Application of APW Pseudopotential Form Factor in the Calculation of Liquid Metal Resistivities.

Transcription

1 Internatinal Jurnal f Pure and Applied Physics. ISSN Vlume 8, Number (01), pp Research India Publicatins Applicatin f APW Pseudptential Frm Factr in the Calculatin f iuid Metal Resistivities. Amah, A.N., Echi, I.M. and Ahemen, I. Department f Physics, University f Agriculture, Makurdi, Benue State, Nigeria Abstract The APW pseudptential frm factr develped by cmparative analysis with the Slaters ld APW matrix elements and presented in a clsed frm was applied in the cmputatin f liuid metal resistivities fr metals. The cmputed values were in perfect agreement with the experimental result shwing that the develped frm factr is uite gd. Keywrds: APW, Pseudptential, Frm-factr, iuid-metal, Resistivity. Intrductin Slater frmulated the Augmented Plane Wave (APW) methd in 197 fr ratinal explanatin f cndensed matter prperties [1]. Six years later, clse t ne hundred APW calculatins n energy bands and Fermi surfaces had been reprted in literature. Cntrary t the expectatins that this accelerated pace wuld cntinue and the applicatin diversified, there was general shift f interest t ther methds f calculatin. This, the critiues argue was nt due t any defect in the frmulatin, but due t (i) the cmplex nature f the matrix elements (ii) the rigrus and extensive calculatins invlved, cnsidering the memry size f the cmputers at that time. Applicatins were als limited t cndensed matter prperties having direct bearing t the frm f the matrix elements nly. The advantage f APW methd is that it can be applied in transitin, nn-transitin metal, nble and allys []. Cmputatinally, the APW methd is demanding because the basis functins are energy dependent and the eigenvalue prblem nnlinear. In 1974, the cnnectin between the APW methd and the Abarenkv mdel pseudptential methd was explred in cnnectin with the electrnic structure and prperties f transitin metals []. Further develpment f the APW methd led t linearizatin f the APW methd (APW) [4, 5], where the energy dependence is remved by selecting a fixed set f suitable muffin-tin radial functins and their energy derivatives. The Full-Ptential inear Augumented Plane

2 114 Amah, A.N. et al Wave (FAPW) methd [6, 7] cnsidered t be the mst accurate electrnic structure calculatin scheme has its rigin in Slater s earliest wrk. In this apprach, there is n sharp apprximatin fr either the charge density r the ptential, and all electrns are treated in the self cnsistent prcess; and the cre electrns are treated fully relativistically and the valence electrns are treated semi-relativistically [8]. The interest t revisit the Slater s riginal APW is mtivated by the current numerical cmputatins f electrnic prperties f cndensed matter which has taken n an increased mmentum with the new age f advanced cmputer algrithms using the giga bytes f even lap tp cmputers. The idea is t apply the APW pseudptential frm factr develped by Amah et al [9] in 006 t calculate the liuid metal resistivity f metals. Thery The APW matrix elements riginally prpsed by Slater [1] have the familiar frm. M Where k ik. rv ( k Ε) S e G = Ω δ 4π v v (1) v ( i. k Ε) ( k Sv ) k ( l + 1) P ( ki. k ) ( k Sv ) R' ( Sv ) R ( Sv ) l= 0 P = legendre plynmials R(Sv) = radial wave functin at vth sphere surface R'(Sv)/R(Sv) = lgarithmic derivative evaluated at the surface f the sphere using radial Schrdinger euatin. by Frm the wrk f Amah, et al [9], the APW matrix elements develped is given ( k Ε) + < x v x > M = Ω δ () Where x a linear cmbinatin f a set r> = APW functin. Cmparatively, euatin () crrespnds t the APW matrix elements in euatin (1) develped by Slater. The third term f euatin () i.e. <xi v x> = (xvx)dω=<k+ v k> crrespnds t the surface integral f the APW matrix elements and is therefre the APW frm factr. This can be defended because it is implicitly dependent n the muffin-tin ptential thrugh the lgarithmic derivatives f the radial party f the muffin-tin functins, which are the slutins f Schrdinger euatin H r>=e r>. Therefre, the APW frm factr becmes

3 Applicatin f APW Pseudptential Frm Factr 115 ik. rv V () = < k+ v(r) k> = 4π Sv e Gv () v In this clsed frm, calculatin can nw be extended t ther cndensed matter prperties like liuid metal resistivities, ptical prperties etc which depend directly n frm factrs. This liuid metal resistivity is given by the expressin[10] 4π z 1 = < + > ρ S( ) k v b k d (4) e k f k f k f Where S () = structure factr <k+ vb k> = frm factr Vb = bare ptential z = valence Kf = Fermi wave vectr Cmputatin f liuid metal resistivity using the frm factr The first step in this applicatin is t screen the APW pseud-ptential frm factr evaluated using the Thmas Fermi dielectric functin. The charge densities, the muffin-tin ptentials that apprximate the actual crystal ptential and the frm factr fr all metals had been calculated [9]. In rder t evaluate the integral in euatin (4) numerically, the euatin is reduced further if we let k f = x and π ρ = 4 e z k f d k f 1 = dx. Then euatin (4) becmes a( x) V sc ( x) x dx z 1 = 4K a( x) V ( ) sc x x dx (5) k f z 1 ρ = 4K f ( x) dx (6) k f Where x sin½θ; 0 x 1 a(x) = structure factr

4 116 Amah, A.N. et al e = electrnic charge V sc = < k f + v b (x) k f > /ε() = screened frm factr ε() = Thmas Fermi dielectric functin f(x) = a(x) v sc (x) x and π K = = in the unit f ћ = e = 1 e By Simpsn s rule [11] h h f ( = N m m i i (7) m dd m even 1 I = x) dx f + f + 4 f + f ( 4 f + f f ) Where there must be an even number f intervals and dd number f pints. T avid the prblem with ε() at x = 0, we started the integratin at x = apprximately. It is in the frm f euatin (5) and (7) that the liuid metal resitivities were calculated using a cmputer prgramme. Table1 shws the cmputed liuid metal resistivities and their crrespnding experimental values. Results and Discussins Table 1: Cmputed resistivity values and crrespnding experimental values S/N Metal ρ(cal. values) Ω -6 cm ρ(exp. values)ω -6 cm 1 ithium * Ptassium Rubidium..5 4 Silver * Cadmium Caesium Gld * Thallium ead Key: * means abslute values [1] The APW methd was develped in 197. The applicatin was restricted t the cmputatin f energy bands and Fermi surfaces due t the cmplex nature f the

5 Applicatin f APW Pseudptential Frm Factr 117 matrix elements. Hwever, with the develpment f APW pseud-ptential frm factr in a clsed frm ut f the matrix elements, the applicatin has been diversified t ther cndensed matter prperties having direct bearing t frm factr. The applicatin t liuid metal resistivities has prved the frm factr t be elegant (i.e table 1). The calculated values f all metals investigated agreed with their experimental values except thse asterisked which have negative values. Cnclusin The applicatin f APW pseudptential frm factr in the calculatin f liuid metal resistivities fr nine metals has prved t be uite adeuate in predicting the liuid metal resistivities fr all metals. This has extended the applicatin f APW methd frm energy bands and Fermi surfaces t liuid metal resistivities. The frm factr can als be used t predict the ptical prperties f materials. References [1] Slater, J.C. (197): Phys. Rev. 51, 846. [] Terry,. (1967): Augmented Plane Wave Methd. W.A. Benamin, Inc. New Yrk, 1 6. [] Heine, V. and Aberenkve, I.V. (1974): Phys. Rev. Bi, [4] Andersn, O.K. (1975): Phys. Rev. 1, 060. [5] Fuchs, M. (1998): Phys. Rev. B57, [6] Perdew, J.P., Burk, K and Ernzerhf, M. (1996): Phys. Rev. ett. 77, 865. [7] Wimmer, E., Krakauer, H., Winert, M and Freeman, A.J. (1981): The full- Ptential inear Augumented Plane Wave Methd. Phys. Rev. B4, 864. [8] Freeman, A.J. (00): Materials by Design and the exciting Rle f Quantum Cmputatin/Simulatin. Jurnal f cmputatinal and applied mathematics, 149, [9] Amah, A.N., Oli, B.A. and Ekpunbi, A.J. (006): The Develpment f APW Pseudptential Frm factr fr the cmputatin f varius cndensed Matter prperties. Jurnal f infrmatin, cmmunicatin and cmputing technlgies. (JICCOTECH) Vl., N. 1, Pp [10] Oli, B.A. and Onniwu, S.J. (1986): Phys. Stat. Sl. B14, 5. [11] Riley, K. F., Hbsn, M.P., and Bence, S.J. (1997): Mathematical Methds fr Physics and Engineering. Cambridge University press, ndn, 958. [1] Faber, T.E. (1977): Intrductin t the Thery f iuid Metals Cambridge University Press, ndn, 151.

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