paper in an indexed journal

1. Applied Mechanics and Materials Vols. 313-314 (2013) pp 324-328
Online available since 2013/Mar/25 at www.scientific.net
© (2013) Trans Tech Publications, Switzerland
doi:10.4028/www.scientific.net/AMM.313-314.324
Infinite 3D Cubic Lattices of Identical Resistors
J. H. Asad
Tabuk University, Dep. of Physics, P. O Box 741, Tabuk 71491 , Saudi Arabia.
Email: jhasad1@yahoo.com.
Keywords- Lattice Green’s Function, Resistors, Simple Cubic Lattice.
Abstract. We expressed the resistance between the origin and any lattice point (l,m,n) in an
infinite perfect Simple Cubic (i.e., SC) network rationally in terms of the known value of the
Lattice Green's Function at the origin (i.e., (0,0,0) o G ), and π . On the other hand, we investigated
the asymptotic behavior of the resistance. Finally, some numerical results has been calculated.
I- Introduction
The calculation of the resistance between two arbitrary grid points in an infinite network of resistors
is a new-old problem in the circuit theory. It attracts the attention of many authors 1−13 .
Focusing on the previous efforts one can see that many methods have been used such as: current
distribution and difference equations method 1,6 , random walk method 2,3 , and recently, an important
educational method based on the LGF have been used 7−13 . In these efforts many lattices has been
studied well for both perfect and perturbed lattices.
The LGF for cubic lattices has been investigated by many authors see for example 14−18 and the
references there and the so-called recurrence formulae which are often used to calculate the LGF at
different sites are presented.
The values of the LGF for the SC have been recently evaluated exactly19 , where these values are
expressed in terms of the known value of the LGF at the origin.
In this paper; we calculate the resistance between two arbitrary points in a perfect infinite SC
using the LGF method.
The LGF presented here is related to the LGF of the Tight-Binding Hamiltonian (TBH) 20 .
II- Perfect SC Lattice
Consider an infinite SC network consisting of identical resistors. The equivalent resistance between
the origin of the SC network and any lattice site ( l,m,n ) can be expressed rationally as 7,19 :
ρ
= ρ + + ρ
(1)
0 2
1 0 2 3
0
R (l,m, n)
g
π
R g
where (0,0,0) 0 0 g = G is the LGF of the SC lattice at the origin.
and 1 2 3 ρ ,ρ ,ρ are related to 1 2 3 r , r , r (i.e 1 2 3 λ ,λ ,λ Duffin and Shelly’s parameters 19,21 ) as
15λ − λ
= 1− = 1− 1 1 ρ r 1 2 12
1 ρ = −r = λ
2 2 2 2
1ρ = −r = λ (2)
3 3 3 3
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2. Applied Mechanics and Materials Vols. 313-314 325
Various values for 1 2 3 ρ ,ρ ,ρ can be obtained directly from Glasser et. al 19 where he presented
different values for 1 2 3 r , r , r ranging from(0,0,0) − (5,5,5) lattice sites. In this work, we calculated
different values for 1 2 3 r , r , r (i.e., 1 2 3 ρ ,ρ ,ρ ) beyond the site (5,5,5) , where we have used the
following relation 22
G (l +1,m,n) + G (l −1,m,n) + G (l,m +1,n) + G (l,m −1, n) = G (l,m, n +1) + G (l,m,n −1) = o o o o o o
2 2 ( , , ) 0 0 0 EG l m n l m n o = − δ δ δ + . (3)
where E = 3.
Our calculated values are presented in Table 1 below.
The value (0,0,0) o G was first evaluated by Watson in his famous paper 23 , where he found that
) (18 12 2 10 3 7 6)[ ( )] 0.505462.
2
(0,0,0) ( = 2 + − − 2 = o o G K k
π
where = (2 − 3)( 3 − 2) o k
= ∫ is the complete elliptic integral of the first kind.
θ
θ
π
2 2
2
0 1
1
( )
k Sin
K k d
−
A similar result was obtained by Glasser and Zucker 24 in terms of gamma function.
Finally, To study the asymptotic behavior of the resistance in a SC network, since
G (l,m,n) →0 o as any of l,m,n goes to infinity then, one can showed that
( , , ) → = o
(0,0,0) 0.505462
R l m n
o G
R
. (4)
III- Results and Discussion
Using the calculated values for 1 2 3 ρ ,ρ ,ρ , and Eq. (1) then we can easily obtained the equivalent
resistance between the origin and any lattice site ( l,m,n ). Our calculated values are presented in
Table 1 above.
Fig. 1. shows the resistance against the site ( l,m,n ) along the [100] direction for a perfect
infinite SC network. It is seen from the figure that the resistance is symmetric (i.e.
R (l,0,0) R ( l,0,0) o o = − ) for the perfect case due to the inversion symmetry of the lattice. As
( l,m,n ) goes away from the origin the resistance approaches its finite value for both cases 8 .
Fig. 2. shows the resistance against the site ( l,m,n ) along the [111] direction for a perfect SC
network. The resistance is symmetric along [111]direction for the perfect network.
Figure Captions
Fig. 1 The resistance of the perfect SC network between the origin and the lattice site (l,0,0) along
the [100] direction as a function of l
Fig. 4 The resistance of the perfect SC network between the origin and the lattice site (l,m,n) along
the [111] direction as a function of (l,m,n)

3. 326 Machinery Electronics and Control Engineering II
-6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6
0.60
0.55
0.50
0.45
0.40
0.35
0.30
0.25
0.20
0.15
0.10
0.05
0.00
R(l,m,n)/R
The site (l,m,n)
Fig. 1 The resistance of the perfect SC network between the origin and the lattice site (l,0,0) along
the [100] direction as a function of l
-6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6
0.55
0.50
0.45
0.40
0.35
0.30
0.25
0.20
0.15
0.10
0.05
0.00
R(l,m,n)/R
The Site (l,m,n)
Fig. 2 The resistance of the perfect SC network between the origin and the lattice site (l,m,n) along
the [111] direction as a function of (l,m,n)

4. Applied Mechanics and Materials Vols. 313-314 327
Table Captions
Table 1: Values of the resistance in a perfect infinite SC lattice for arbitrary sites.
Site
lmn 1 ρ 2 ρ 3 ρ
ρ
R l m n
0 ( , , ) ρ
3
= ρ + +
0
2
2
1 0
π
g
g
R
000 0 0 0 0
100 0 0 1/3 0.333333
110 7/12 1/2 0 0.395079
111 9/8 -3/4 0 0.418305
200 -7/3 -2 2 0.419683
210 5/8 9/4 -1/3 0.433598
211 5/3 -2 0 0.441531
222 3/8 27/20 0 0.460159
300 -33/2 -21 13 0.450371
310 115/36 85/6 -4 0.454415
311 15/4 -21/2 2/3 0.457396
320 -271/48 119/8 1/3 0.461311
321 161/36 -269/30 0 0.463146
332 -26/9 1012/105 0 0.471757
333 51/16 -1587/280 0 0.475023
400 -985/9 -542/3 92 0.464885
410 531/16 879/8 -115/3 0.466418
441 4197/32 -919353/2800 0 0.477814
442 -2927/48 31231/200 0 0.479027
443 571/32 -119271/2800 0 0.480700
444 -69/8 186003/7700 0 0.482570
500 -9275/12 -3005/2 2077/3 0.473263
510 11653/36 138331/150 -348 0.473986
511 -271/4 -5751/10 150 0.474646
5620 -2881/16 15123/200 229/3 0.475807
521 949/12 -27059/350 -24 0.476341
522 -501/8 4209/28 2 0.477766
530 -3571/18 1993883/3675 -8 0.478166
531 1337/8 -297981/700 4/3 0.478565
532 -2519/36 187777/1050 0 0.479693
533 2281/48 -164399/1400 0 0.481253
540 -18439/32 28493109/19600 1/3 0.480653
554 -24251/312 -1527851/7700 0 0.485921
555 9459/208 -12099711/107800 0 0.487123
600 -34937/6 -313079/25 5454 0.478749
610 71939/24 160009/20 -9355/3 0.479137
633 18552/72 -747654/1155 0 0.483209
644 -388051/1872 23950043/46200 0 0.486209
655 13157/78 -5698667/13475 0 0.488325
700 -553847/12 5281913/50 44505 0.482685
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6. Machinery Electronics and Control Engineering II
10.4028/www.scientific.net/AMM.313-314
Infinite 3D Cubic Lattices of Identical Resistors
10.4028/www.scientific.net/AMM.313-314.324