Volume 53A. number 2 PHYSICS LLTTERS 2 June1975

PHONON DISPERSION IN SCANDIUM AND YTTRIUM

J.SINGH and S. PRAKASHPhysicsDepartment,Panjab University,Chandigarh-160014,India

Received3 March 1975

Phonon dispersionrelations for Sc and Y arecalculatedalong [00011 and [01101 symmetry directionsusing Ani-

malu transition metal model potential.

RecentlyAniinalu [1] hasput forwarda transition to theelementsof the dynamicalmatrix arising frommetal model potential(TMMP) which yields reasona- (i) ion-ion Coulombpart, (ii) ion-ion exchangerepul-ble results for phonondispersionrelationsin fcc and sion and(iii) the ion-electron-ioninteraction.The firstbce tiansition metalsbut this hasnot yet beentested is evaluatedby Ewald [2] methodand thesecondistoi hcp metals.In this lettei we reportthe calculation neglectedsincethe size of thecore is small as pointedof phonon frequenciesof hcp metalsSc andY, which out by Animalu [1]. The third contributionis eva!-heloiig~to 3d and4d seriesof transitionmetals, re- uatedby usingAnimalu TMMP in conjuctionwith thespcctively. freeelectrondielectricfunction wheretheexchange

In theself-consistentfield methodof calculating and correlationcorrectionsareincorporateddue tothe phononfrequencies,thereare threecontributions Singwi et al. [3]. The phononfrequenciesare calcu-

oa[otTo] [0001]

00.6zUi

0Ui

S S• TOl-

SU

(I) . -—.—

£ ...—-~

ILo ~ ____--___~Q ATOll ~ ~ LO I ~ - • • . - 0 -C’,

z ~- .~&t ~ ~o

‘ I30.2 ~T~~A~LA LA7

ioj_ a LA0 TAi. ~ TA• ~~ TAll

I I I~~•• I

A TOIl~ ________________________________

M 0.8 0.6 0.4 0.2 0.2 0.4 0.6 0.8 A

maxI. Phonondispersioncurvesol Sc. Presentparameters, Animalu parameters, • S ~ A experimental

points.

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Volume53A, number2 PHYSICSLEUERS 2 June1975

I I I I

E0li0~ _-—-—

0.7 ~.

[0001]—-

101

o --~ ....ONz —-

w ~ S0.6 ~ —-——:~ • __.1r_.1P...._1II___R~_~_~lNi:i..~.1L>NIL.,~ 0.5 U~••~~-~j•Ui

t~.-S

2 0.4 ~ “..In

~-Ja.

~ 0.3 A1~ \T0ll /U)I.-i

0.2 ~ A\

o.I~N\\/0~z ULO •L0.• 101 LA•T0A TA II

________________________ I I OTA£10 ItM 0.8 0.6 0.4 0.2 r 0.2 0.4 0.6 0.8 A

Fig. 2. Phonondispersioncurvesof Y. Thedescriptionis the sameas thatof fig. 1.

latedalong [0001] and [0110] directionsin a manner by solid linesin figs. 1 and 2 andthe renormalizedpa-describedby PrakashandJoshi [4]. Theseresultsare rametersaregiven in table 1. The agreementin theshown by dashdot linesin figs. 1 and 2 for Sc andY, longitudinalbranchesof [0001] direction is goodrespectively,wheretheexperimentalresultsdue to with presentparametersbut transversebrancheshaveSinhaet al. [5] arealso presented.All the phonon still lower values.In the [0110] directionthe longitu-frequenciesarefoundto be real. The calculatedpho- dinal modesandthe transversemodesI to thebasalnon frequenciesare lowerby about 10%alonglongi- planeshow reasonablygoodagreementbut transversetudinalbranchesand35%alongtransversebranches modes Ito the basalplanehavelower values.Compar-thantheexperimentalvaluesin Sc in boththesym- ing the two calculationsit is evidentthat the renor-metry directions.In Y, the phononfrequenciesare malizedparametersyield betteragreementwith thehigherthan theexperimentalvaluesapproximately experimentalvalues.Thereforewe concludethat theby 17%for longitudinalbranchesalongboththe sym- renormalizedAnimalu TMMP canbe usedto repro-metry directionsandtransversemodesI to thebasal ducethe phonondispersionrelationsof otherhcpplanein [0110] directionsandareloweralong the transitionmetalstoo. However, to obtaina morerestof thebranchesby approximatelythesame closeagreementbetweenthe calculatedandexperi-amount. mentalvaluesalongall thebranches,theanisotropy

To obtaina betteragreementwith theexperimen- of Sc andY andthe role of d electrons,whicharetal values,we renormalizedtheparametersA0 and partially localizedin transitionmetals,in the dielec-Rm of Animalu TMMP by fitting the phononfrequen- tric screening[6] shouldbe included.ciesat theZone boundaryin the [0001] directionalongthelongitudinal branch.Theseresultsareshown Theauthorsarethankfulto CSIR,New Delhi

for financial assistance.

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Volume53A, number2 PHYSICSLETTERS 2 June1975

Table 1The renormalizedAnimalu TMMP parameters.All thequantitiesarein atomic units except IE~Iwhich is in rydberg.

~4o /l~ A~ Rm tt(~ Z m* R~ °eff IE~

Sc 1.1556 1.65 1.40 L6 168.7 3 1.0 1.531 0.045 0.090Y 0.85 1.30 1.10 1.75 223.1 3 1.0 1.739 0.049 0.087

References 151 S.K. Sinha,T.O. llrun, L.D. Muhiesteinand J. Sakurai,Phys.Rev. Bl (1970) 2430 (for Y);N. Wakabayashi,S.K. SinhaandI .H. Spedding,Phys.

Ill A.O.F. Animalu, Phys.Rev. B8 (1973)3542-62. Rev. B4 (1971) 2398 (for Sc).12] PP.Lwald,Ann. Phys.64 (1921) 253. [6J N. SinghandS. Prakash,Phys.Rev. B8 (1973)5532;131 K.S. Singwi, A. Sjölander,M.P. Tosi and R.H. Land, W.R. Hanke,Phys.Rev. B8 (1973)4583-96;

Phys.Rev. BI U970) 1044.N. Singh,J. SinghandS. Prakash,Phys.Rev. Bi, to be141 5. PrakashandS.K. Joshi,Phys.Rev. B! (1970) 1468.published.

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