Khanna, K N

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J. r»Kv«. 54As, 502,-5^ 0Pseudopotential calculations for simple metalsK. N. K i i a n n aDe/parlmml oj Phyniv-s, V D College, Kanpur-2i)^{yd2 {UecGwad H  Novemhar 1977)T}\() nuniol p^uud(>x)ottmiial i>ioj)ufiod l)y tho autl'or ih I'.sud to cal- ;;ulalo tlio ])iiiding oiicji'gy, cfjmprosaibility and monot^acatity resistivity (>r simple metals The screened form factors of the potential are presented for Na, K, Rb. Ch and aio coJiLi^ arefl with, those obtained by Sliyu H al1. I ntbodttotion \Pseudopoteiitial theory has been remarkably Sitceisssfnl in the calculation \of atojiiic propei’tics of simple metals Tlie general procedure (Harrison 190 )^ of determining thoHe propiu ties is to start with th(^  total energy wliicli is the snV of tluee contributions, viz , tl','i modifuvd friio election energy, th,e (dectrostatic Hiieigy and tluv band structuie energy Ti\e band stnicturo energy represents tile deviation of tlie energy of the eondui tion elections fiom that of a free electron gas It is calculated from the riecond-ordor jicitiirbation eneigy of a conduction electron in an expansion ol the ps(tndopotential and can be wiitteii as/7b. -- (I)where F{(jl) is called the energy wave mmilKU' characteristic, G  a reciiiioeal lattice Vector and the prime on tJie sum mdieafycs that the term with G =  0 is excludedDuring the last low years, considerable information has lieeii obtained for the electron ion potential fiy emiiirical fitting of the Fermi surface data and many model pseudox»oteiitials have been suggested by various workers How­ever, it IS well known that one can always find a suitable model potential to interpret specific properties because each property is related to the Fourier transform of the potential in a certain range of momentum transfer2 Th e  Model PsbddopotentialThe ijoteiitial Ji.a,s generally been expressed in the model form containing few adjustable parameters. In an eai’lior papeir Klranna (1977) a smiple expression552for baro-io„ paoudopotontial p,op„,sod ,o dcsenl-o 11,o «lorln,n-.,„ int,«-a„t,o„ ill aimplb motttiH of tlu> lori)iPseudopotential calculations fo r  sim ple metals 553i (2) - MU(fP f) \ -.020 (/-ia , /  \ ' • •  (2)wUoto Z, ft, rc JiT'o valoiico, jitoitiu; voluim* and. tln^  radius <'i loii (“ovo Tliu fiisi tonn arisos from tlw louj) (-o iloiuli liidd of ti\o iu’>i mu iliar i^o /o  ThoSuiioiid tonn roprosouis ilm oontnliuli )u fjoio ilv* mu <;o»i' loj i^oii Tl\o iumtin^ valno of Wb(2 ) a,s q gOfvs to zoio i,^  — iTrZt-fLhf vliirli satihlios tho corrout limit roquin^i by diargu nenUality Tl'.o paivuu^ do.r ba,i boon ditrimiiuwl by fitting V{q) —  W b {q )j E {q )  to tho oxpoi miontally (lotoriiiukd point at q oipial to tho (1 1 0 ) and (1 1 1 ) reupioral latlioo roctois loi b o o  and (o o  motals rosjionlivojy (Soitz (it al 1970) TUo loim O'Xji (— 0 2 (i(/^ ) Ua,s boon mohulod iioio to imiirovo tho oonvorgoiioo ol tbo potonlial for hi},]utr v.iluo.s of q Tlio oooiruiont in tho oxponont was arbitrarily olioso’’ so , Uat 1 1 >,o ibnii lartor dooi’isisos rapidly aftor K f To taki^  into ajooant tb.o offoot ol sorooning, tbi^  Hartii'O diolootru: innotion has boon u,st>d Tlio ovohango and oorrolation olloots aro tiOiiiSidorod on i ] w  linos of Hhmi (1905) l'\ii A?(i7 ), tlv. lullowing for inula was ustnlJ^ {9) -- 1 ! [1 1- 1Lu \ f r I ‘f I 2/\j.’+7 I 1~^ KfQ I 2/1/,’- 7  J(3)whom E f  and A> fho muwgy and wav(^  nunduM ol an (dodioii at tho I'Viin suifaLX> ro,spootivoly, K s  is tl».o Tluanas-Fornn ,soio(ming vector3 Binding Entiroy ATfn CompiuoshiriutyOon,siduring tJio frw^  olootron approxmiation and tlio jioidurliation thooiy, tho bnding onorgy and oompmsHibility vvort^  oahvdatod using tint oxinossion givon by Saxona at a l  (1909) Tlv^  ovialap ropnlsion Ootnoon ion ooros wornrogloolod0-737 2  /0-91(id I 792Z^ "^^'/'5“ 31h -\ . - l ''\ 1 \ rs ^■ 105 i-0 031 In r s3 J  _ ^]07r“'(127r3)i^ 3^  'r/?'Tn(7)P ,,, 11 r n: 1 -- T I(4)^^ 0 _  M  0-089rfiH2(0-9HW-l 792^‘^ ^^ )r,s*-4-42 'K 22 1 L9 1327r^ (127r2)>/-‘ r,s’“ ,}^ i)whom, T  q l ^ h p2 1W M I U _  i„ i . ] ± l  i\] . . .  (5)’l_7u  I' ! 1 -T llJ4. Resistivity of Vacancies of Simple MetalsIt Ls sinipJo to calculate) tho inoiiovacancy rosistivity within the frame work of iiyeuflopoteiitial formaliHrn as has boon attempted by various workers Shyu et al (1971), Behaii (1973). The expreSiSion is554 K. N. Kbannawhere,1 2Kfwhere, Vp is tho Fermi velocity and V{g) is the psciidojiotential matrix element .screoued by dielecdnc constant5 Results and Discussions \[Jsmg this model potential, the form factors of elfoctivo ele(*tron-ion potential are determined for alkali inetaLs and shown ui Fig. 1. This form of pseudopotentialLs superior to various existing psendopotentialb in tho sense that the oscillations in form factors decreases eonsidorably for large momentum transfer The form factors of Na and K of Shyn ei (1971) are sliown in Fig 2 Tt can be seen that in tho wave vector region between q — 0 and 2Kf their plots arc m broad agroe^  merit with those of the autlior The contiibittion from this region plays a tlomi- nant part in the calculation of most of tho dynamical properties.P seu do]^oien tial calculations f o r  s im p le  m etals 5&0F ir 2 Cc-mptu’iHoti of oiii'form’ factors Shyu et i Klianiia^fnll linos, Shvii et al broken linos,Table 2tuie 1 V'Oie- 1 .wiius in a uMotalCoro ifulius VcISIa 2'lfiK 2-G3Kb 271Cs 2 00Al 1-50Pb 1-30aiul (‘xpciPimental values o f bmdinjcompusKsibilityMeta'RindiuR energy in liy/electron { — Kb)Coinjji oBsibiliiy K /K ,ThoorcticalvaluesExporimontal vain osThooroticalvaluesExjjor'inentulvaiuosNtt ■ 474 460 1-32 1-60K -396 390 0-92 1-00Kb •370 ■ 306 0 84 0-82Ck ■ 340 345 0 76 0-78Tho binding energy and (!ompn>;s,sibility were (!aJn[lat(Hl from otjiS (4) and (5) roispeciively and are listed in tabl«» 2 along with tJn» experimental v^ aliies The calculated values are ni good agreement with tlio exporimoiital values556 K . N. KJiannaMoii()va,t;aiiuy rosistivity of sim])lc metals are preMented in table 3. No iviasonablc oomparjhun betwecji theory aiKl experiment is possible here due to the liiL-lv ol (iX])i,i imental results For rornparLson, thooretioal losults obtained by Shyn d al (1071) using Hartieo sereemng are also bstiid, Hamson (1966) has also (jaleulated pv in A] to -8 pQ. cm/at% using Ha*rtree soroonmg Some diserepamuos may anse here from the neglect of lattice distfirtion and assumes the vacano}  ^ ooruientration to be small so that one lias essentially a collection of simply isolatts:! va( aiiiuesTable 3 Resistivity of vacancies in simple metals in pLlM etal ProHOnt calculation Results of Shyu ct al (‘alculationNa 0*67 0*77K 0-98 1*00Kb 1-06Ch 1 21Al 0*44 0 52I’b 0-86 i 64®, 0 56® Sec* B(4ian (1973).6 C o n c l u s io n sft may be coiicJudiKl tliat this model iiotontial is able to interpret tJie dyna­mical jiropuities in a satisfactory manner By iiitrodiicing a dampmg factor we have been able to bring its form factors more m aei oidame with the physically acoojjtablo situation.A c k n o w l e d g m e n t sTJie author IS thanliful to Dr P K Sharnia. Allahabad Ciiiveisity for valuable suggestionsR e f e r e n c e sBehan J 1973 J. q/ Physica F3, 1553Hamsoii W A 1906 Pseudopotm hal m  theory of rtiPialu (New Yorlt : Benjaniin) Harrison W A 1903 Phj/6 Pav. 129, 2503 2512.Khanna K  N. 1977 Ind ia n  J .  Pure A pp l. Physics 15, 733.Saxona V. K  , Kapoor Q S. <fc Bhattacharya D L. 1969 P hys Sta tus Soluh  34, 145 Roiiz F & Tiimbull D 1970 Solid State Physics 24, 183 (Acatleinu* Pioss) iSham L J 1965 Proc. Ji Soc A383, 33,Phyii W M . Gaspan G D , Wohhnp J. H & Cords M. 1971 P hys  i?r« D4. 1802

Pseudopotential Calculations for Simple Metals

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Pseudopotential Calculations for Simple Metals

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