lSVESTlGACIÓNREVISTA MEXICANA DE FíSICA "'7 (h) 5::!1--5::!4

Cohesion energy ealculations for ternary ionie novel erystalsG. Vázquez-PoloJ and E. Cabrera

Instituto de Física, U//iI'ersidad Nacio//al AlItOllOI1U1 de MéxicoApartado posta! 20-364, OIO()() México, D.F., M('.rit.o

('-11/(/;1: I \'a¿qll(';@.tis;c.//I/(/II/.lIlx

R.R. MijangosCellfm de 11I1'('srigaciófl ('11flúea, Unil'l'I"sid(/{1 d(' SOl/ora

Apartado pos!aI5.88, 83 IYO HernlOsillo, SOIl., Alexieo

DIClEt\.lBRE 2(X) I

E. V"ldezFS{'1fcJ(f Nacio/lal de ESllldio,\' Pndc'ciOl/(/les Acm/tÍll, Ullil'(,l"sidllll Naciollal AlItOIlO/l/(/ de México

SanUl Cmz Acm/dll, Nllllca/¡1lI1I, A1exic()

c. DllartcDepartamento de Geolog(a, Uni\'ersid{ul de Sonora

83000 !lCnllO.Ül!O, SOIl., Mexico

Recihido el 11 de mayu de lOO1; w:cplaUl) ellO oc agoslo de lOO1

Thc prcscnl work .:akulalcs IllCvalue 01"the link energy 01"a crystallinc ternary \trll.:llIl'l.' Ilc\Vlyformcd hy alkalí halides. The ternary \tructureprcparcd \\"ilh dilfcrenl colll:elltrations 01' KCI~.KBr.tlRhCl: mailltains a very good miscibility and s{ahility. The calculation is ba~ed on the11.~l'nI"" ~l'neralization of the Vegaru's law (which generally is valid for binary compounds) rol' ca1culaling the vallles 01'the lauice constanl:llId the rcpulsivc 'In expom::nt. The value uf tlle laltice parameter t!i\'cn hy X-ray difractornetry agITes with Ihe elose approximalion uf thel';llculated valuc of lhc llletllnd uscd. It also compares the value nI"ellergy cuhesiulI ohtained hy tllc HUnl c""prcssion with more complex,Ippnlxilnations.

¡';ITll'onl.\: Ternary alkali halide eompounds: cohesion energy

En el presente trahajo se calcula el valor de la energía de amarre de una estruclllra ternaria cristalina llueva I"ormada por halogenuros alcalinos.La c...•truclUra ternaria preparada con diferentes concentraciones KCJ,¡.KBr,¡RhCI, mantiene una muy buena estahilidad y miseibilidad. Eldkulo "e haS~len el uso de una generalización de la ley de VeganJ (que en general es v¡,lid,1 para cOllllHlestos hinarios) para calcular losv,lIores de la constante de red y el exponente repulsivo///'. El valor lid par;í.metl"Ode red medido por difractometría de rayos X, concuerdaen huena aproximación con el valor calculado por c1método usado. Tamhién se COlllp~lrael valor de la enl'l"~ía de cohesión obtenido por lacxpl"esilín de 130m con aproximaciones m,-lscomplejas.

Iksl.ri¡)/(Jr(',\: 11,llogcnuro alcalino ternario; energía de Cllllcsióll

l'ACS: J 1.1O.H.; J 1.25.-Eh

2. Thcorctical cousidcratious

Dne of the most important thermodynmnic properties in thephysical ionic linkagc is "the energy of the latticc", which isdefined as Ihe change of cnergy dE, ncedcd to canvert anemolc of a crystallinc ionic compound into thc gaseous ion s atinfinite separatioll. For a 1;1 compound this process is

of lhe generalizatioll 01' lhe Veg,lrd's law on the new ternarycompound, Il1at \Ve are proposing for the first time with thiswork,

Thc ailll of Ihis work is to bring out the importance afIhe existence of ionic crystallinc syslems, which are novelmaterials. that open Ilew prospects for the study of alkalihaliete cryslals, and Ihis is possible using the Vegard's (awfor ternary compounds.

1. Illtrodllctiou

Thcrc are so me reports in lhe !iterature about binary al-kali halidc compollnds slIch as, (KCI)(l - ;¡:) :(KBr).¡; and(KCIl(l - :¡;):(RbCI):¡; [1] these are miscible in all mol"r(onceOlrations 01' its components, setting up a crystal withfee~rruclure.

Recently we llave grown an alkali halide novel mixture¡,:rystalline with three cOlllponents KC:¡, KBrll RbCI:; by theC1.ocl1ralski techniquc [2-4]. The knowledge ofthe existence01' these crystals awoke the necessity of performing the cal-culations for the value of the cohesion energy, which keepstogether these crystalJographic structures. We used theoreli-cal values for lhe lattice constant ro and the 11/.constant ofIhe repulsive force for rhese calculations, and assllllled validlúr 1"(1 tlle generalized Yegard's law. \Ve show thar lhe ob-lailll.'d villuc:\ fOf lhe 1arrice constan{ using rhis law are ingond ,lgrccmem with lhe valllcs obtaincd by X.ray powderdilTraCIOllll'lry rhe rcsults show more evidcntly Ihe validity --tlJl'IPlallilll' -t .'1"+ + B'l~. (1)

522 COHESION ENERGY C'ALCULATIONS FOR TERNARY IONIC NOVEL CRYSTALS

where 1'0 is Ihe cryslal iOlls eqllilibriulIl position, Replacingthe IJ valllc in Eq. (51. \Ve nbtaincd

111urder lO procecd furtl1el we shall adopl (he Lisler's nola.tion [ti]. and thus, thc Eq. (8) will be represcnted as

(7)

('J)

- ,:,)

..YA .-111 +,,- ('111= -----

.liTl"IIII"(\-m .

[j, = ~ (_(2_).r" (2 + M

['., =

fOI"the last expression one thus oblains the n value:

'!'he Eq. (8) is known as the Boro expression for ca!cu-lafion of lauicc cllergy. If the inlerionic distancc ro and therepulsive exponelll /TI are knmv, Gne of the cOnlriburiolls ofthis work is lO c;¡ll"ulate its values ro and 111 applying onegeneraJiwtioll of Vcgarcl's la\\' as il is cxplained in Scc. 4.

3. Cnllsidcralinlls ahnlll Ihc Ihcnrclicaldc\'c1npmclIl

(it:l~eral.ly in solid-state physic, ionic mixed compoundscrystalllze In several structurcs, \vhich are different in theca(ions ;lnd anions arrangemcnt. BOll1's lheery of the lauiceClll'rgy is based on Ihe asslllllption that the crystals underconsideration are built up by positive and negative ions. IrWl' as",llme thal rhe charge dislriblltion in the ions is spheri-cally Sy'lllmClric, lile force hel\vcen t\VO slIch ions dcpcndsollly 011t!leir dislance apan and it is indcpendent of dircctioll.ror e"amplc. in the rock-salt structures, as adopted by lIlilllYl'lllllpOunds 1: I sllch as NaCI, each cation is surrounded by"i" anions at a distance r. \••..hich results fmm the addition ofilllion ¡¡nd cation nldii. This intcraetion is auraetive and sup-plies ,1Coulomb encrgy E(" per eation, equal to

GlI+1/-e2E - ---- (2)..•.( - 47l"f

nr

where 1/+ <lne!11- rcpresent lile Illllllber of calion ami anionchargc" respectivL'ly. /.' is tlle valuc of tlle deerrical chargeand 1" Ihe vaCllUll1 penllittivily. The effeet in Ihe Coulombencrg)' to secondary neighbors, is given by rhe cffeet ofIwclvc ¡,:ations at a distallce 01"J2 1', lo third neighbors thet:1Tcct is eigllt aniolh al a distance of J3 r .1I11J so on. TheCoulomb em:rgy oí' this ion in the field of al! other iOlls is(berelore

In Eq. (4), IJ ¡¡mi /11 are constants. s¡nce repulsive forcesdept:nd :-.0"rrongly on the distance betwcen the panicles, IherL'Jnil....ive eIlergy Eq. (..l) is mainly detennined by lhe ncare:-.tnL'ighhors of thL' L.cntral ion.

1:¡l\¡¡lly the lotal energy 01"the cryslalline laltice \vill be¡!le additioll 01"Coulomb energ:y plus replllsivc energy, that

101'one mole i"¡\' \..-1n+n-":! IJ

l', = ",",+ E = - ' + -, ()1" 4íTfnr ,.11I '

\,.llL'rc A i" [\1adelung's constanl and ¡YA is Avogradro 's nUIll-~cr. 11'we cOllsider the crystal in equilibrilllll conditions itnL'Cl'S"ary that Ur tu be a minilllulll, which will be the caserOl" tllL' equilibriulIl value 7. = ro . when: ro represcnts tlleinlerionie distance in the crystal. Por this l1linil1lul11,

TIlL' ,,~rie~ in Eq. (3) show~ convergen ce and gets a lhed\'allle. The yaluc ol" the series in Eq, (3) is caHed the Made-lllllg':-, constanl and it has been calculated wilh high precisiollrot' ,111nI' lhe COllllllon crystal stmetmes [5].

()n the otber hand. the rcpulsive effect of Ihe interacrionamonoS clcctron clouds of the n:fcrenee ion ami adjacent ions,Ibo Illusl bl' cakulated, these forces are dominant at a shortdísl,lIllT. Born llladc the simple ;¡ssumption thal the rcpulsivcl'IlLT,SYhet\\.:een t\\'() ions as fUllerían of their separation couldIw ex prl'sscd by

( 12)

(11)

( lO)

(2 = (/1, - 1),11,

J'

hcrc JI value is "no\\'1l in Eq. (101 amI Ihe vallle of m, whichis the repulsive exponent is. knO\vn.for alkali halides. So weare ¡¡ble tu ca1clllate the theoretical valllc of 111 for the temaryCOlllpound lIsing l!le generalization of Vcgard's law as it isdetaincd ín Seco 4, Thus \i:C are able to calculate Ihe q vatuelIsing the cxprcs"ioll in Eq. ( 12 l.

The advantagc 01"llsing Listcr's l1lcthod is that Illore (;0111-plex equations can be considered ami express them in a sim-ple formo r:or example. for lhe Horn-Mayer equalion

where \;1 is the molar \'olllllle for Ihe crystal and do is thecompressibility. holll values at lera pressure. Values of \ ~land ;Jo are ver)' sp"rse lhercfore. it is not easy to know thelll.Nevertheless it is pos"ible lo calculate tht.: (j value will10UIlIsing \;1 and /Jo- I)ue (hat Eqs. (~) ami (<)) represent the sameellcrgy we llave

and

whcre

(4)[]

E,=-., 1'01

(dr,'!, )-- = (J,d,.

1"='.11

(6) ( 13)

R{'\'. Mi',l, Fú. -17 (6) (2001) 521-52-t

G. VAZQUEZ.POLO. R.R. MIJANGOS. E. CABRERA. E. VAl.DEZ. AND C. DUARTE523

".,,.. -''''' u,..., ,~

éO ".,>- ".,•... ".,Vi ,~.7-

'" noo '.•... 1100 -:7-- """% '''''O¡:: ,,.,U ".,;2 '''''~ ,= ~l~ ...,O>- mJO -:

« 1200 -.:

'" 1100 -

X ''''' ."" .

••'00

""='00

Fl(jl'kl l . .\.-r;¡y dilfracltlllll'lry IYIX'r\aClllf ~1~crY<'lal. !-.how"':l:-'11l~lcpha<,c

TllL' Eq. \ 14) i•..n:l;llt:d lo Eq. (9) ami il uses lhe samc paralll-l.'lers. (J ;l1ld ,H. \VI..'can procl'cd in lhe same way with morec(l111plicalL'd rcjali(H1S rOl" binding cncrgy. sllch tha! BOrJ1-Van

del Waa]:..,. whic!J has Ihe lúnn

\\'¡lh Lj<,lt:r <, llolaliOIl iI is COllvcrtc:d into

. .\/ ( (1 + M )I~11 = 0, (1 + 2M .

. -:..l.. ((1 + 25D + DC¿¡M)( (. - --------,." (1 + M + :\GD .

lile \'¡tllIes lor lhe 1/1 e.\pOnenl for lhe repulsive energy andthe r(j par,lIndel' 01' lhe ne\\' compolllld ternary structure.

Fm X-ray analysis a ditTr;¡ctomcter Bruciker Advanced 8\vas USL'd, 111 ¡:i~.I \\'L' SIlO\\' ;¡ diffractogram of the mixcdalkali-halidc crystal KCI.,(I:KBr:!!",:RbCL2", on it \Ve can...•el' a cryslallogl'¡¡phic structure lypc Icc, 1'01' this n:ason\\le adoptL'sd the stl'ucture known as rock-salt ror the ourcompotllld lemary, an thc Madclllng constant A is takcnfrolll NaCl .1 = 1.,.\".,1; IX].

TIlL' IllL'orctical L'aiL'ulalitlll 01' Ihe 1'1) ami thc 1/1 value \VasIll;lCh: llsill),! lile Ve~ard's la\\' gcneralizcd fol' temar)' (0111-

POllllds, thal \\'1.' proposed. \Ve add Ihe product 01' the molarpCITcntagl'.I', ni each binar)' illvolvl'd in tht: formation ofthcmÍ"; and ll1ultiplying it \\'ith the corresponding "oi al' 11li cum-poncnl singlt: valuc:

( 16)

( 1))

(14)

('.\1 I!{'(.:= - +

,. ,."2

II i•..COllvcr!cd lo Ihe: C"IJrcssion

,Illlllhl' 130m i\ layer cquation wilh lhe Van der \Vaals tenn

I = :..l..((/+.\1 + f)((¿¡JI +:m+GD'/.\/) (17)n 1"11 (J + '2.\/ + .1'2D .

1111

(Lr",J,).(LIlJ,r,) 11 S)

[1\ lht: Eq ...•.1 16) and t 17\. J) i•..('/ ,.;;.whcrc e is lhe samep;¡r;llllcter ¡ha! in thL' Eq. ( 1:)J. Thl.' valul's for e cm be (on-"ullcd in Re!. 7.

.l. RcslIlIs dcvdol'll1clIl amI cOlldusiolls

Tlle,.e ¡¡re lhree things \Ve musl taKc inlo aC(,Ollllt 10 lllakc thecah:u\¡ltiun: The v~lIl1es lleeded rOl' Ihe r-.,'ladelung constant A.

Tabh: I ctllltains the \allleS (Ir Ihe rcplIIsive exponclll /1/,

Ihe 1;¡t1ice C(llhtallts /"(1' {""'II' {'II ••." tllHl{',.xp of Ihe sin-1:!k L'OIIlPOI1L'Il\~,Ir ',,,!!I reprL'sL'1l\ the Coulomb contributionlo IhL' cohesi\l' l'lll'rgy ;1I1dL:111I'''lhe theoretical cohesivc CI1-l'rgy, ohtaillL'd by lllllltiplying {',,,,d by (1 - 1//11). In Table 11can be o\lsL'l'\'ed Ihe \'atuL'S L';¡1t:uialed in Illis \Vork and it alsoshow.'i lhat Ihe Ialliú~ UlllSl;ull oblilincd lIsing gcncralized Ve-gard's I;l\v :llld X-ray dilr;H:lollll'lry tlrL' in good agrecment\vith lhe L'xpL'rillll'lItal valucs rcpnrted in [2, !)). ln '[~lblc lB

N{'\~ Me.\". Fis ..H (())¡lOOII5~I-)~.J

524 COHESION ENERGY CALCULATIONS FOR TERNARY fONIC NOVEL CRYSTALS

T!\BI.E 1. V:dtlc." orIlle WIl .••lant .••u.••cd ror the calculalion .••.whcrcl'nlumn." ("'!l,'" ;uHllí"",1' are in Re!". X.

T.\BLE 11.Values ohtained ror the a\'eraged constalUs, where "o"X])i..•in Rd ••.2 and 1).

l/ti" 1'0 /'n,.xl'

(A) (Al

~II KCII~: KBrt't :RhCI:n $.89 :1.24 :12:1

\t2 KC1,-,u:KBr:r,:RbCI,2r, S.8.1 321 :121

;\l.' KCI \\:KBR l.o:RhCI n7 8.85 3.24 323

af\: lhe cohc:-.ivc cnergy values oblaincd in this work ror Ihelemary compotlllds. obtaíned by Born's express ion and ~lrccOlllpared willl more complex approximations.ln lhe last col-ullln. considering valid the thermodynamic principie of Iheadditivc encrgy. appears the ;'l\'erage values (U<lV")' using Ihe"alucs of pure components sl1O\\' in Table 1.

\Ve ohserved lhal

(/l lh!.? Ihcorctícal valuc for the ro balance position ob-lajned IEq. (1 S)] matches Ihe ene oblaincd experi-Illcntally through X-ray diffractolllelry (Fig. l and Ta-hlclVI.

IJ) \Ve noticed lhat Ihe energy va¡ue calculated using theBorn approximation is the closest to U;WI', in all cases.

The cohercncc of tlle calclllations rcsults based in the Gcner-alizt'd Ycg~ml's la\\', givc a strong argull1cnt of his application.

l. T. Olmo, ./, C'-y,\.t, (jt"(}wth 91 (llJXX) 576.

2. R.R. t\lijall~(lS i't (/1.. Phys. Li'U. ,1 245 (1998) 123.

.,. R.R. ~li.ian~o~, E. Camarillo. G. V¡ízt)lIcz-Polu, ami V.M. Cas-tao. Rc\'. Mn. Fh. 45 (SI) (1999) X5.

J. R.R. \lijangos ('t l/l.• Phys. Status ,""olidiB 220 (2000) 6X7.

d. T.e. WaddinglOn,,\ch'. Illorg. Rac/iocheltl. 1(1959) 157.

(j. 1\1.W. Li."ter. 'lh('l"1II0C!l('IIl. Aeta X ( 1974) 341.

TABI.!: 111. Valucs 01' thc rni ni lllll 111encrgy Wilh diffcrellt potcminls.rn. the Ia••t t'Olullln. cnn ••idl'rin~ valid Ihe thermouynamic principlcnI (hc addltl~'C energy appc;lrs tlll' a\"er~lf!cu va!Lle•• 01' iJ"VI(' lIsinglhc \"allle••(It lile JlureCOlllpllJlcnt ••. illIU/mol).

Born Bnfl] \layer Bum V;ll1 Born-\I,I)Cr U"Vl'.dn \\';[;t1.~Van der \Vaab

~11 -(jSi.l:l -GG5.;);j -(i.",~J..13 -G70.ji:) -047.80.~12 -(¡(¡.'J, i~ -(;7.1..1:2 -(¡(¡S.O;) -(¡79.:)] -658.79~IJ -(¡:')(j.()() -GG-l.;>;J -(;58.31 -G69,fil -G52.77

TAHLE IV. COlllplclllcnt;lry illrl)rrn~ltion tI) Ji tlral{)gram in Fig. ( 1).

1"1) Intl'll ••ilY h 1..lA) '¡(

:~.¿I~l'2 JIU) 2 11 "'2n."d:1 (it! 2 2 Ol.x.",G:J'2 '2(l 2 2 21.8(li~;;) " 4 O "J..1:IS:.:I Ir, .1 2 OL:ll:l22 11I 4 2 2Ll3580 .) 4 .1 O10719:/ .1 G O O1.01010 .) O 2 O

rvlorc dclaikd experilllcl1t:t1 e\'idetKC ¡¡bOUl the genenllilcdYegard ..••l;¡w was publishcd H:celllly [10].

Tl1cre is all c"ídent l1t'l'd rOl"carrying out Illorc studiesabolll Il1i:-.1100cl crystal. whicll opens a !le\\' inlcrcsting fleld01' :-.tudy.

Acknowlcdgments

\Ve are gratcflll to f\1r. R. Gucrrero ror growing lhe cryslal.

l. J,E. ~laycr • ./. C!l('HI. I'hy.\. I (1933) 27S.

R. N.\\'. Asl1crof1 and !\.D. ~krlllin. Solid Stah' l'IIy.\'il"s. (Hoh •({lllehan allJ \\'in~lOn, 191)7),Ch. 20, p. '¡OX.

9. R.R, ~tij:lIl~os('1 al.. R('I'. Mn. F{s . ..u (SJ) (199X) 171.lll. R.R. f\lij;mgo.", A. Conlt:m-Borhoil. E. Alvarez. ¡¡nd M. Cer-

val1tC". I'/'.\'.\". I.-d/. 1\ 2X3 (20() 1) 195.

RCI'. M('x. F{\ . .t7 (6) (2001) 521-524