef Cserti, G abor Sz echenyi an Gyula D avi, J. Phys. A, Math. Theor. 0). 7. M.Q. Owaiat, Am. J. Phys. Accepte). 8. Chaikin P M an Lubensky T C 995 Principles of Conense Matter Physics Cambrige,Englan: Cambrige University Press) 9. Ashcroft N W an Mermin N D 976 Soli State Physics Philaelphia, PA: Sauners College). 0.Ziman J M 97 Principles of The Theory of Solis Cambrige, Englan: Cambrige University Press).. Kittel C 986 Introuction to Soli State Physics 6th e New York: John Wiley an Sons). 5

16 Appenix. elation between the ecorate simple cubic lattice an the usual simple cubic lattice of resistor networks In this appenix we summarize the results for the resistance,, ) on a ecorate simple cubic lattice in terms of the resistances,, ) sc ) on an usual simple cubic lattice:,, ) sc,, ) A. sc,, ) sc,, ),, ) 4,, ) sc,, ) sc,, ) sc A. sc,, ) sc,, ),, ) 4,, ) sc,, ) sc,, ) sc A. sc,, ) sc,, ),, ) 4,, ) sc,, ) sc,, ) 44 sc A.4,, ) sc,, ) sc,, ) A.5,, ) sc,, ) sc,, ) A.6 4,, ) sc,, ) sc,, ) A.7 sc,, ) sc,, ),, ) sc,, ) sc,, ) A.8 sc,, ) sc,, ),, ) sc,, ) sc,, ) 4 A.9 sc,, ) sc,, ),, ) sc,, ) sc,, ) 4 A.0 6

17 Figure captions Fig.. Infinite strip of the triangular lattice of resistors as an example of a one imensional network Fig..The moifie square lattice of the resistor network. Sites within a unit cell are labele as shown. Fig..The Union Jack tetrakis) lattice of the resistor network. Sites within a unit cell are labele as shown. Fig.4. The so-calle ecorate simple cubic lattice. Sites within a unit cell are labele as shown. Fig. 5. Base-centere lattice structure of the resistor network. Each unit cell consists of two sites. Fig. Fig. 7

18 Fig. Fig. a 4 a a Fig.4 8

19 Fig. 5 9