Resistance calculation of the face-centered cubic lattice: Theory and experimentM. Q. Owaidat Citation: American Journal of Physics 81, 918 (2013); doi: 10.1119/1.4826256 View online: http://dx.doi.org/10.1119/1.4826256 View Table of Contents: http://scitation.aip.org/content/aapt/journal/ajp/81/12?ver=pdfcov Published by the American Association of Physics Teachers

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Resistance calculation of the face-centered cubic lattice: Theory andexperiment

M. Q. Owaidata)

Department of Physics, Al-Hussein Bin Talal University, Ma’an 71111, Jordan

(Received 11 March 2013; accepted 7 October 2013)

The effective resistance between two arbitrary lattice points in an infinite, face-centered cubic

lattice network of identical resistors is calculated using the lattice Green’s function method.

Theoretical results have been verified experimentally by constructing actual finite networks of

resistors. This problem could be useful in undergraduate courses (e.g., advanced mathematical

methods course) and would provide a good example for introducing the concept of Green’s

function. VC 2013 American Association of Physics Teachers.

[http://dx.doi.org/10.1119/1.4826256]

I. INTRODUCTION

The determination of the effective resistance between anytwo nodes on an infinite lattice network of identical resistorsis a problem of considerable interest in electric circuittheory. The problem has been studied extensively by manyauthors using several methods.1–5 Recently, Cserti5 pre-sented an elegant method to calculate the resistance betweentwo arbitrary nodes for infinite lattice networks. Thisapproach is based on the lattice Green’s function of theLaplacian operator on the difference equations governed byOhm’s and Kirchhoff’s laws. Cserti et al.6 have also appliedthe Green’s function method to the problem of a perturbednetwork obtained by removing one bond from the perfectlattice.

The Green’s function is a versatile tool in many areas oftheoretical physics. It provides, for example, a powerfulmethod for solving linear problems involving a differentialequation. (Excellent introductions to Green functions can befound in Barton,7 Duffy,8 and Economou.9) The latticeGreen’s function is often used in condensed matter,10 withgood discussions of such uses in the aforementioned referen-ces.5,6 Green’s functions also appear when the finite differ-ence approximation is used to solve partial differentialequations.

Based on the lattice Green’s function method,6 other per-turbations to the resistor network are considered, such aschanging the value of one resistor in the perfect lattice11 andintroducing an extra resistor between two arbitrary nodes inthe perfect lattice.12 The Green’s function method is also auseful tool for calculating the capacitance of a perfect anda perturbed capacitor network.13,14 For a finite resistornetwork, Wu15 has developed a general formulation for com-puting two-point resistances in a network in terms of theeigenvalues and eigenfunctions of the Laplacian matrix.More recently, Cserti et al.16 have presented a general for-malism for calculating the resistance between any two latticepoints in any infinite lattice structure of resistors that is aperiodic tiling of space using lattice Green’s function.

In this work, we present an application of the generallattice Green’s function method;16 namely, we calculate theresistance between any two grid points in a face-centeredcubic (FCC) lattice network constructed of electrical resis-tors (see Fig. 1). In addition to resistors on the edges of thecubes, there are resistors between the center of each face ofthe cube and its corners. The unit cell is a cube containingfour lattice points: one at one of the corners of the cube and

the others at the three face-centers of the cube nearest thecorner lattice point.

A difference between our calculation and that of Asadet al.17 pertains to exactly which lattice points are connectedby resistors. Their system consisted of an FCC resistor net-work formed using resistors only between the center of eachface of the cube and its corners, with none on the edges of thecube (see Fig. 2). Their unit cell contains one lattice pointand they calculated the two-point resistance using the exactvalues for the FCC lattice Green’s functions given in Ref. 18.

The FCC resistor network in this paper can systematicallybe treated by the Laplacian matrix of the difference equa-tions governed by Ohm’s and Kirchhoff’s laws, as we willshow in the next section. The lattice Green’s function is thendefined as the solution to these difference equations corre-sponding to the Laplacian matrix and can be related to the re-sistance between two arbitrary nodes on a resistor network.We believe that the Green’s function method is a highlyeffective technique for the present problem, even in caseswhen other methods1–4 face extreme difficulties. Further,networks of resistors may serve as a didactic example forintroducing the Green’s function method, as well as otherbasic concepts (such as the Brillouin zone) used in solid statephysics.

The paper is organized as follows. In Sec. II, we presentanalogies between a resistor network and a crystal lattice. InSec. III, the Green’s function and the resistance for an FCCresistor network are presented. In Sec. IV, some numerical andexperimental results regarding the resistance are presented anddiscussed. Finally, a brief conclusion is given in Sec. V.

II. AN ANALOGY BETWEEN A RESISTOR

NETWORK AND CRYSTAL

In this section, we define many basic analogies between aresistor network and a crystal, such as the lattice point, theunit cell, the reciprocal lattice and the Brillouin zone.19–21 Alattice point is a node on a resistor network where two ormore resistors meet. A unit cell of resistors is the smallest col-lection of lattice points that can be repeated by translationsymmetry to create a periodic resistor network. The positionvector of any lattice point of the d-dimensional regular resistornetwork is given by r ¼ n1a1 þ n2a2 þ � � � þ ndad, where ni

are integers and a1; a2;…; ad are the unit cell vectors.As in solid state physics, in order to define the Brillouin

zone in a periodic resistor network we need to first define thereciprocal lattice. Assuming periodic boundary conditions

918 Am. J. Phys. 81 (12), December 2013 http://aapt.org/ajp VC 2013 American Association of Physics Teachers 918

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requires that the Fourier series expansion of the potential andthe current at any lattice point are given in terms of planewaves eik�r.

The lattice in the Fourier space (or k-space) is called the“reciprocal lattice” and is also perfectly periodic with atranslation symmetry of the set fbig, defining arbitrary recip-rocal lattice vectors as k ¼ m1b1 þ m2b2 þ � � � þ mdbd,where mi are integers and the reciprocal lattice vectors bi aredefined by ai � bj ¼ 2pdij (for i; j ¼ 1; 2;…; d). The firstBrillouin zone is defined as the unit cell in the reciprocal lat-tice. The zone boundaries are the wave vectors ki between�p=ai and p=ai.

III. GREEN’S FUNCTIONS AND RESISTANCES FOR

FCC NETWORK

Consider an infinite, face-centered cubic lattice composedof identical resistances R as shown in Fig. 1. The unit cell

contains four lattice points labeled by a ¼ A;B;C;D. Thelattice points can be specified by the lattice vectorr ¼ la1 þ ma2 þ na3, where a1, a2, and a3 are the unit cellvectors and l, m, and n are integers. We let fr; ag denote anarbitrary lattice point, where r and a specify the unit cell andthe lattice point in the given unit cell, and IaðrÞ and VaðrÞdenote the current and potential at point fr; ag. It is assumedthat a net current IaðrÞ enters the site fr; ag from a sourceoutside the lattice.

Applying Kirchhoff’s current rule to node fr;Ag (seeFig. 3) gives

IAðrÞ ¼ I1 þ I2 þ I3 þ � � � þ I18: (1)

Using Ohm’s law, Eq. (1) can be written as the finite differ-ence equation

RIAðrÞ ¼ ½VAðrÞ � VAðrþ a1Þ� þ ½VAðrÞ � VAðr� a1Þ�þ ½VAðrÞ � VAðrþ a2Þ� þ ½VAðrÞ � VAðr� a2Þ�þ ½VAðrÞ � VAðrþ a3Þ� þ ½VAðrÞ � VAðr� a3Þ�þ ½VAðrÞ � VBðrÞ� þ ½VAðrÞ � VBðr� a1Þ�þ ½VAðrÞ � VBðr� a2Þ�þ ½VAðrÞ � VBðr� a1 � a2Þ�þ ½VAðrÞ � VCðrÞ� þ ½VAðrÞ � VCðr� a2Þ�þ ½VAðrÞ � VCðr� a3Þ�þ ½VAðrÞ � VCðr� a2 � a3Þ�þ ½VAðrÞ � VDðrÞ� þ ½VAðrÞ � VDðr� a1Þ�þ ½VAðrÞ � VDðr� a3Þ�þ ½VAðrÞ � VDðr� a1 � a3Þ�: (2)

In a similar manner, the currents at nodes fr;Bg, fr;Cg, andfr;Dg are given by

RIBðrÞ ¼ ½VBðrÞ � VAðrÞ� þ ½VBðrÞ � VAðrþ a1Þ�þ ½VBðrÞ � VAðrþ a2Þ�þ ½VBðrÞ � VAðrþ a1 þ a2Þ�; (3)

Fig. 2. A face-centered cubic lattice resistor network with no resistors along

the cube edges. In the unit cell there is one lattice point at a corner of the

cube. The three unit vectors point from a corner of the cube to the centers of

the three cubic faces.

Fig. 1. The face-centered cubic (FCC) lattice resistor network considered in

this paper. In addition to resistors on the edges of the cube, there are resistors

between the center of each face and its four corners. Each unit cell contains

four lattice points labeled by a ¼ A;B;C;D. The unit cell vectors are a1, a2,

and a3.

Fig. 3. The current IAðrÞ enters the node fr;Ag from a source outside the lat-

tice and passes through the 18 resistors connected to the point fr;Ag.

919 Am. J. Phys., Vol. 81, No. 12, December 2013 M. Q. Owaidat 919

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RICðrÞ ¼ ½VCðrÞ � VAðrÞ� þ ½VCðrÞ � VAðrþ a2Þ�

þ ½VCðrÞ � VAðrþ a3Þ�

þ ½VCðrÞ � VAðrþ a2 þ a3Þ�; (4)

and

RIDðrÞ ¼ ½VDðrÞ � VAðrÞ� þ ½VDðrÞ � VAðrþ a1Þ�

þ ½VDðrÞ � VAðrþ a3Þ�

þ ½VDðrÞ � VAðrþ a1 þ a3Þ�: (5)

Equations (2)–(5) can be written in the form

Xr0;b

Labðr� r0ÞVbðr0Þ ¼ �IaðrÞ; (6)

where Labðr� r0Þ is a 4� 4 matrix called the Laplacian ma-trix, with a, b ¼ A;B;C;D. The discrete Fourier transformsof the current and potential are

IaðkÞ ¼X

r

IaðrÞe�ik�r (7)

and

VaðkÞ ¼X

r

VaðrÞe�ik�r; (8)

where a ¼ A;B;C;D, and k is the wave vector in the recipro-cal lattice and is limited to the first Brillouin zone.19–21

The general expressions for the inverse Fourier transformof the current and potential are given by

IaðrÞ ¼ =�1½IaðkÞ� ¼X0

ð2pÞ3ðp=a1

�p=a1

ðp=a2

�p=a2

ðp=a3

�p=a3

IaðkÞeik�rdk

(9)

and

VaðrÞ ¼ =�1½VaðkÞ�

¼ X0

ð2pÞ3ðp=a1

�p=a1

ðp=a2

�p=a2

ðp=a3

�p=a3

VaðkÞeik�rdk; (10)

where X0 ¼ a1a2a3 is the volume of the unit cell. Thus, wemay rewrite Eqs. (2)–(5) as

LðkÞ

VAðkÞ

VBðkÞ

VCðkÞ

VDðkÞ

26666664

37777775¼ �

IAðkÞ

IBðkÞ

ICðkÞ

IDðkÞ

26666664

37777775; (11)

where LðkÞ is the Fourier transform of the Laplacian matrix,given by the 4� 4 matrix

LðkÞ ¼ 1

R

x� 18 ðc1c2Þ� ðc2c3Þ� ðc1c3Þ�

c1c2 �4 0 0

c2c3 0 �4 0

c1c3 0 0 �4

2666664

3777775; (12)

where x¼2ðcos k �a1þcos k �a2þcos k �a3Þ and cj¼1þeik�aj

(with j¼1;2;3).The lattice Green’s function GðkÞ is a 4� 4 matrix corre-

sponding to the Fourier transform of the Laplacian matrixLðkÞ, defined by GðkÞ ¼ �L�1ðkÞ. The matrix GðkÞ can becalculated to give

GðkÞ ¼ 4

DðkÞ

16 4c�12 4c�23 4c�13

4c12 72� 4x� c13c�13 � c23c�23 c12c�23 c12c�13

4c23 c�12c23 72� 4x� c12c�12 � c13c�13 c23c�13

4c13 c�12c13 c13c�23 72� 4x� c12c�12 � c23c�23

26666664

37777775; (13)

where c12 ¼ c1c2, c23 ¼ c2c3, c13 ¼ c1c3 and DðkÞ ¼ 16ð72� 4x� c12c�12 � c13c�13 � c23c�23Þ=R is the determinant of matrixLðkÞ.

To compute the resistance between the origin f0; ag ¼ f0; 0; 0; ag and the site fl;m; n; bg in an infinite FCC lattice, we usethe general resistance formula derived in Ref. 16 (with d¼ 3), given by

Rabðl;m; nÞ ¼ðp

�p

dh1

2p

ðp

�p

dh2

2p

ðp

�p

dh3

2p

�Gaaðh1; h2; h3Þ þ Gbbðh1; h2; h3Þ�Gabðh1; h2; h3Þe�iðlh1þmh2þnh3Þ

� Gbaðh1; h2; h3Þeiðlh1þmh2þnh3Þg; (14)

where hi ¼ k � ai (with i ¼ 1; 2; 3) and Gabðh1; h2; h3Þ (with a; b ¼ A;B;C;D), as given in Eq. (13).

920 Am. J. Phys., Vol. 81, No. 12, December 2013 M. Q. Owaidat 920

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IV. RESULTS AND DISCUSSION

A. Numerical results

The resistance between any two nodes can be calculated numerically from Eq. (14). As an example, the resistance betweensecond neighbor nodes f0;Ag and f1; 0; 0;Ag that belong to adjacent unit cells—the resistance across the edge of a cube—isgiven by

RAAð1; 0; 0Þ ¼R

ðp

�p

dh1

2p

ðp

�p

dh2

2p

ðp

�p

dh3

2p

� 1� cos h1

9� cos h1 � cos h2 � cos h3 � 2 cos2h1

2cos2 h2

2� 2 cos2 h1

2cos2 h3

2� 2 cos2 h2

2cos2 h3

2

: (15)

This integral can be evaluated to give RAAð1; 0; 0Þ ¼ 0:129686R. It is worth mentioning that this cube-edge resistanceRAAð1; 0; 0Þ is much smaller than that in the simple cubic lattice4,5 (R=3) and also less than that in the body-centered cubic lat-tice16 (0:1481R). This is expected because the current has more branches in the face-centered cubic lattice. As another exam-ple, the resistance between nearest neighbor points f0;Ag and f0;Bg—between lattice points A and B that belong to the sameunit cell—is

RABð0; 0; 0Þ ¼R

ðp

�p

dh1

2p

ðp

�p

dh2

2p

ðp

�p

dh3

2p

�11� cos h1 � cos h2 � cos h3 � 2 cos2 h1

2cos2 h3

2� 2 cos2 h2

2cos2 h3

2� 4 cos

h1 þ h2

2

� �cos

h1

2cos

h2

2

4 9� cos h1 � cos h2 � cos h3 � 2 cos2h1

2cos2 h2

2� 2 cos2 h1

2cos2 h3

2� 2 cos2 h2

2cos2 h3

2

� � ;

(16)

which evaluates to 0:300912R. From the lattice symmetry,we have RABð0; 0; 0Þ ¼ RACð0; 0; 0Þ ¼ RADð0; 0; 0Þ, andRBCð0; 0; 0Þ ¼ RBDð0; 0; 0Þ ¼ RCDð0; 0; 0Þ. Numerical valuesfor some additional effective resistances are listed in Table I.

B. Experimental results

To experimentally study this situation, we constructed twofinite FCC resistor networks of different sizes (5� 5� 5 and7� 7� 7) using 1-kX (65%) resistors. We began by using ahigh-precision multimeter to measure the individual resistan-ces of 2000 resistors for the 5� 5� 5 grid and 4000 resistorsfor the 7� 7� 7 grid, obtaining averages and standarddeviations of 0:975 6 0:007 kX and 0:978 6 0:005 kX,respectively. Resistance measurements are then performedbetween the origin f0; 0; 0; ag and the site fl;m; n; bg in theFCC networks.

Table I. Numerical and experimental values of the resistances Rabðl;m; nÞfor an FCC lattice of resistors of value R (values given in terms of R). The

numerical values are for an infinite lattice, and the experimental values

are for finite 5� 5� 5 and 7� 7� 7 FCC lattices. In the experiment, the

individual resistances are R ¼ 0:975 kX for the 5� 5� 5 FCC lattice and

R ¼ 0:978 kX for the 7� 7� 7 FCC lattice. The values in parentheses are

the deviations from the infinite lattice values.

Rabðl;m; nÞ InfiniteExperimental results

(in units of R) lattice 5� 5� 5 7� 7� 7

RABð0; 0; 0Þ 0.300912 0.309744 (2.94%) 0.301389 (0.16%)

RBCð0; 0; 0Þ 0.521738 0.533333 (2.22%) 0.522068 (0.06%)

RAAð1; 0; 0Þ 0.129686 0.135385 (4.39%) 0.130772 (0.84%)

RBBð1; 0; 0Þ 0.523657 0.541539 (3.42%) 0.525133 (0.28%)

RAAð1; 1; 0Þ 0.147921 0.157949 (6.78%) 0.150184 (1.53%)

RAAð1; 1; 1Þ 0.155672 0.176410 (13.32%) 0.159379 (2.38%)

RABð1; 0; 0Þ 0.348226 0.366154 (5.15%) 0.352472 (1.22%)

RABð1; 1; 0Þ 0.358347 0.390769 (9.05%) 0.364732 (1.78%)

RAAð2; 0; 0Þ 0.157623 0.194872 (23.63%) 0.163466 (3.71%)

RAAð2; 1; 0Þ 0.16132 0.205128 (27.16%) 0.179812 (11.46%)

RAAð3; 0; 0Þ 0.167211 — 0.204332 (22.20%)

Fig. 4. A chart showing the data in Table I.

921 Am. J. Phys., Vol. 81, No. 12, December 2013 M. Q. Owaidat 921

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Table I summarizes our experimental measurements andcompares them to the numerical values for the infinitelattice; these data are also presented graphically in Fig. 4.In both the table and the figure, these data have beennormalized by the individual resistance values of R ¼0:975 kX for the 5� 5� 5 grid and R ¼ 0:978 kX for the7� 7� 7 grid. It can be seen in the table that the calcu-lated and measured resistances are in reasonable agreementnear the lattice origin, but become worse as one of the sitesgets closer to the boundaries of the finite lattice. Thisdiscrepancy is due to the finite size of experimental lattice,which causes the effective resistances to be larger than thevalues for an infinite network. One can also see that as thesize of the lattice increases, the agreement with the infinitelattice improves.

V. CONCLUSION

Using the general Green’s function approach,16 we calcu-lated the effective resistance between any two lattice sites inan infinite face-centered cubic network lattice. We comparednumerical results with the measured resistances of an actualfinite FCC network of resistors. We found the theoretical andexperimental results to be consistent within the estimatederror bounds.

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