Resistance Calculation for an infinite Simple Cubic Lattice Application of Green s Function
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1 Resistance Calculatin fr an infinite Simple Cubic Lattice Applicatin f Green s Functin J. H. Asad *, R. S. Hijjawi, A. Sakaj and J. M. Khalifeh * Department f Phsics, Universit f Jrdan, Amman-1194, Jrdan. Phsics Department, Mutah Universit, Jrdan. Phsics Department, Ajman Universit, UAE. Abstract It is shwn that the resistance between the rigin and an lattice pint ( l, in an infinite perfect Simple Cubic (SC) is epressible ratinall in terms f the knwn value f G (,,). The resistance between arbitrar sites in a SC is als studied and calculated when ne f the resistrs is remved frm the perfect lattice. Finall, the asmpttic behavir f the resistance fr bth the perfect and perturbed SC is als investigated. Ke wrds: Lattice Green s Functin, Resistrs, Simple Cubic Lattice. Crrespnding authr jhasad1@ah.cm.
2 1. Intrductin The calculatin f the resistance between tw arbitrar grid 1 7 pints f infinite netwrks f resistrs is a new-ld subject. Recentl, Cserti 8 and Cserti et. al 9 studied the prblem where the intrduced fr the first time a methd based n the Lattice Green s Functin (LGF) which is an alternative apprach t using the superpsitin f current distributins presented b Venezian and Atkinsn et. al 4. The LGF fr cubic lattices has been investigated b man 1 authrs, and the s-called recurrence frmulae which are ften used t calculate the LGF f the SC at different sites are presented 1,. The values f the LGF fr the SC have been recentl evaluated eactl, where these values are epressed in terms f the knwn value f the LGF at the rigin. In this paper; we calculate the resistance between tw arbitrar pints in a perfect and perturbed (i.e. a bnd is remved) infinite SC using the methd presented b Cserti 8 and Cserti et. al 9. The LGF presented here is related t the LGF f the Tight-Binding Hamiltnian (TBH) 4.. Perfect SC Lattice In this sectin we epress the resistance in an infinite SC netwrk f identical resistrs between the rigin and an lattice site ( l n 8, ratinall, where it can be easil shwn that, ) R R g 1 g (1) where g G (,,) is the LGF at the rigin. and 1,, are related t r 1, r, r (i.e 1,, Duffin and Shell s, 5 parameters ) as r r () r 1
3 Varius values f r1, r, r are shwn in Glasser et. al [Table 1] fr ranging frm (,,) (5,5,5 ). T btain ther values f r1, r, r ne has t use the fllwing relatin G ( l 1, G G n 1) G ( l 1, G n 1) m 1, G l m n EG m 1, () ; E In sme cases ne ma need t use the recurrence frmulae (i.e. Eq. ()) tw r three times, and b the methd eplained abve we calculate different values fr r 1, r, r fr ( l, bend ( 5,5,5 ). Varius values f 1,, are shwn in Table 1. The value f the LGF at the rigin (i.e. G (,,) ) was first evaluated b Watsn in his famus paper 6, where he fund that G (,,) ( ) ( )[ K( k where k ( )( ) )] K( k) d is the cmplete elliptic integral 1 k Sin f the first kind. A similar result was btained b Glasser and Zucker 7 in terms f gamma functin. T stud the asmpttic behavir f the resistance in a SC, ne can shw that 8 as an f l, n ges t infinit then,. Thus R G (,,). (4) R. Perturbed SC Lattice In this sectin we calculate the resistance between an tw lattice sites in a SC, when ne f the resistrs (i.e. bnds) between the sites i i, i, i ) and j j ) is brken, where 9 ( z ( z G [ R ( i, j ) R ( j, i ) R ( i, i ) R ( j, j )] R( i, R ( i, (5) 4[ R R ( i )]
4 4 As an eample; let us assume that the bnd between i (,,) and j (1,,) is brken. S, we calculate the resistance between an tw sites. Our results are arranged in Table, and fr eample: The resistance between the sites i (,,) and j (1,, ) is R R(1,,). (6) i.e. the resistance between the tw ends f the brken bnd is R, which is a predictable result 9. Nw, if the brken bnd is shifted t be between the sites i (1,, ) and j (,,), then ne can find the resistance between an tw sites (i.e. i ( i, i, iz ) and j ( j z ) ). Using Eq. (5) again ne btains the results arranged in Table. Fr large values f i and j the resistance in a perturbed SC, becmes R( i, R ( i, g. (7) R R We cnclude that fr large separatin between the tw sites the perturbed resistance appraches the perfect ne (i.e. it appraches a finite value). 4. Results and Discussin Fig. 1. shws the resistance against the site ( l, n ) alng the [1] directin fr bth a perfect infinite and perturbed SC (i.e the bnd between i (,,) and j (1,, ) is brke. It is seen frm the figure that the resistance is smmetric (i.e. R,) R,) ) fr the perfect case due t the inversin smmetr f the lattice while fr the perturbed case the smmetr is brken s, the resistance is nt smmetric. As ( l, n ) ges awa frm the rigin the resistance appraches its finite value fr bth cases 8. Fig.. shws the resistance against the site ( l, n ) alng the [1] directin fr a perfect infinite and perturbed SC (i.e the bnd between i (,,) and j (1,, ) is brke. The figure shws that the resistance is smmetric fr the perfect and perturbed case, since there is n brken 4
5 5 bnd alng this directin. As ( l, n ) ges awa frm the rigin the resistance appraches its finite value fr bth cases 8. Fig.. shws the resistance against the site ( l, n ) alng the [111] directin fr a perfect and fr a perturbed SC (i.e the bnd between i (,,) and j (1,, ) is brke. The resistance is smmetric alng [ 111] directin fr bth the perfect and perturbed cases. Figures same as the abve figures ecept that the brken bnd is shifted (i.e the bnd between i (1,, ) and j (,,) is brke. The resistance alng [ 1] directin is nt smmetric in the perturbed case since the brken bnd is taken t be alng that directin. Frm Figs. 1-6, as the brken bnd is shifted frm the rigin alng [1] directin then the resistance f the perturbed SC appraches that f the perfect lattice. Als, ne can see that the perturbed resistance is alwas larger than the perfect ne. Measurement f the resistance f a finite SC is under investigatin in rder t cmpare results. 5
6 6 References: 1- B. Van der Pl and H. Bremmer, Operatinal Calculus Based n the Tw-Sided Laplace integral (Cambridge Universit Press, England, 1955) nd ed., p P. G. Dle and J. L. Snell, Randm walks and Electric Netwrks, (The Carus Mathematical Mngraph, series, The Mathematical Assciatin f America, USA, 1984) pp Venezian, G Am. J. Phs. 6, Atkinsn, D. and Van Steenwijk, F. J Am. J. Phs. 67, R. E. Aitchisn Am. J. Phs., F. J. Bartis Am. J. Phs. 5, Mnwhea Jeng.. Am. J. Phs. 68(1), Cserti, J.. Am.J.Phs.68, Cserti, J. Gula, D. and Attila P.. Am. J. Phs, 7, Mrita, T. and Hriguchi, T J. phs. C 8, L. 11- Jce, G. S J. Math. Phs. 1, Sakaji, A. Hijjawi, R. S. Shawagfeh, N. and Khalifeh, J. M.. J. f Math. Phs. 4(1), Hijjawi, R. S. and Khalifeh, J. M.. J. f The. Phs. 41(9), Sakaji, A. Hijjawi, R. S. Shawagfeh, N. and Khalifeh, J. M.. J. f The. Phs. 41(5), Hijjawi, R. S. and Khalifeh, J. M. J. f The. Phs. in press. 16-Mrita, T. and Hrigucih, T J. f Math. Phs. 1(6), Inue, M J. f Math. Phs. 16(4), Man, K J. f Math. Phs. 16(9), Katsura, S. and Hriguchi, T J. f Math. Phs. 1(),. - Glasser, M. L J. f Math. Phs. 1(8), Mrita, T J. Phs. A 8, Hriguchi, T J. Phs. Sc. Japan, Glasser, M. L. and Bersma, J.. J. Phs. A: Math. Gen., N. 8, Ecnmu, E. N. Green s Functin in Quantum Phsics Spriger-Verlag, Berlin. 5- Duffin, R. J and Shell, E. P Duke Math. J. 5, Watsn, G. N Quart. J. Math. (Ofrd) 1, Glasser, M. L. and Zuker, I. J Prc. Natl. Acad. Sci. USA, 74, 18. Figure Captins 6
7 R(i,/R 7 Fig. 1 The resistance n the perfect (squares) and the perturbed (circles) SC between i (,,) and j ( j,,) alng the [1] directin as a functin f j. The ends f the remved bnd are i (,,) and j (1,, ). Fig. The resistance n the perfect (squares) and the perturbed (circles) SC between i (,,) and j (,) alng the [1] directin as a functin f j. The ends f the remved bnd are i (,,) and j (1,, ). Fig. The resistance n the perfect (squares) and the perturbed (circles) SC between i (,,) and j j ) alng the [111] directin as a functin f j. The ends f the ( z remved bnd are i (,,) and j (1,, ). Fig. 4 The resistance n the perfect (squares) and the perturbed (circles) SC between i (,,) and j ( j,,) alng the [1] directin as a functin f j. The ends f the remved bnd are i (1,, ) and i (,,). Fig. 5 The resistance n the perfect (squares) and the perturbed (circles) SC between i (,,) and j (,) alng the [1] directin as a functin f j. The ends f the remved bnd are i (1,, ) and i (,,). Fig..6 The resistance n the perfect (squares) and the perturbed (circles) SC between i (,,) and j j ) alng the [111] directin as a functin f j. The ends f the ( z remved bnd are i (1,, ) and i (,,) The Site (i, Fig. 1 7
8 R(i,/R R(i,/R The Site (i, Fig The Site (i, Fig. 8
9 R(i,/R R(i,/R The Site (i, Fig The Site (i, Fig. 5 9
10 R(i,/R The Site (i, Fig. 6 Table Captins Table 1: Varius values f the resistance in a perfect infinite SC fr arbitrar sites. Table : Calculated values f the resistance between the sites i (,,) and j j ), fr a perturbed SC ( the bnd between i (,,) and j (1,, ) is brke. Table : Calculated values f the resistance between the sites i (,,) and j ( j z ), fr a perturbed SC ( the bnd between i (1,, ) and j (,,) is brke. ( z Table 1 lmn 1 1 1/. 11 7/1 1/ /8 -/ R R g 1 g 1
11 11-7/ /8 9/4-1/ / /6 9/ /16-1/ /8 7/ / /6 85/ /4-1/ / /48 119/8 1/ /6-69/ /16 1/ / 146/ / -148/ /9 11/ / / /9-54/ /16 879/8-115/ / -57/ /7 19/ /16-151/ / 14/ /48 849/ -1/ / / / 4617/ /7-889/ /6 6161/ / -9195/ /48 11/ / / /8 186/ /1-5/ 77/ /6 181/ /4-5751/ /16 151/ 9/ /1-759/ /8 49/ / / / /7 4/ / / / / / 84919/196 1/ / -8674/ / / / / / 561/ / / / / / 57189/ / / / / / / /6-179/ /4 169/ -955/
12 / / / / / / / / Table j ( j z ) R ( i, / R j ( j z ) R ( i, / R (1,,).5 (-1,,).568 (,,).4857 (-,,).4541 (,,).56 (-,,) (4,,).5157 (-4,,).4677 (5,,) (,-1,).699 (,1,).699 (,-,) (,,) (,-,).491 (,,).491 (,-4,) (,4,) (,-5,) (,5,) (,,-1).699 (,,1).699 (,,-) (,,) (,,-).491 (,,).491 (,,-4) (,,4) (,,-5) (,,5) (-1,-1,-1) (1,1,1) (-,-,-).59 (,,).5597 (-,-,-) (,,) (-4,-4,-4).5777 (4,4,4).5475 (-5,-5,-5) (-5,,) (5,5,5)
13 1 Table j ( j z ) R ( i, / R j ( j z ) R ( i, / R (1,,).568 (-1,,).4495 (,,).4857 (-,,) (,,) (-,,).4565 (4,,).471 (-4,,).4677 (5,,) (-5,,) (,1,).4191 (,-1,).4191 (,,).4155 (,-,).4155 (,,).4578 (,-,)
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