Incommensurate instability and lattice dynamics of potassium selenate within a semiempirical rigid-ion model

PHYSICAL REVIE% BVOLUME 42, NUMBER 13 1NOVEMBER 1990 Incommensu a e ins abili y and la ice dynamics o po assium selena e wi hin asemiempi ical igid-ion model I.E xeba ia and J.M. Pe ez-Ma o Depa amen o de Fssica de la Ma e ia Condensada, Facul ad de Ciencias, Uni e sidad del Pais Vasco, Apa ado Pos al 644, 48080 Bilbao, Spai n A. C iado Depa amen o de Fi'sica de la Ma e ia Condensada, Facul ad de Ciencias Fi'sicas, Uni e sidad de Se illa, Apa ado Pos al 1065, 41080Se illa, Spain (Recei ed 23 May 1990) The la ice dynamics o po assium selena e is analyzed using a igid-ion model wi h he selena e g oups educed o igid bodies. The in e a omic o ces ha e been adjus ed only using s a ic s uc u al da a. The numbe o adjus able pa ame e s a ies om wo o i e. Such asimple model is al eady su icien o ep oduce semiquan i a i ely he phonon dynamics o he eal sys em. In pa icula , he model exhibi s he la ice ins abili y leading o he exis ence o an incommensu a e phase. The cha ac e is ics o he esul ing so mode ag ee wi h hose obse ed expe imen ally. The calcula ed eigen ec o , in excellen ag eemen wi h he expe imen al one, is a he unsensi i e o he de ails o he in e ac ions. This explains he s ong simila i ies o he incommensu a e modu- la ions in mos A2BX4 compounds. On he o he hand, he o m o he so -phonon b anch s ong- ly depends on he o ce model. I is su icien o i he model o he s a ic s uc u e obse ed a 145 Kins ead o he one a oom empe a u e, o p o oke aconspicuous so ening o he b anch. The b anch minimum is specially sensi i e o some po assium-oxygen in e ac ions. The ela i e size o he ca ions plays an essen ial ole in he o igin o he incommensu a e ins abili y. Fo compa ison he esul s o asimila analysis o Cs2Se04 a e p esen ed. In his case, he uns able o "so " cha - ac e o he lowes X2 b anch disappea s. I. INTRODUCTION No mal phase (Pnam ):INC phase X2, k=a(T)a =lockin phase k=(m /n )a* This gene al scheme is ep oduced in all compounds, bu some pa icula ea u es can a y conside ably. Fo ins ance, T, is 130 Kin KzSe04, bu 553 Kin KzZnC14. The empe a u e ange o he INC phase changes om 15 Kin Cs2CdB 4 o 155 Kin R12ZnB 4. The com- pounds o Zn do no exhibi ap ope so -phonon Aconside able numbe o ma e ials o he ype A2BX4 (K2Se04, Rb2ZnC14, Rb2ZnB 4, Rb2CoC14, Rb2Co8 4, K2ZnC14, e c.)exhibi acommon sequence o phase an- si ions as he empe a u e is a ied. A ace ain empe - a u e ange hey a e isomo phous o P-K2SO4, ha ing space g oup Pnam and apseudohexagonal axis along he adi ec ion. Below some empe a u e T;, his s uc u e is uns able and he compounds ans o m in o an incom- mensu a ely modula ed (INC) phase wi h i s wa e ec o along he aaxis (in mos cases close o he alue —, 'a'). The symme y o he p ima y dis o ing mode (o de pa- ame e ) is X2. A lowe empe a u es, asecond phase ansi ion in o acommensu a e phase akes place and he wa e ec o locks in o acommensu a e alue ( o a comp ehensi e e iew see Re s. 1and 2). Thus, ascheme o he co n non ansi ion sequence is b anch, while in K2Se04 he exis ence o aXz so mode was clea ly obse ed. 'In he Zn compounds, he e ahed al g oups BX4 a e diso de ed in he no mal phase wi h wo equi alen posi ions ela ed by ami o plane. In con as , x- ay s uc u e de e mina ions o he no mal phase o K2Se04 ha e been success ul using a nondiso de ed model. In his las case, howe e , la ge he mal ellipsoids indica e s ong lib a ions o he ionic g oups and ecen ly some spec oscopic anomalies in his compound ha e been in e p e ed in e ms o adiso de ed model. ' Despi e hese di e ences, i is easonable o assume ha he no mal-INC ins abili y obse ed in all hese compounds has essen ially acommon o igin. Acompa - ison o he eigen ec o s desc ibing he modula ion o dis- o ing mode in hei INC phase suppo s his iewpoin . This eigen ec o is shown in Table I o se e al A2BX4 ma e ials. The a omic modula ions co esponding o a oms belonging o he BX4 e ahed al g oups ha e been adjus ed o a igid-body model and in he able only he i ed Fou ie ampli udes o he sy n ne y-allowed o a- ions and displacemen s o hese g oups a e lis ed. A s ong simila i y among he di e en dis o ions is clea ly obse ed (see also Re . 8 o acompa ison conside ing all a oms independen ly). Only in he case o KzSe04, a signi ica i e di e ence in he ela i e weigh o he ampli- udes o he BX4 o a ions along he aand baxes wi h espec o he o he compounds is de ec ed. The INC modula ion is he e o e no e y sensi i e o he de ails o 8482 1990 The Ame ican Physical Socie y 42 INCOMMENSURATE INSTABILITY AND LA j. IICE DYNAMICS. ..8483 TABLE I. Pola iza ion ec o o he modes desc ibing he INC modula ion o some A&BX4 com- pounds. Findica es he deg ee o i ing o igid-body displacemen s. Fo each ion o ionic g oup in he asymme ic uni , he ampli ude o he displacemen s ( ela i e uni s X10 )and/o o a ions (deg) o- ge he wi h he co esponding phase (/2m) a e gi en. A(a) A(P) BX4 T. T. T. R„ Ry K2Se04 69/0. 102 92/0.321 58/0.825 2.27/0.036 3.68/0.643 92% RbqZnC14 69/0. 101 80/0.231 64/0. 771 5.30/0.024 2.55/0.654 89% Rb2ZnB 4 143/0. 110 147/0.238 118/0.778 9.66/0.025 4.27/0.654 92% KqZnC14 377/0.097 430/0.287 262/0. 828 14.49/0.013 9.71/0.626 73% Cs~CdB 4 83/0.067 84/0.078 43/0. 695 6.67/0.016 1.13/0.695 84% he in e a omic in e ac ions and seems o o igina e essen- ially in some o he common s uc u al ea u es o all hese ma e ials. This ype o conclusion is consis en wi h some ecen a gumen s, which in oduce he con- cep o "la en symme y" (a ce ain impe ec symme y o he s uc u e) as he ul ima e cause o he appea ance o he INC modula ion. Howe e , aclea unde s anding o he mic oscopic mechanism o he INC ins abili y and i s connec ion wi h he s a ic s uc u al p ope ies o hese s uc u es is s ill missing. Apa om he phenomenological ea men s in he ame o Landau heo y, only a ew wo ks ha e a - emp ed o explain he INC ins abili y om amic oscop- ic iewpoin . Iizumi e al. sugges ha he ins abili y in K2Se04 is he esul o acompe i ion be ween he i s -, second-, and hi d-neighbo in e laye e ec i e in e ac- ions in aone-dimensional local-mode laye model along he xdi ec ion, wi h he e ec i e cell co esponding o hal he apa ame e . This model implies he conside a- ion ha he local mode has s onge e ec i e in e ac- 0 ions wi h a oms a dis ances o abou 10 A han wi h a oms a dis ances smalle han 3.8A. This s ong as- sump ion has been no obs acle o he model o be widely accep ed and he a gumen has been ep oduced in many subsequen wo ks. Mo e sophis ica ed models in he ame o mean- ield app oxima ion educe he in e ac- ions o i s - and second-neighbo laye s. 'O he s udies p opose he anha monic pola izabili y o he Xiona as he o igin o he s uc u al ins abili y (a leas in he case o KzSe04)."Finally, some ecen ly epo ed in es iga- ions on he s abili y o hese ma e ials, including he in- oduc ion o he men ioned concep o la en symme y, ha e been achie ed h ough he analysis o he ha monic la ice dynamics o igid-ion models wi h in e - a omic cen al o ces. ' ' These models seem o be su icien o ep oduce he essen ial ea u es o he ene - ge ic and ib a ional p ope ies o hese ma e ials. In pa icula , he la ice dynamics o po assium selen- a e has been s udied om a heo e ical poin o iew by Haque and Ha dy. 'Using a igid-ion model wi h ap io is ic alues o he ionic cha ges, hese au ho s de- i ed he heo e ical phonon b anches along he Xline. The necessa y pa ame e s o he epulsi e pai wise po- en ials we e de e mined analy ically using he se o equa ions esul ing om he equilib ium condi ion o he s uc u e obse ed a oom empe a u e. Fo he pai s oxygen-oxygen and oxygen-selenium no unc ional o m was assumed and only he necessa y i s and second de i a i es a he dis ances co esponding o he ac ual bonds we e de e mined. The esul ing posi i e sign o he i s de i a i e o he epulsi e o ce be ween some o he a oms pai s was, howe e , incohe en wi h i s nomi- nal epulsi e cha ac e . Fo he pai s oxygen-po assium aBo n-Maye po en ial was conside ed wi h some ad hoc modi ica ions o i s second de i a i es. The in e nal de- g ees o eedom o he selena e g oups we e included in he analysis. The ob ained highe - equency ex e nal modes de ia ed sys ema ically o highe alues han hose obse ed expe imen ally. The model, howe e , was able o ep oduce he exis ence o he so -phonon b anch by a ying he second de i a i e o he epulsi e o ce o a speci ic pai o oxygen a oms, and he calcula ed so - mode eigen ec o ag eed quan i a i ely wi h he expe i- men al da a. ''The o he simula ions o he la ice dy- namics o A2BX4 compounds ha e been pe o med using "ab ini io" po en ials. ' In he p esen pape anew analysis o he la ice dy- namics o K2Se04 and i s ela ion wi h he la ice INC in- s abili y is epo ed using asemiempi ical igid-ion mod- el. We cen e ou s udy on he compound K2Se04, as i is one o he bes known om he expe imen al poin o iew, bu he main aim is o cla i y he la ice-dynamical o igin o he s uc u al ins abili y leading o he INC phase in he whole A2BX4 amily. One o he ques ions o elucida e e e s o he abo e- men ioned concep o la en symme y. This concep has been in oduced in la ice-dynamics s udies ha simul- aneously s ess he necessi y o ab ini io speci ic in e - a o nic po en ial o ep oducing he INC ins abili y. Howe e , i he INC ins abili y can be aced back o a he gene al pseudosymme ic ea u es o he s uc u e o he no mal phase, i would be mo e easonable o ex- pec ha he la ice dynamics o any model o in e a om- ic o ces compa ible wi h he obse ed s uc u e should exhibi *'pa hological" dynamical cha ac e is ics leading o he men ioned s uc u al ins abili y and he obse ed INC modula ion. To check his poin , he igid-ion mod- el used in he p esen s udy only conside s semiempi ical in e a omic po en ials. In addi ion, we expec ha he use o a i ed semiempi ical model can lead o mo e ea1- is ic esul s han i s -p inciples models, as he i s ones can abso b a leas pa ially c ys al- ield e ec s. In he same line o simpli ying he model wi hou los- ing he essen ial s uc u al and dynamical ea u es o he 8484 I.ETXEBARRIA, J.M. PEREZ-NATO, AND A. CRIADO 42 ma e ial, asecond impo an cha ac e is ic o he p esen analysis is ha he BX4 g oups a e conside ed igid uni s. This con as s wi h p e ious ib a ional s udies ha ha e included he in e ac ions be ween he a oms inside he e ahed al g oups. ''The in e nal in e ac ion B-X has been e en conside ed o some ele ance in he ansi- ion mechanism in he case o Rb2ZnC14. As can be seen in Table I, he e ahed al g oups BX~ a e app oxima ely igid in he no mal-incommensu a e ansi ion in all he compounds. The e o e, hei in e nal deg ees o eedom a e expec ed o be essen ially ine and i ele an in he ansi ion. Only he ex e nal ib a ions should play a majo ole in he la ice-dynamical model o he INC in- s abili y. By his means, he numbe o pa ame e s o he in e a omic o ce model o be i ed is educed o wo o i e depending on he model used (see nex sec ion), which compa es a o ably wi h he 14 pa ame e s o he s a ic equilib ium and mo e han 20 addi ional pa ame- e s i ed using expe imen al ib a ional da a, which a e conside ed in Re . 12. The model pa ame e s o he in e a omic o ces o be used in he dynamical analysis we e op imized using he condi ion ha he esul ing a omic equilib ium con igu a ion should be agood app oxima ion o he ob- se ed s uc u e (see Table II). As a omic equilib ium con igu a ion, we mean he a omic posi ions and he uni -cell pa ame e s, which minimize he po en ial ene gy wi h espec o any s uc u al dis o ion compa ible wi h he ixed o ho hombic Pnam symme y. Consequen ly, o his con igu a ion he de i a i es o he o al po en ial ene gy wi h espec o any s uc u al a iable, which is homogeneous in each cell and compa ible wi h he Pnam symme y, is ze o. This equi emen is necessa y bu ob- iously no su icien o ensu e ha he s uc u e is s able wi h espec o modula ed dis o ions o symme y- b eaking modes in gene al. I is, howe e , enough o de e mine he alues o he ene ge ic pa ame e s desc ib- ing he model o in e a omic o ces. Once he model is chosen unde he abo e condi ions, he ex e nal phonon dispe sion b anches in he ha monic app oxima ion ha e been calcula ed o he di e en models. E en ual ins abili ies o he s uc u e wi h espec o symme y-b eaking modes will be accompanied by he exis ence o imagina y equencies in one o mo e dispe sion b anches. We analyze ho oughly he e en ual p esence o hese uns able modes as hey can be con- side ed he main ac o leading o a he mal s uc u al in- s abili y o he Pnam s uc u e. The eigen ec o o he mos uns able la ice mode can hen be compa ed wi h he obse ed so mode in he eal sys em. By a ying speci ic in e a omic in e ac ions we also in es iga e he essen ial ac o s ha a e esponsible o he INC ins abil- i y and he deg ee o sensi i i y o he eigen ec o o he modula ion o he in e ac ion model. Fo compa ison, we p esen he esul s o asimila analysis o Cs2Se04, which demons a e he c ucial ole played by he e ec i e olume o he Aca ions on he o igin o he INC ins abil- i y. Ob iously, he p esen analysis, al hough mic oscopic, adop s amechanical pic u e o he ma e ial. The mal e ec s a e only conside ed om aphenomenological poin o iew, in he sense ha we assume ha he mal eno maliza ion o he mode equencies can be s ong enough o s abilize a high enough empe a u es he se o modes ha a e uns able in amechanical pic u e. The model o adjus able in e a omic o ces is pa ially adap - ed o he he mal e ec s, since i is he s uc u e a a ini e empe a u e ha is aken as he obse ed s uc u e in he i ing p ocess. In his sense, i can be conside ed a se o he mal e ec i e o ces. II. INTERATOMIC FORCES We expec ha he p esence o he INC ins abili y is essen ially independen o he de ails o he in e a omic o ces. The e o e, he model is s ongly simpli ied. In pa icula , only Coulomb igid-ion and Bo n-Maye epulsi e o ces a e in oduced. In con as wi h p e i- ous s udies, he Se04 g oups a e aken as igid uni s in he oom- empe a u e con igu a ion. The in e a omic po en ials V( , )a e aken as V, ( ,i)+V„, ( , ), wi h V, ( ,j}=Q, Q, i ,i and V„,~( ,i)=A;, exp( B,J j ).— TABLE II. Expe imen al s uc u e o po assium selena e a oom empe a u e and a 145 K, and cesium selena e a oom empe - 0 a u e. The la ice cons an s a e gi en in A, and he a omic posi ions in ela i e uni s (X10 ). The da a a e aken om Re s. 5, 17, and 23, espec i ely. K2Se04 (293 K) 7.661 10.466 6.003 K,Se04 (145 K) 7.597 10.375 5.975 Cs,Se04 (293 K) 8.377 11.276 6.434 Se K(a) K(P) O(1) 0(2) 0(3) 2242 1705 — 57 2931 3024 126 4200 843 7095 3471 5644 4251 2500 2500 2500 271 2500 2500 2228 1670 — 35 2918 3083 102 4195 823 7076 3440 5638 4274 2500 2500 2500 274 2500 2500 Se Cs(n) Cs(P) 0(1) 0(2) 0(3) 2348 1768 — 105 3032 3022 392 4188 900 7061 3525 5546 4178 2500 2500 2500 418 2500 2500 42 INCOMMENSURATE INSTABILITY AND LA IIICE DYNAMICS. . . 8485 As he Se04 e ahed a a e conside ed igid, hei in e - nal in e a omic in e ac ions a e no included. The epul- si e o ces be ween he selenium a oms and a oms ex e - nal o he e ahed a can also be easonably neglec ed. Ob iously he oxygen-oxygen in e ac ion is conside ed only be ween a oms belonging o di e en Se04 g oups. The epulsi e in e ac ion K-K was included in p elimi- na y calcula ions, bu i s in luence on he esul s was e y weak, as hey in ol e compa a i ely long in e a omic dis- ances. Consequen ly, ene ge ic con ibu ions o he K-K epulsi e o ces we e neglec ed in u he calcula ions. The po assium ions we e aken o be ully ionized, QK =+le. Thus, he cha ge o he oxygen ion was ela - ed o he one o he selenium ions by he neu ali y condi- ion Qs, +4Qo =— 2e. As epulsi e pa ame e s Ao o and 8~~, we ook hose p oposed in Re . 15 (Aoo=55X10 kcal/mol, 8o=3.96 A'). The alue o Bzowas ixed a 3.76 A, ob ained om he alues o BzK and Bo& and he app oxima e empi ical ule 8;J.=(8;;+B~J)/2. In his ela ion, he alue o Bx& 0 was aken o be 3.56 A'. This alue esul s om con- side ing he alues o BK „and B„„gi en in Re . 16 and he empi ical ule men ioned abo e. I is well known ha he pa ame e s 8; a e s ongly co ela ed wi h he A; alues, so ha simila po en ials can be desc ibed by di e en se s o bo h pa ame e s. The e o e he alues gi en ap io i o he 8,¶me e s a e no ade e minan es ic ion in he p esen s udy. The emaining ene ge ic pa ame e s (Qs„A~o)we e adjus ed, as explained in he p eceding sec ion, using he condi ion ha he esul ing a omic equilib ium con igu a ion should be agood app oxima ion o he ob- se ed s uc u e ei he a oom empe a u e, o a 145 K jus abo e T, (Re . 17) (see Table II). We used o his p ocess ap og am based on aSIMPLEX algo i hm' ha sea ched he ene ge ic pa ame e s minimizing he s uc u al di e ences be ween he co esponding "equi- lib ium con igu a ion" and he obse ed s uc u e. The "equilib ium con igu a ion" was calcula ed a each s ep o he SIMPLEX p ocess by " elaxing" acon enien ini ial s uc u e (in mos cases he expe imen al one a oom empe a u e). The p og am wMIN wo king in mode 1 (New on-Raphson ene gy minimiza ion mode)' was used o his pu pose. The nume ical ac o ha was in ac minimized and quan i a i ely measu ed he di e ence be- ween he obse ed and elaxed s uc u e is de ined by R=(I/X) g[(b,a, /a, )100] i=1,3 +gQ[bx, ;a, (10)]2 (3) whe e ha, and hx, a e, espec i ely, he de ia ions o he la ice cons an s and o he a omic ac ional posi- ions om he obse ed alues, while a,.a e he expe i- men al la ice cons an s in A. The sum in jex ends o all he a oms in he asymme ic uni , and he sum in iis e- s ic ed o hose componen s ha a e no ixed by sym- me y equi emen s. Nis he numbe o e ms in he sum. Acco ding o i s de ini ion, his pa ame e is equal o one in he hypo he ical case ha all uni -cell pa ame- e s de ia e 1% and all he a omic coo dina es di e in 0.1A om he expe imen al obse a ion. This o de o magni ude o he disc epancies be ween he simula ed s uc u al model and he eal obse a ion is no mally conside ed accep able in his ype o s udy. In he p esen s udy, he ac o Ra ained alues o he o de o 0.1-0.3. I can be no ed ha hese alues a e 1o de o magni ude smalle han he one (R =1.42) co esponding o he elaxed s uc u e ob ained o Rb2ZnC14 in Re . 9 using an ab-ini io o ce model. The signi icance o he quan i a i e alue o Rwas in- es iga ed by de e mining i s alue o he s uc u al di e ence be ween wo elaxed s uc u es co esponding o o ce models di e ing only in he cha ge alues an amoun b,Qs, =O.le (EQo=0.025e). Typically R ook alues be ween 0.04 and 0.09. The bes i was inally ob ained o Qs, =+l.30e (Qo= — 0.825e) and Ax a=97.7X10 Kcal/mol wi h R=0,159. The elaxed s uc u e is p esen ed in Table III. In he ollowing, we shall call he esul ing model o in e a omic o ces, model 1. The whole se o model pa- TABLE III. Relaxed s uc u e o po assium selena e o model 1, model 2(293 K), and model 2(145 K), and o cesium selena e o 0 model 1. The la ice cons an s a e gi en in A, and he a omic posi ions in ela i e uni s (X10 ). K2Se04 model 1 a7.662 b10.426 c6.074 K2Se04 model 2(293 K) 7.611 10.413 6.070 K&Se04 model 2(145 K) 7.556 10.335 6.028 Cs2Se04 model 1 8.352 11. 219 6.496 Se K(a) K(P) 0(1) 0(2) 0(3) 2333 4181 1644 856 — 76 6963 3006 3440 3156 5620 207 4261 2500 2500 2500 297 2500 2500 2319 4190 1657 887 — 95 7023 3030 3467 3071 5652 189 4214 2500 2500 2291 2500 1629 — 68 296 2997 25QO 3127 152 4191 2500 879 2500 7022 2500 3440 293 5648 2500 4252 2500 Se Cs(a) Cs(P) 0(1) 0(2) 0(3) 2363 4179 2500 1684 895 2500 — 111 6971 2500 3036 3506 437 3065 5537 2500 401 4191 2500 8486 I.ETXEBARRIA, J.M. PEREZ-MATO, AND A. CRIADO o TABLE IV. Pa ame e o he di e en o ce models conside ed (Ao p=55X10' Kcal/mol, Boo=3.96 A o all models). R Qs.(e) Qo(e) BK.-O (A ) ~K(a)-o(1-2) (10' Kcal/mol) ~K(a)-O(3) ~K(/3)-O(1-2) ~K( 3)-0(3) Model 1 (K2Se04) 0.159 1.30 — 0.825 3.76 97.7 Model 2 (K,Se04-293 K) 0.132 1.34 — 0.835 3.76 114.7 102.6 83.3 118.5 Model 2 (K,Se0 -145 K) 0.105 1~34 — 0.835 3.76 106.6 97.4 75.4 100.0 Bcs-0 Model 1 (Cs2Se04) 0.109 1.44 — 0.86 3.50 122.5 ame e s is lis ed in Table IV. Achange o he selenium cha ge o O. le did no change signi ican ly he quali y o he adjus men . Fixing Qs, o +1.6e is, howe e , enough o double he R alue. Any a emp o ix he cha ge o he selenium and oxygens o highe alues ailed. The wMIN p og am in mode 1was unable in hese cases o e- lax p ope ly om he expe imen al s uc u e in o an equilib ium con igu a ion, indica ing ha he la e does no exis o di e s conside ably om he expe imen al one. This esul con as s wi h he analysis o he same compound in Re . 12, whe e he cha ge o he selenium ion was ixed o +2e (Qo =— le). On he o he hand, a ecen molecula dynamics s udy o he ela ed com- pound LiKSO& conside ed an oxygen cha ge e en smalle han hose ob ained in he p esen s udy. 21 The esul s o he la ice-dynamics analysis o o ce model 1, desc ibed in he nex sec ion, led us subsequen - ly o inc ease he complexi y o he o ce model by as- suming speci ic epulsi e in e ac ions o he pai s K-0 depending on he c ys allog aphic ype o po assium o oxygen conside ed. The e a e wo ypes o inequi alen po assium a oms in he s uc u e ha occupy e y di e en posi ions om a opological poin o iew (see Fig. 1). The K(P) occupy aca i y o app oxima e hex- agonal symme y along he aaxis and is su ounded by six e ahed a. I s sepa a ion om he oxygen a oms O(1) and O(2), on he ace o he e ahed a pe pendicula o he aaxis a ies om 2.74 o 2.82 A, while he sho es dis ance o an oxygen on he o he e ex, O(3), is al eady 2.97 A. In con as , he K(a) a e si ua ed app oxima ely on some o he e ices o he hexagonal ca i ies (see Fig. 1), such ha hei sho es con ac co esponds o an oxy- oo oo oo oo oo K(P) I I K(a) oo oo oo oo oo FIG. 1. P ojec ion along he aaxis o he s uc u e o K2Se04 a oom empe a u e showing he wo kinds o inequi alen po assi- um a oms. 42 INCOMMENSURATE INSTABILITY AND LATTICE DYNAMICS ~ ~ ~8487 gen O(3) a 2.62 A, while he nex ones a e al eady a dis- ances a ying om 3.00 o 3.26 Aand co espond o ox- ygens o he ype O(1} o O(2). I is well known ha e ec i e epulsi e po en ials can a y conside ably when di e en anges o a omic dis ances a e conside ed, so ha hey a e desc ibed by di e en Bo n-Maye po en- ials. Also, ab-ini io in e ac ions like Go don-Ki n po en- ials a e gi en by di e en analy ical exp essions o di e en dis ance anges. "I is he e o e easonable o assume ase o ou di e en Bo n-Maye K-0 po en- ials, i we ake in o accoun he e y di e en in e a om- ic dis ances ele an in each case. Acco dingly, we spli he ini ial unique K-0 Bo n-Maye po en ial in o ou di e en po en ials depending on he a oms conside ed: K(a)-O(1-2), K(a)-O(3), K(P)-O(1-2), and K(P)-O(3). The numbe o adjus able pa ame e s in his gene al- ized model inc eases o i e: he e ec i e cha ge o selenium and he ou pa ame e s Ax o( he pa ame e BK pis kep he same o he ou di e en pai s and has he alue gi en abo e). An op imal se o hese pa ame- e s was de e mined by means o a i ing p ocess analo- gous o he p eceding one. The p ocess con e ged o a alue o he Se cha ge o +1.6e (Qo= — 0.9e) wi h a i ing ac o R=0.109. Howe e lowe alues such as +1.3e did no de e io a e he adjus men signi ican ly and Rinc eased only up o R=0.136. The subsequen no mal-mode analysis indica ed ha wi h Se cha ges as high as +1.6e, he sys em was uns able along he Xline. The e o e he Se cha ge was ixed o alowe alue (+1.34e )and only he pa ame e s AKowe e op imized. The esul ing model, hence o h e e ed o as model 2 (293 K), is lis ed in Table IV. In Table III, he co e- sponding Pnam elaxed s uc u e is also p esen ed. As in he p e ious case, when highe alues han 1.6e we e conside ed no p ope Pnam elaxed s uc u e close o he expe imen al one could be ound. The esul ing con igu a ional ene gy o he equilib ium s uc u e [— 416.39 Kcal/mol o model 1, — 416.34 Kcal/mol o model 2(293 K)] is close o he expe imen- al cohesion ene gy (— 413.76 Kcal/mol). A hi d model [model 2(145 K) in Table IV] was con- side ed using as obse ed equilib ium con igu a ion he s uc u e de e mined a 145 K(Re . 17) (see Table II) and main aining Qs, =+1.34e. Essen ially, as aconsequence o he he mal con ac ion, adec ease o he e ec i e K- 0 epulsi e in e ac ions wi h espec o he oom- empe a u e model is ob ained. Finally, o Cs~Se04, asimila p ocess was ollowed. A simple model wi h asingle Cs-0 in e ac ion analogous o model 1 o KzSe04 was i ed ollowing he same p o- cedu e. As obse ed s uc u e we conside ed he ecen ly de e mined s uc u e a oom empe a u e (see Table II). The O-O in e ac ion was kep he same as in po assi- um selena e, while he alue o Bc,pwas ixed o 3.50 A'. The esul s a e also gi en in Tables III and IV. The inal op imized alues o Qs, and Ac, owe e +1.44e and 122.5X10 Kcal/mol, espec i ely, wi h R=0.109. Thus, he epulsi e in e ac ion A-0 inc eases conside - ably wi h espec o po assium selena e, in acco dance wi h he co esponding inc ease o he uni -cell olume and he e ec i e a omic size o he Aa oms. III. LATTICE DYNAMICS A. In oduc ion Using he same no a ion o he i educible ep esen a- ions as in Re . 3, he symme y decomposi ion a he cen e o he B illouin zone o he ex e nal no mal modes 1s g ]gag 3g+ 5 A„+5B &u +7Bz„+7B3u Along he Xline (q =au '} he decomposi ion becomes 14K]+10K~+ 10K +14K and a he poin X(q=—, 'a' )is 14X)+10Xq . The no a ion o he i educible ep esen a ions a he cen e o he B illouin coincides wi h he one o Re s. 6 and 7. The numbe o b anches Xz and X3 is he same, and he same happens o X, and X4. The compa ibili y ela ions a e such ha each pai o hem a e degene a e a he poin X. The e o e, in an ex ended zone scheme each pai o b anches Xz-X3, o X,-X4, educes o asingle b anch, and he o al numbe o b anches educes o 24. In he ollowing, we will use his ex ended zone scheme o he g aphical ep esen a ion o he phonon b anches. Iizumi e al. de e mined by means o inelas ic neu on sca e ing he exis ence o aso -mode X~ b anch con- nec ed wi h aX3 acous ic b anch in KzSe04. The ins a- bili y o his Xz b anch a apoin close o 0.31a* akes place a a empe a u e consis en wi h he known ansi- ion empe a u e T, =130K. Se e al Raman and in a ed spec oscopic measu e- men s o KzSe04 ha e been epo ed in he las ew yea s. ''The numbe o in e nal modes de ec ed by his means in he no mal phase di e s om hose co e- sponding o he usual s uc u al model. As apossible cause, i has been sugges ed ha he selena e e ahed a ha e some kind o weak diso de , which canno be de ec ed by x- ay me hods. 'On he o he hand, he ex e nal Raman and in a ed-ac i e modes (Ag, 8&g, Bzs, 83g and 8&„,Bz„,83„, espec i ely) ha e been iden i ied and measu ed wi hou much p oblem in he ame o he symme y analysis abo e. 'The expe imen al da a e- po ed in Re . 6will be used o compa ison wi h he p esen simula ion. Asigni ican ac can be deduced om hese s udies: appa en ly, he so Xz b anch be- comes an op ically inac i e A„mode a he cen e o he B illouin zone. This can be de i ed indi ec ly om he knowledge ha aXz b anch is only compa ible wi h A„ o 83g symme ies a he zone cen e , and he lowes 8, measu ed Raman equency is a he ange o 45 cm while om he neu on da a o Iizu ni e al, he end o he so Xz b anch has alues lowe han 25 cm B. Phonon b anches calcula ion The dispe sion cu es we e calcula ed unde he ha - monic app oxima ion using agene al p og am, which al- 8488 I.ETXEBARRIA, J.M. PEREZ-MATO, AND A. CRIADO 42 D Rl CY Q ghJ o he expe imen al so -mode eigen ec o in Table I. The ag eemen be ween he phases is especially signi ican, bu also he di8'e en a omic ampli udes ol- low in gene al e ms he expe imen al pa e n. Only a signi ican de ia ion o la ge alues o he ampli ude o he selena e o a ion a ound he xaxis and a oo small selena e z ansla ion can be clea ly obse ed. In con as wi h he expe imen al si ua ion, he X2 mode a he zone cen e has B3g symme y. An inc ease o he Se cha ge leads, apa om he changes in he equilib ium s uc- u e, o as onge ins abili y o he s uc u e, wi h he so -mode imagina y b anch aking la ge absolu e alues. REDUCED WAVE VECTOR 0.82. Model 2(293 EC) FIG. 2. Calcula ed dispe sion cu es plo ed in he ex ended zone scheme co esponding o he model 1o K&Se04 (solid line) and Cs2Se04 (dashed line). Fo imagina y alues o he e- quency, — ~co~ is ep esen ed. lows us o conside igid molecula uni s and he e o e can es ic he calcula ions o he ex e nal modes. The ex e nal Bo n- on Ka man o malism, in which he o ce cons an s a e ob ained in e ms o molecula o a- ions and ansla ions, is used. The dynamical ma ix is buil up using he me hod o Pawley whe e, using a pai wise po en ial model, o ce cons an s a e calcula ed analy ically as a unc ion o o a ional and ansla ional coo dina es. "Sel - e ms" o o ce cons an s ela ing wo displacemen s o he same molecula uni a e ob ained om he c ys al ansla ional and o a ional in a iance condi ions. Sho - ange o ce cons an s we e calcula ed o molec- ula pai s inside acu o adius dis ance (8 A). Fo Coulombic in e ac ions howe e , gi en hei long- ange cha ac e , he Ewald con e gence me hod was used. C. Resul s 1. Model 1 This model is uns able o ib a ions along he Xline, namely, an uns able dispe sion b anch o X2-X3 symme y appea s as shown in Fig. 2. No o he b anch along his line akes imagina y alues. Aclea minimum wi h a a he la ge dep h (5.32 meV) can be obse ed a apoin q=0.28a'. The mode eigen ec o a his minimum lis ed in Table Val eady ep oduces mos o he cha ac e is ics The sys em is s able wi h espec o all no mal modes. Again he lowes nonacous ic b anch along he Xline has X2 symme y, bu in his case he equencies ha e eal alues. This b anch (see Fig. 3) is a he la and p esen s aweak minimum a q=0.22a'. The mode eigen ec o a his poin ag ees wi h he expe imen al one much be e han o model 1(see Tables Iand V). The p e ious disc epancies obse ed in he ampli udes o A„and 1, o he selena e g oup a e co ec ed, bu he de ia ion o he ' , ampli ude o he K(a) a om su e s asligh inc ease. The phases o he di e en a oms seem o be weakly dependen on he de ails o he in e ac ions. They sca ce- ly change be ween he wo models and ha e an excellen ag eemen wi h he expe imen al alues in Table I. The b anch zone cen e has A„symme y. Howe e , o la ge alues o he Se cha ge (an inc ease o 0.02e is su icien ) he mode changes o 8& symme y. The eason o his ex eme sensi i i y o he symme y o he zone-cen e mode o he de ails o he in e a omic o ces is caused by he p oximi y o ano he Xz b anch a ound he zone cen e . The wo b anches may "in e ac " and can easily in e change he symme y o hei zone-cen e modes. The equency o he lowes B3& mode is specially sensi i e o changes o he ionic cha ges and s ongly de- c eases o la ge cha ges. A Qs, =+1.36e he in e - change o he ex emes o he wo X2 b anches akes place. In addi ion, his lowe ing o he B3g mode causes, by asimila e ec , he des abiliza ion (nega i e alues o C~) o he acous ic b anches along he yand zdi ec ions co esponding o he shea e„which has he same sym- me y. Fu he inc eases o he Se cha ge leads o lowe equencies all along he lowes X2 b anch and s eng hens he p esence o aminimum a alues close o TABLE V. Eigen ec o s o he mode co esponding o he minimum in he so b anch o K&Se04 o (a) model 1, (b) model 2(293 K), (c) model 2(145 K), (d) modi ied model 2(145 K) changing he Az opa ame e o he in e ac ion K(a)-O(1-2). (e) Eigen ec o o he mode X2 a 0.31a* o Cs&Se04. The da a a e gi en in he same manne as in Table I. K(a) K(P) Se04 T. T. T. R„ Ry (a) 61.2/0. 110 72.8/0. 318 38.2/0. 811 4.06/0.012 3.06/0.610 (b) 79.6/0. 135 80.3/0.320 68.0/0.816 2.23/ — 0.003 3.58/0.621 (c) 75.2/0. 126 83.3/0.319 62.6/0. 820 2.30/0.002 3.66/0.617 72.1/0.119 80.8/0.320 56.5/0. 830 2.70/0.008 3.63/0.613 (e) 64.6/0. 118 81.2/0. 319 45.3/0.824 0.96/ — 0.025 2.46/0. 614 42 INCOMMENSURATE INSTABILITY AND LA IlICE DYNAMICS. ..8489 6 VCV a bl CY O00.20.40.60.8 —, 'a'. Fo alues o Qs, a ound +1.6e, which was he op- imum alue esul ing om s uc u al conside a ions (see Sec. II), he Xz b anch is comple ely uns able, wi h a simila o m o he one o Fig. 2 o model 1. This was he eason o ix he alue o Qs, in his model o 1.34e. Thus, he condi ion o comple e s abili y o he sys em p ac ically o ced he A„symme y a he zone cen e o he b anch, as obse ed expe imen ally. The elas ic cons an s and Raman equencies calcula - ed o model 2(293 K) a e lis ed in Tables VI and VII, e- spec i ely, whe e expe imen al alues a e also gi en. Ag aphical compa ison can be seen in Fig. 4. In gene al, he ag eemen is wi hin 5— 20%, which is excellen con- side ing ha he o ce model has only been adjus ed us- ing s a ic s uc u al da a. Howe e , aB, mode ( he lowes one) s ongly de ia es om he expe imen al alue. This s ong de ia ion can be due o asymme y in e change e ec simila o he one commen ed abo e be ween he lowes modes o A„and B3g symme y. In his case, bo h B,gand Bz„modes a e compa ible wi h he b anches o X4 symme y. In he simula ion, he lowes X4 op ic b anch ends a he zone cen e wi h aBz„ mode (see Fig. 5). This symme y could be in e changed wi h he B]g symme y o he mode a he end o he nex X4 b anch i he o ce model is a ied. Fo he sake o compa ison wi h p e ious s udies, and also as a e e ence o e en ual subsequen expe imen al REDUCED i%AVE VECTOR FIG. 3. Calcula ed dispe sion cu es o K&Se04 plo ed in he ex ended zone scheme o model 2(293 K) (solid line), mod- el 2(145 K) (dashed line) and model 2(145 K) dec easing he in- e ac ion o he K(a)-O(1-2) pai s (poin line). The expe imen- al poin s a 145 K(Re . 3) a e also shown. s udies, he whole se o ex e nal phonon b anches wi h his model is shown in Fig. 5. The densi y o s a es co e- sponding o he ex e nal modes is also shown in Fig. 4. The highes ex e nal equencies ha e alues ha a e signi ican ly smalle han hose ob ained in Re . 12, wi h abe e ag eemen wi h he expe imen al da a. The global ag eemen o he mode equencies can be con i med by compa ing he esul ing speci ic-hea cu e o he expe imen al esul s. This is done in Fig. 6, whe e he e ec o he in e nal equencies o he selena e g oups ha e been included by adding o he densi y o s a es o Fig. 4 ou Eins ein- ype con ibu ions, cen e ed a he app oxima e expe imen al alues o he selena e in e nal equencies. The expe imen al speci ic-hea cu e has been aken om Re . 34. The ag eemen be- ween expe imen al and heo e ical alues is excellen . Only an ob ious and expec ed disc epancy exis s close o he ansi ion empe a u e. The de ia ion o he expe i- men al alues a highe empe a u es is due o he usual disc epancy be ween he expe imen al speci ic hea a cons an p essu e and he heo e ical one a cons an olume, which is caused by anha monic e ec s. This de- ia ion can be i ed o he usual app oxima e linea em- pe a u e dependence. Thus, model 2 ep oduces quan i a i ely up o an ex- cellen deg ee many o he known physical p ope ies o po assium selena e. I should be no ed ha he model was no chosen o i he expe imen al dynamical p ope - ies and only he condi ion o mechanical s abili y o he oom- empe a u e s uc u e was used o adjus he model pa ame e s. 9. Model 2(145K) As explained in Sec. II, his model was ob ained as a modi ica ion o model 2(293 K), using as obse ed s uc- u e he one de e mined a 145 K(Re . 17}(see Table II) o adjus he model pa ame e s. In he i ing p ocess (see Sec. II) he selenium cha ge was kep ixed o he alue o model 2(293 K}. An examina ion o Table IV indica es ha he essen ial di e ence be ween his model and mod- el 2(293 K) is ace ain dec ease o he e ec i e epulsi e in e ac ions K-O. This dec ease is cong uen wi h he olume con ac ion o he s uc u e a 145 K. The mos conspicuous change in he phonon b anches ob ained o his model wi h espec o model 2(293 K) can be seen in Fig. 3, whe e he lowes X3-Xz b anch is shown. The ex- pe imen al da a a 145 K(Re . 3) a e also shown in he igu e o compa ison. The sub le change in he epulsi e K-0 pa ame e s is su icien o p oduce aclea so ening o he Xz b anch, which has now as ong minimum a q=0.25a'. The co esponding mode eigen ec o lis ed in Table Vag ees e en be e han he one o model 2 TABLE VI. Elas ic cons an s (X10' dyn cm )o KzSe04..expe imen al (Re . 32) and calcula ed alues o model 2(293 K). Exp . Calc. 50.5 58.747.9 53.040.7 49.28.1 3.6 Css 14.2 16.3 C„ 15.4 18.1 C&z 15 17.1 Cz 19 16.9 C 17 17.1 8490 I.ETXEBARRIA, J.M. PEREZ-MATO, AND A. CRIADO 42 TABLE VII. Raman equencies (cm ') o K2Se04. expe imen al (Re . 6) and calcula ed alues o model 2(293 K). ~exp 41 75 97 105 121 144 162 ca e 43 88 94 106 125 130 162 Disc epancy {%) 4.7 17.3 3.1 1.0 3.3 9.7 0 ~exp 47 74 96 109 120 143 160 ~ca]c 81 88 102 128 146 152 162 Disc epancy (%) 72.3 18.9 6.2 17.4 21.7 6.3 1.2 50 75 95 119 146 57 81 85 114 154 B2g 14.0 8.0 10.5 4.2 5.5 44 75 93 119 142 40 71 82 113 122 B3g 10.0 5.3 11.8 5.0 14.1 (293 K) wi h he expe imen al one. The so ening o he X2 b anch is specially sensi i e o a a ia ion o he epulsi e pa ame e A~~ ~0~, 2~. The phonon b anch ob ained a e adec ease o less han 4% (102.6X 10 Kcal/mol) o his pa ame e wi h espec o i s alue in model 2(145 K) is also shown in Fig. 3. Wha is ema kable is he esul ing deep minimum o he b anch a apoin (q=0.27a' ), which is now close o he expe imen al alue o he so -mode wa e ec o . These esul s con as wi h hose epo ed in Re . 12, whe e he o ma ion o he deep minimum a alues close o —, 'a' was a ibu ed o aspeci ic O-O in e ac ion. We pe o med addi ional calcula ions changing he O-O epulsi e o ce. The esul s demons a ed ha in ou 0 Ag Blg B2g 83g 50 FREQUENCY(cm ) FIG. 4. Densi y o s a es o K2Se04 co esponding o he ex e nal modes calcula ed wi h model 2(293 K). Also ag aphical co n- pa ison o he calcula ed Ra nan equencies {dashed lines) and he expe imen al ones (Re . 6) is shown.