Analytical Treatment of the Isotropic and Tetragonal Lattice Green Functions for the Face-centered Cubic,Body-centered Cubic and Simple Cubic Lattices

B.A.Mamedov

1　Introduction

The lattice Green functions plays a decisive role in the theory of solid state physics[Economou(1983);Morita and Horiguchi(1972)].These functions arise not only in their own right,but are also central to the calculation of the lattice statistical problems[Berlin and Kac(1952);Economou(1983);Kobelev,Kolomeisky and Fisher(2002);Montroll and Weiss(1965);Tewary and Read(2004);Tewary and Vaudin(2011);Yakhno and Ozdek(2012)].In the literature,various efficient methods have been proposed for improving the evaluation of the lattice Green functions[Economou(1983);Berlin and Kac(1952);Kobelev,Kolomeisky and Fisher(2002);Montroll and Weiss(1965);Tewary and Read(2004);Tewary and Vaudin(2011);Yakhno and Ozdek(2012)].In literature,most of the studies on lattice functions are based on elliptic integral and recurrence relations[Borwein,Glasser,McPhedran,Wan and Zucker(2013);Inoue(1974);Iwata(1969);Morita and Horiguchi(1971);Morita(1975)].Unfortunately,for most of these purely elliptic integrals and recurrence relations,there are some limitations in their applicability despite the huge development in the computational methods.The reproduce properties of the recurrence relation schemes can lead to a decrease in the accuracy of calculation results.Therefore,it is desirable to use the binomial expansion theorems from which the problems of evaluation of lattice Green functions do not arise.Simple yet accurate analytical formulae have proposed to compute anisotropic lattice Green functions for FCC,BCC and SC lattices[Guseinov and Mamedov(2007);Mamedov and Askerov(2008)].Notice that,the obtained simple analytical formulas for the lattice Green functions are completely general fort≥3.In the present article we propose the series expression formulas occur as one in finite sum and in terms ofInbasic integral,which make possible the fast and accurate evaluation of the isotropic and tetragonal lattice Green functions.This simplification and the use of the computer memory for calculation of binomial coefficients may extend the limits of large arguments to the calculators and result in speedier calculation,should such limits be reached in practice.The new analytical approach for evaluating the isotropic and tetragonal lattice Green functions for FCC,BCC and SC lattices is conceptually simpler than existing methods in the literature.

2　Definition and basic formulas

The isotropic and tetragonal lattice Green functions are defined as

wheretis a complex number,which is described in terms of energy in solid state physics,and(l,m,n)is a set of integers such that the suml+m+nis an even number[Morita(1975)].γ is the parameter which is unity for the isotropic lattice.If γ 6=1 lattice may be called tetragonal lattice[Morita(1975)].The parameters ω(x,y,z)are de fined as follows:

for FCC lattice

for BCC lattice

for SC lattice

In order to establish expressions for the lattice Green functions we shall first consider well known binomial expansion theorems for an arbitrary real or complexnand|x|>|y|[Gradshteyn and Ryzhik(1980)],

HereNis the upper limit of summations andFm(n)are binomial coefficients defined by

We notice that form<0 the binomial coefficientFm(n)in Eq.(7)is zero and the positive integernterms with negative factorials do not contribute to the summation.Taking into account Eq.(6)we obtain for the function(t−ω)−1occurring in Eq.(1)the following series expansion relations:

Thus,substituting Eq.(8)into Eq.(1),we obtain the series expansion formulas for the isotropic and tetragonal lattice Green functions in terms of binomial coefficients and basic integrals,respectively

for FCC lattice

for BCC lattice

for SC lattice

By using the proposed method,we can obtain alternative series formulas for lattice Green functions,respectively:

for FCC lattice

for SC lattice

The quantitiesJn(k)occurring in Eqs.(9)-(15)are determined by the relation

The basic integralsLn(k)andInoccurring in Eq.(16)are determined from the following relations,respectively

and

In Eqs.(9)-(15)the indexesN,N0,Mand M0are the upper limits of summations.In the present work,we propose an alternative accurate method for the analytical evaluation of the lattice Green functions for FCC,BCC and SC lattices.The obtained formulas are practically simple and they offer some advantages over currently available methods.

3　Numerical results and discussion

We have presented a new approach to the calculations of isotropic and tetragonal lattice Green functions using binomial expansion theorems.The analytical results are validated by the numerical calculations for each lattice Green function.The numerical computation of the lattice Green functions has been performed by using the scientific software Mathematica 7.0.Comparisons of the numerical and analytical results are presented in Tables 1,2 and 3.It is clear from these tables that the results from the literature and Mathematica numerical integration and the analytical method proposed in this article are satisfactory for all sets of the parameters.The computer time required for the calculation of lattice Green functions is not given in tables due to the fact that the comparison cannot be made as computer times are different because different computers have been used in various studies reported in literature.It is seen from the algorithm presented for lattice Green functions that our CPU times are satisfactory.For instance,for lattice Green functions with setst=2.7;l=4;m=2;n=0;γ=1;N0=80 the CPU times taken are about 0.015 s and 0.038 s by using formulas Eq.(10)and Eq.(3.2)in[Morita(1975)],respectively.The calculations have been made on a Pentium 4 PC at 800MHz with 128

In Eq.(17)the indexE[k/2]is the upper limit of summation de fined byMB of RAM.The results show that the three methods almost have the satisfactory precision,but the CPU time of the presented method is less than those of the other methods.In the Tables 4 and 5 list partial summations,corresponding to progressively increasing upper summations limits of equations(9)and(10).Using the new decomposition,the obtained results are presented in Tables 4 and 5 to demonstrate the improvements in convergence rates.The reason for empty columns in Tables 1 and 2 is that the indicated equations(Eqs.(9)and(10))are not valid for the value of the lattice Green functions parameters.We expect that our new formulae for the FCC,BCC and SC lattice Green functions will be useful,in particular,in the calculations of various lattice structures of solids.

Table 1:The comparative values of FCC isotropic lattice Green function for N=120.

Table 2:The comparative values of BCC isotropic lattice Green function for N=80.

Table 3:The comparative values of SC isotropic lattice Green function for N=80.

Table 4:Convergence of derived expression for FCC lattice(Eq.(9))as a function of summation limit N for l=4;m=5;n=5;γ=1.

Table 5:Convergence of derived expression for BCC lattice(Eq.(10))as a function of summation limit N for l=6;m=6;n=6.

Berlin,T.H.;Kac,M.(1952):The Spherical Model of a Ferromagnet.Phys Rev,vol.86,pp.821-834.

Borwein,J.M.;Glasser,M.L.;McPhedran,R.C.;Wan,J.G.;Zucker,I.J.(2013):Lattice sums then and now.Cambridge University Press,London,pp.100-145.

Economou,E.N.(1983):Green Function in Quantum Physics.Springer,Berlin,pp.45-130.

Gradshteyn,I.S.;Ryzhik,I.M.(1980)Tables of Integrals,Sums,Series and Products,4thed.,Academic Press,New York,pp.485-490.

Guseinov,I.I.;Mamedov,B.A.(2007):A unified treatment of the lattice Green function,generalized Watson integral and associated logarithmic integral for thed-dimensional hypercubic lattice.Philos Mag,vol.87,pp.1107-1112.

Inoue,M.(1974):Lattice Green’s function for the face centered cubic lattice.J Math Phys,vol.15,pp.704-707.

Iwata,G.(1969):Evaluation of the Watson integral of a face-centered lattice.Nat Sci Rep Ochanomizu Univ,vol.20,pp.13-18.

Morita,T.;Horiguchi,T.(1972):Analytic properties of the lattice Green function.J.Phys A:Gen Phys,Vol.5,pp.67-77.

Kobelev,V.;Kolomeisky,A.B.;Fisher,M.E.(2002):Lattice models of ionic systems.J Chem Phys,vol.116,pp.7589-7598.

Montroll,E.W.;Weiss,G.H.(1965):Random Walks on Lattices.II.J Math Phys.Vol.6,pp.167-181.

Morita,T.;Horiguchi,T.(1971):Calculation of the Lattice Green’s Function for the bcc,fcc,and Rectangular Lattices.J Math Phys,vol.12,pp.986-991.

Morita,T.(1975):Use of a recurrence formula in computing the lattice Green function.J Phys A:Math Gen,vol.8,pp.478-789.

Mamedov,B.A.;Askerov,I.M.(2008):Accurate Evaluation of the Cubic Lattice Green Functions Using Binomial Expansion Theorems.Inter J Theor Phys,vol.47,pp.2945-2951.

Tewary,V.K.;Read,D.T.(2004):Integrated Green’s Function Molecular Dynamics Method for Multiscale Modeling of Nanostructures:Application to Au Nanoisland in Cu.CMES:Computer Modeling in Engineering&Sciences,vol.6,pp.359-372.

Tewary,V.K.;Vaudin,M.D.(2011):A Semicontinuum Model for SixGe1−xAlloys:Calculation of Their Elastic Characteristics and the Strain Field at the Free Surface of a Semi-In finite Alloy.CMC:Computers,Materials&Continua,vol.25,pp.265-290.

Yakhno,V.G.;Ozdek,D.(2012):Computation of the Time-Dependent Green’s Function for the Longitudinal Vibration of Multi-Step Rod,CMES:Computer Modeling in Engineering&Sciences,vol.85,pp.157-176.