Copyright 2014 Tech Science Press CMC, vol.43, no.2, pp.87-95, 2014

Transcription

1 Copyrght 2014 Tech Scence Press CMC, vol.43, no.2, pp.87-95, 2014 Analytcal Treatment of the Isotropc and Tetragonal Lattce Green Functons for the Face-centered Cubc, Body-centered Cubc and Smple Cubc Lattces B.A. Mamedov 1 Abstract: In ths paper, we propose an effcent method to calculate the sotropc and tetragonal lattce Green functons for the face-centered cubc (FCC), bodycentered cubc (BCC) and smple cubc (SC) lattces. The method s based on bnomal expanson theorems, whch provde us wth analytcal formulae through basc ntegrals. The resultng seres present better convergence rates. Several acceleraton technques are combned to further mprove the effcency of the establshed formulas. The obtaned results for the lattce Green functons are n good agreement wth the known numercal calculaton results. Keywords: Isotropc lattce Green functons, tetragonal lattce Green functons, face-centered cubc lattce, body-centered cubc lattce, smple cubc lattce. 1 Introducton The lattce Green functons plays a decsve role n the theory of sold state physcs [Economou (1983); Morta and Horguch (1972)]. These functons arse not only n ther own rght, but are also central to the calculaton of the lattce statstcal problems [Berln and Kac (1952); Economou (1983); Kobelev, Kolomesky and Fsher (2002); Montroll and Wess (1965); Tewary and Read (2004); Tewary and Vaudn (2011); Yakhno and Ozdek (2012)]. In the lterature, varous effcent methods have been proposed for mprovng the evaluaton of the lattce Green functons [Economou (1983); Berln and Kac (1952); Kobelev, Kolomesky and Fsher (2002); Montroll and Wess (1965); Tewary and Read (2004); Tewary and Vaudn (2011); Yakhno and Ozdek (2012)]. In lterature, most of the studes on lattce functons are based on ellptc ntegral and recurrence relatons [Borwen, Glasser, McPhedran, Wan and Zucker (2013); Inoue (1974); Iwata (1969); Morta and Horguch (1971); Morta (1975)]. Unfortunately, for most of these purely 1 Department of Physcs, Faculty of Arts and Scences, Gazosmanpaşa Unversty, Tokat, Turkey (Turkye)

2 88 Copyrght 2014 Tech Scence Press CMC, vol.43, no.2, pp.87-95, 2014 ellptc ntegrals and recurrence relatons, there are some lmtatons n ther applcablty despte the huge development n the computatonal methods. The reproduce propertes of the recurrence relaton schemes can lead to a decrease n the accuracy of calculaton results. Therefore, t s desrable to use the bnomal expanson theorems from whch the problems of evaluaton of lattce Green functons do not arse. Smple yet accurate analytcal formulae have proposed to compute ansotropc lattce Green functons for FCC, BCC and SC lattces [Gusenov and Mamedov (2007); Mamedov and Askerov (2008)]. Notce that, the obtaned smple analytcal formulas for the lattce Green functons are completely general for t 3. In the present artcle we propose the seres expresson formulas occur as one nfnte sum and n terms of I n basc ntegral, whch make possble the fast and accurate evaluaton of the sotropc and tetragonal lattce Green functons. Ths smplfcaton and the use of the computer memory for calculaton of bnomal coeffcents may extend the lmts of large arguments to the calculators and result n speeder calculaton, should such lmts be reached n practce. The new analytcal approach for evaluatng the sotropc and tetragonal lattce Green functons for FCC, BCC and SC lattces s conceptually smpler than exstng methods n the lterature. 2 Defnton and basc formulas The sotropc and tetragonal lattce Green functons are defned as G(t,l,m,n) = 1 π 3 π 0 π π 0 0 coslxcosmycosnz t ω(x,y,z) dxdydz (1) where t s a complex number, whch s descrbed n terms of energy n sold state physcs, and (l,m,n) s a set of ntegers such that the sum l + m + n s an even number [Morta (1975)]. γ s the parameter whch s unty for the sotropc lattce. If γ 1 lattce may be called tetragonal lattce [Morta (1975)]. The parameters ω(x,y,z) are defned as follows: for FCC lattce ω(x,y,z) = γ cosxcosy + cosycosz + coszcosx (2) for BCC lattce ω(x,y,z) = cosxcosycosz (3) for SC lattce ω(x,y,z) = cosx + cosy + γ cosz (4)

3 Analytcal Treatment of the Isotropc and Tetragonal Lattce Green Functons 89 ω(x,y,z) = 2cosxcosy + γ cosz (5) In order to establsh expressons for the lattce Green functons we shall frst consder well known bnomal expanson theorems for an arbtrary real or complex n and x > y [Gradshteyn and Ryzhk (1980)], (x ± y) n = lm N N m=0 (±1) m F m (n)x n m y m. (6) Here N s the upper lmt of summatons andf m (n) are bnomal coeffcents defned by F m (n) = { n(n 1)...(n m+1) ( 1) m Γ(m n) m!γ( n) m! for nteger n for nonnteger n (7) We notce that for m < 0 the bnomal coeffcent F m (n) n Eq. (7) s zero and the postve nteger n terms wth negatve factorals do not contrbute to the summaton. Takng nto account Eq. (6) we obtan for the functon (t ω) 1 occurrng n Eq. (1) the followng seres expanson relatons: (t ω) 1 { ( 1) = F ( 1) t 1 ω for ω t ( 1) +1 ω 1 t (8) for 0 t ω Thus, substtutng Eq. (8) nto Eq. (1), we obtan the seres expanson formulas for the sotropc and tetragonal lattce Green functons n terms of bnomal coeffcents and basc ntegrals, respectvely for FCC lattce G(t,l,m,n) = 1 π 3 lm N N ( 1) F ( 1)t 1 J j+k (l)j k (m)j j (n) for t 3, F j ()γ j j k=0 F k ( j) (9) for BCC lattce G(t,l,m,n) = 1 π 3 lm for SC lattce N N ( 1) F ( 1)t 1 J (l)j (m)j (n) for t 1, (10) G(t,l,m,n) = 1 π 3 lm M M ( 1) F ( 1)t 1 J j (m)j j (n) f or t 3, F j ()2 j γ j J j (l) (11)

4 90 Copyrght 2014 Tech Scence Press CMC, vol.43, no.2, pp.87-95, 2014 G(t,l,m,n) = 1 π 3 lm M M ( 1) F ( 1)t 1 J j k (m)j k (n) f or t 3 F j () j k=0 F k ( j)γ k J j (l) (12) By usng the proposed method, we can obtan alternatve seres formulas for lattce Green functons, respectvely: for FCC lattce G(t,l,m,n) = 1 π 3 lm N L N ( 1) F ( 1)t 1 J j+k (l)j j+ k (m)j (n)for t 3 L ( 1) j F j ( 1 )γ j t j k=0 F k () (13) for SC lattce G(t,l,m,n) = 1 π 3 lm N L N ( 1) F ( 1)t 1 J j (l)j k (m)j k (n) f or t 3 L ( 1) j F j ( 1 )t j F k () k=0 (14) G(t,l,m,n) = 1 π 3 lm N L N J j (m)j (n) f or t 3 ( 1) F ( 1)γ t 1 L ( 1) j F j ( 1 )2 j t j J j (l) The quanttes J n (k) occurrng n Eqs. (9)-(15) are determned by the relaton (15) I n for k = 0 L J n (k) = n (k) for k > 1 I n+1 for k = 1 0 for k > n or k + n odd (16)

5 Analytcal Treatment of the Isotropc and Tetragonal Lattce Green Functons 91 The basc ntegrals L n (k) and I n occurrng n Eq. (16) are determned from the followng relatons, respectvely π L n (k) = 0 coskxcos n xdx =2 k 1 I k+n + k E[k/2] ( 1) 2 k 2 1 F 1 (k 1)I k+n 2 =1 (17) and π I n = 0 cos n ϕ dϕ = 0, f n odd Γ( π n+1 2 ) Γ( n +1), f n even (18) 2 In Eq. (17) the ndex E[k/2] s the upper lmt of summaton defned by E(n/2) = n [1 ( 1)n ]. (19) In Eqs. (9)-(15) the ndexes N, N, M and M are the upper lmts of summatons. In the present work, we propose an alternatve accurate method for the analytcal evaluaton of the lattce Green functons for FCC, BCC and SC lattces. The obtaned formulas are practcally smple and they offer some advantages over currently avalable methods. 3 Numercal results and dscusson We have presented a new approach to the calculatons of sotropc and tetragonal lattce Green functons usng bnomal expanson theorems. The analytcal results are valdated by the numercal calculatons for each lattce Green functon. The numercal computaton of the lattce Green functons has been performed by usng the scentfc software Mathematca 7.0. Comparsons of the numercal and analytcal results are presented n Tables 1, 2 and 3. It s clear from these tables that the results from the lterature and Mathematca numercal ntegraton and the analytcal method proposed n ths artcle are satsfactory for all sets of the parameters. The computer tme requred for the calculaton of lattce Green functons s not gven n tables due to the fact that the comparson cannot be made as computer tmes are dfferent because dfferent computers have been used n varous studes reported n lterature. It s seen from the algorthm presented for lattce Green functons that our CPU tmes are satsfactory. For nstance, for lattce Green functons wth sets t = 2.7; l = 4; m = 2; n = 0; γ = 1; N = 80 the CPU tmes taken are about s and s by usng formulas Eq. (10) and Eq. (3.2) n [Morta (1975)], respectvely. The calculatons have been made on a Pentum 4 PC at 800MHz wth 128

6 92 Copyrght 2014 Tech Scence Press CMC, vol.43, no.2, pp.87-95, 2014 Table 1: The comparatve values of FCC sotropc lattce Green functon for N = 120. t l m n Eq. (9) Mathematca numercal ntegraton results E E E E E E E E E E E E E E E E E E E-13 Table 2: The comparatve values of BCC sotropc lattce Green functon for N = 80. t l m n Eq.(10) Mathematca numercal ntegraton results E E E E E E E E E-06 Table 3: The comparatve values of SC sotropc lattce Green functon for N = 80. t l m n Eq. (11) Borwen, Glasser, McPhedran, Wan and Zucker (2013) E E E E E E E E E E E E E E-35

7 Analytcal Treatment of the Isotropc and Tetragonal Lattce Green Functons 93 Table 4: Convergence of derved expresson for FCC lattce (Eq.(9)) as a functon of summaton lmt N for l = 4; m = 5; n = 5; γ = 1. N t = 3.6 t = E E E E E E E E E E E E E-05 Table 5: Convergence of derved expresson for BCC lattce (Eq.(10)) as a functon of summaton lmt N for l = 6; m = 6; n = 6. N t = 2.3 t = E E E E E E E E E E E-08 MB of RAM. The results show that the three methods almost have the satsfactory precson, but the CPU tme of the presented method s less than those of the other methods. In the Tables 4 and 5 lst partal summatons, correspondng to progressvely ncreasng upper summatons lmts of equatons (9) and (10). Usng the new decomposton, the obtaned results are presented n Tables 4 and 5 to demonstrate the mprovements n convergence rates. The reason for empty columns n Tables 1 and 2 s that the ndcated equatons (Eqs. (9) and (10)) are not vald for the value of the lattce Green functons parameters. We expect that our new formulae for the FCC, BCC and SC lattce Green functons wll be useful, n partcular, n the calculatons of varous lattce structures of solds. References Berln, T. H.; Kac, M. (1952): The Sphercal Model of a Ferromagnet. Phys Rev, vol. 86, pp

8 94 Copyrght 2014 Tech Scence Press CMC, vol.43, no.2, pp.87-95, 2014 Borwen, J. M.; Glasser, M. L.; McPhedran, R. C.; Wan, J. G.; Zucker, I. J. (2013): Lattce sums then and now. Cambrdge Unversty Press, London, pp Economou, E. N. (1983): Green Functon n Quantum Physcs. Sprnger, Berln, pp Gradshteyn, I. S.; Ryzhk, I. M. (1980) Tables of Integrals, Sums, Seres and Products, 4 th ed., Academc Press, New York, pp Gusenov, I. I.; Mamedov, B. A. (2007): A unfed treatment of the lattce Green functon, generalzed Watson ntegral and assocated logarthmc ntegral for the d-dmensonal hypercubc lattce. Phlos Mag, vol. 87, pp Inoue, M. (1974): Lattce Green s functon for the face centered cubc lattce. J Math Phys, vol. 15, pp Iwata, G. (1969): Evaluaton of the Watson ntegral of a face-centered lattce. Nat Sc Rep Ochanomzu Unv, vol. 20, pp Morta, T.; Horguch, T. (1972): Analytc propertes of the lattce Green functon. J. Phys A: Gen Phys, Vol. 5, pp Kobelev, V.; Kolomesky, A. B.; Fsher, M.E. (2002): Lattce models of onc systems. J Chem Phys, vol. 116, pp Montroll, E. W.; Wess, G. H. (1965): Random Walks on Lattces. II. J Math Phys. Vol. 6, pp Morta, T.; Horguch, T. (1971): Calculaton of the Lattce Green s Functon for the bcc, fcc, and Rectangular Lattces. J Math Phys, vol. 12, pp Morta, T. (1975): Use of a recurrence formula n computng the lattce Green functon. J Phys A: Math Gen, vol. 8, pp Mamedov, B. A.; Askerov, I. M. (2008): Accurate Evaluaton of the Cubc Lattce Green Functons Usng Bnomal Expanson Theorems. Inter J Theor Phys, vol. 47, pp Tewary, V. K.; Read, D. T. (2004): Integrated Green s Functon Molecular Dynamcs Method for Multscale Modelng of Nanostructures: Applcaton to Au Nanosland n Cu. CMES: Computer Modelng n Engneerng & Scences, vol. 6, pp Tewary, V. K.; Vaudn, M. D. (2011): A Semcontnuum Model for S x Ge 1 x Alloys: Calculaton of Ther Elastc Characterstcs and the Stran Feld at the Free Surface of a Sem-Infnte Alloy. CMC: Computers, Materals & Contnua, vol. 25, pp Yakhno, V. G.; Ozdek, D. (2012): Computaton of the Tme-Dependent Green s

9 Analytcal Treatment of the Isotropc and Tetragonal Lattce Green Functons 95 Functon for the Longtudnal Vbraton of Mult-Step Rod, CMES: Computer Modelng n Engneerng & Scences, vol. 85, pp

10