Infinite Body Centered Cubic Network of Identical Resistors. Abstract. and any other lattice site ( n

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1 Infinite Boy Centere Cubic Network of Ientical esistors J. H. Asa,*, A. A. Diab,. S. Hijjawi, an J. M. Khalifeh Dep. of Physics- Tabuk University- P. O Box 7- Tabuk 79- Saui Arabia General Stuies Department- Yanbu Inustrial College- P. O Box 06- Yanbu Inustrial City- KSA Dep. of Physics- Mutah University- Karak- Joran Dep. of Physics-Joran University- Amman 9 - Joran Abstract We express the equivalent resistance between the origin ( 0,0,0) an any other lattice site ( n, n, n) in an infinite Boy Centere Cubic (BCC) network consisting of ientical resistors each of resistance rationally in terms of known valuesb o an π. The equivalent resistance is then calculate. Finally, for large separation between the origin ( 0,0,0) an the lattice site ( n, n, n) two asymptotic formulas for the resistance are presente an some numerical results with analysis are given. Keywors: Lattice Green s Function, Infinite network, BCC lattices, Ientical esistors. * Corresponing author jhasa@yahoo.com.

2 . Introuction The Lattice Green s Function (LGF) is a basic concept in physics. Many quantities of interest in soli-state physics can be expresse in terms of it. For example, statistical moel of ferromagnetism such as Ising moel [], Heisenberg moel [], spherical moel [], lattice ynamics [], ranom walk theory [5, 6], an ban structure [7, 8]. In Economou s book [9] one can fin an excellent introuction to the LGF, where a review of the LGF of the so-calle tight-bining Hamiltonian (TBH) use for escribing the electronic ban structures of crystal lattices is presente. The LGF efine in this paper is relate to the GF of the TBH. Many efforts have been pai on stuying the LGF of cubic lattices [0-5]. The LGF for the BCC lattice has been expresse as a sum of simple integrals of the complete elliptic integral of the first kin [0], Morita an Horiguci [] presente formulas which are convenient for the evaluation of the LGF for the Face Centere Cubic (FCC), BCC an rectangular lattices. These formulas involve the complete elliptic integral of the first kin with complex moulus. Morita [] erive a recurrence relation, which gives the values of the LGF along the iagonal irection from a couple of the elliptic integrals of the first an secon kin for the square lattice with iscussions of how to apply the result to the BCC lattice. Finally, Glasser an Boersma [] expresse the values of the LGF of the BCC lattice rationally. One can fin more works in these works an references within them. The calculation of the equivalent resistance in infinite networks is a basic problem in the electric circuit theory. It is of so interest for physicists an electrical engineering. There are mainly three approaches use to solve such a problem. The superposition of current istribution has been use to calculate the effective resistance between ajacent sites on infinite networks [6-8]. A mapping between ranom walk problems an resistor networks problems have been use by Monwhea Jeng [9], where his metho was use to calculate the effective resistance between any two sites in an infinite two-imensional square lattice of unit resistors. A thir eucational important metho base on the Lattice Green s Function (LGF) of the lattices has been use in calculating the equivalent resistance [0-8]. This metho has been applie to both perfect an perturbe square, simple cubic (SC) networks an recently to the FCC network The present work is organize as follows:

3 In Sec. we briefly introuce the basic formulas of interest for the LGF of the BCC network. In Sec. an application to the LGF of the BCC network has been applie to express the equivalent resistance between the origin an the lattice site ( n, n, n) in the infinite BCC network rationally in terms of some constants, an the asymptotic behavior for the resistance is also investigate as the separation between the two sites goes to infinity. Finally, we close this paper (Sec. ) with a iscussion for the results obtaine.. Basic Definitions an Preliminaries The LGF for the BCC lattice appears in many areas in physics (e.g., Ising moel [, 9-0], Heisenberg moel [- ] an spherical moels [- 6] an it is efine as [, ] : π π π Cos( nu) Cos( nv) Cos( nw) ( E; n, n, n) = uvw. () E CosuCosvCosw B π Where E n, n, an n are either all even or all o integers. The LGF for the BCC lattice at the site ( 0,0,0) which represents the origin of the lattice for E = (i.e., B ( ;0,0,0) =bo ) was of so interests in physics an it was carrie out first by Van Peijpe [7] an later on by Watson [8]. They showe that: b o = B ;0,0,0) = [ K( )] π Γ ( ) = π ( = () Where K is the complete elliptic integral of the first kin, an Γ is the Gamma function In a recent work the LGF for the infinite BCC lattice has been expresse rationally as []: σ B ( ; n, n, n) = σ bo + + σ. () π bo Whereσ, σ an σ are rational numbers.

4 . Application: Evaluation of the resistance ( n, n, n) in an Infinite BCC Network The aim of this section is to express the equivalent resistance between the origin ( 0,0,0) an the lattice site ( n, n, n) in the infinite BCC network which is consisting from ientical resistors rationally in terms of b o an π. First of all, it has been showe that for a D infinite network consisting of ientical resistors each of resistance, the equivalent resistance between the origin an any other lattice site is [0]: ( r) [ G(0) G( r)] =. () Where r is the position vector of the lattices point, an for a - imensional lattice it has the following form: r = n a + na n a. (5) With n, n,..., n are integers, an a, a,..., a are inepenent primitive translation vectors. Also, the equivalent resistance between the origin an any other lattice site is can be expresse in an integral form as [0]: ( n, n,..., n (6) ) = π π x... π π π x π exp( in x i= + in x ( Cosx ) in On the other han, the LGF for a D hypercube rea as [0]: G( n, n,..., n (7) ) = π π x... π π π x π exp( in x + in x i= in ( Cosx ) For cubic lattices =. Then substituting = into Eqs. (6) an (7) an comparing them with Eq. () one get: ( n, n, n) = [ bo B(; n, n, n)]. (8) Now make use of Eq. () an Eq. (8) one yiels i i x ). x ).

5 ( n, n, n) = r b r o + + r π bo. (9) Where r = σ, r = σ an r = σ are rational numbers. These rational numbers, for the sites from ( 0,0,0) to ( 8,8,8), can be gathere from [, appenix A]. In Table below we present these rational numbers. Base on the recurrence formula presente in [, Eq. (5.8)] we have calculate aitional rational values for the sites from ( 9,, ) to ( 0,0,0) an arrange them in Table below. Since the LGF is an even function (i.e., B( ; n, n, n ) = B(; n, n, n ) ) an ue to the fact that the infinite BCC network is pure an symmetric, then as a result ( n, n, n) = ( n, n, n). Finally, it is interesting to stuy the asymptotic behavior of the equivalent resistance for large separation between the origin ( 0,0,0) an any other lattice site ( n, n, n). The asymptotic form of B ( ;0,0, n), as n, is given as []: 7 B ( ;0,0,n ) ( + ) π n. (0) 8n 8n 0n 6 While, for large value of + n n n = n + it has been shown that [] B ; n, n, ) has the following asymptotic formula ( n B(; n, n, n) n [ n + n ( n + n + n π 8 8 ) n ( n n + n n + n n )]. () Inserting Eq. (0) an Eq. () into Eq. (8), one gets the following two equations: (0,0,n o ) b o πn ( 8n + 8n 7 0n 6 ). () o ( n, n, n) 6 bo ( n + n + n n π 9 [ n n 8 ) 5

6 5 6 n ( n n + n n + n n )]. () The last asymptotic formula agrees with Eq. () for n = 0, n = 0 an for n = n. In aition, the above two asymptotic formulas can be use to check the results obtaine in Table below. For example, o (8,0,0).5; o (8,8,6).; o (8,8,8 ).778; o (9,9,9).57; o (0,0,0).986. (5) From the above two asymptotic formulas, one can see that as n, or as n then the resistance goes to a finite value (i.e., goes tob o ).. esults an Discussion In this work we have expresse the equivalent resistance between the origin ( 0,0,0) an any other lattice site ( n, n, n) in an infinite BCC network consisting of ientical resistors each of resistance rationally in terms of the two known values b o an π. The rational number r, r an r presente in Eq. () were calculate using some recurrence formulas. In Fig. an Fig. the equivalent resistance is plotte against the lattice site. Figure shows the resistance in an infinite BCC lattice against the site ( n, n, n) along the [00] irection. From this figure it is clear that the resistance is symmetric. Figure shows the resistance in an infinite BCC lattice against the site ( n, n, n) along the [] irection. From this figure it is clear that the resistance is symmetric. The above figures inicate that as the separation between the origin an the lattice site ( n, n, n) increases, then the equivalent resistance approaches a finite value (i.e., b o = ) as explaine above. 6

7 Finally, it is worth mention that for the case of Face Centere Cubic (FCC) network the equivalent resistance as the separation between the origin an any other lattice site approaches a finite value ( f o = 0.809) [7] where f o is the LGF at the origin for the FCC lattice, while for the case of an infinite SC network it goes to the finite value g o = [0, ], where g o is the LGF at the origin for the SC lattice. Whereas for the infinite square lattice it goes to infinity [0, ] 7

8 Table Captions Table : Values for selecte rational numbers r, r, r an ( n, n, n ) for sites ( 0,0,0) to (0,0,0) Table : ( n, n, n) r r ( n, n, n) / / / / / / / / / / / / / / /9 96/ /9 / /9-6/ /9 -/ /9 7980/ / /9 9596/ / / /9-9088/ /9-7/ /6 0856/ /6-9888/ /6-50/ /9-750/ / / / / /6 96/ /6-8876/ /6-9969/ / /76-76/ /76-68/ / / /76-770/

9 / / / / / / / / /76 76/ /76 580/ /76 95/ / / / / / 7/ / -79/ /9 086/ / 6666/ / 77776/ / /6-799/ /6-7806/ /6 559/ / / 9685/ / / / 09575/ / / / / /

10 Figure Captions Fig.: esistance between the origin ( 0,0,0) an the site (n,0,0) along [00] irection for BCC network. Fig.: esistance between the origin ( 0,0,0) an the site ( n, n, n) along [] irection for BCC network (n,0,0)/ The Site (n,0,0) Fig (n,n,n)/ The site (n,n,n) Fig. 0

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