First- and Second-Order Thermal Diffuse Scattering (TDS) Intensity in Molecular Crystals: Influence on Crystal Structure Parameters

491 Ac a C ys . (1985). A41, 491-494 Fi s - and Second-O de The mal Di use Sca e ing (TDS) In ensi y in Molecula C ys als: In luence on C ys al S uc u e Pa ame e s BY A. CR~ADO, A. CONDE AND R. MARQUEZ Depa amen o de Op ica y Secci6n de F sica Cen o Coo dinado del CSIC, Uni e sidad de Se illa, Spain (Recei ed 24 July 1984; accep ed 22 Ap il 1985) Abs ac Fi s - and second-o de he mal di use sca e ing (TDS) in ensi ies a e calcula ed in he long-wa e app oxima ion allowing o dispe sion (LWD) in monoclinic pheno hiazine om pola iza ion ec o s and la ice-mode equencies ob ained om la ice dynamical calcula ions wi hin he ha monic app oxi- ma ion and he ex e nal Bo n- on K~i m in o malism using an a om-a om po en ial unc ion in he o m V( ) =-A/ 6+ B exp (-C ). The in luence o i s - and second-o de TDS in ensi y on elec onic densi y maps is analysed and compa ed. Leas -squa es e ine- men s o posi ional and he mal pa ame e s a e ca ied ou in di e en anges o sin 0/A aking in o accoun bo h i s - and second-o de TDS con ibu- ions and he esul s a e discussed. In oduc ion In p e ious pape s (C iado, Conde & M~i quez, 1985a, b) we epo ed a compu a ional p ocedu e o calcula e i s - and second-o de he mal di use sca - e ing in ensi y o molecula c ys als om a la ice dynamical calcula ion, using he ex e nal Bo n-yon K i m~in o malism in he ha monic app oxima ion. An a om-a om po en ial- unc ion model was u ilized as a sum o pai wise con ibu ions in he o m V( ) = -A/ 6 + B exp (- C ) and he p ocedu e was applied o monoclinic pheno hiazine. Fi s -o de co ec ion ac o s o in eg a ed B agg in ensi ies due o he mal di use sca e ing (TDS) in ensi y we e calcula ed wi h he ull la ice dynamical o mulae and in he long-wa e (LWD) (Bo n & Huang, 1968) app oxima- ion allowing o dispe sion and bo h esul s we e p ac ically iden ical. The in luence o i s -o de TDS in ensi y on elec onic densi y maps was s udied; he main con ibu ion was a posi i e addi ional densi y concen a ion a ound he a omic posi ions. A leas - squa es p ocess was ca ied ou o check he in luence o i s -o de TDS in ensi y upon a iable pa ame e s; he main in luence was ound o e he he mal pa am- e e s, which unde wen a dec ease wi h espec o hei ue alue. Calcula ions ha e al eady been pe o med (Helmold & Vos, 1977) on he in luence o he i s - 0108-7673/85/050491-04501.50 o de TDS con ibu ion upon s uc u al pa ame e s and ou pu pose he e is o s udy he in luence o he second-o de one. Basic heo y Fi s -o de TDS in ensi y has been calcula ed using he LWD app oxima ion in which he exp ession o he in ensi y a a poin S o he ecip ocal space such ha S = G -q, whe e G = 2~ H and H is a ecip ocal- la ice poin , is gi en by (Coch an, 1963; Coch an & Pawley, 1964; Bo n & I-Iuang, 1968) dll(S = G-q)/dq = NS 2 F(G) 2 ~ [Ej(q)/w~(q)]{s. eLW(qj)}2, (1) Jac whe e N is he numbe o uni cells in he c ys al, jac s ands o he di e en acous ic modes wi h wa e ec o q, o~j(q) is he angula equency o mode (q j), Ej(q) is i s ene gy, s is a uni ec o along S, F(G) is he B agg s uc u e ac o and eLW(qj) is he pola iza ion ec o o mode (q j) in he LW app oxi- ma ion; i can be ound by sol ing he eigen alue equa ion (Bo n & Huang, 1968; Ma adudin, Mon oll, Weiss & Ipa o a, 1971) D(q)U(q) = o~2(q)U(q), (2) whe e D(q) is he dynamical ma ix. The second-o de TDS in ensi y should be calcu- la ed in he exac la ice dynamical p ocedu e bu compu ing imes a e p ohibi i e. Because o his, we ha e also adop ed he LWD app oxima ion o calcu- la ing second-o de TDS in ensi y. Wi h his app oxi- ma ion, he exp ession o i is dI2(S = G-q)/dq n 2 2 -(S2/2)IF(G)I2E E E {[Ej,(q )Ej,,(q )/o)j,(q )o)j,,(q )] q' Jac Jae ×[s. eLW(q'j')s, eLW(q'~]")]2}, (3) whe e q' uns o e all he allowed wa e ec o s inside he B illouin zone and q = q'+q". This sum akes a long compu ing ime o a c ys al o a ini e size and, as we a e only in e es ed in he beha iou o densi y maps and s uc u al pa ame e s wi h espec o he © 1985 In e na ional Union o C ys allog aphy 492 FIRST- AND SECOND-ORDER THERMAL DIFFUSE SCATTERING ac o S 4, we ha e adop ed a second simpli ying app oxima ion (Ramachand an & Woos e , 1951), which consis s o a heo e ical e alua ion o he sum (in eg al o a la ge c ys al) o e q' assuming a linea ela ion coj(q) = ~q (LW app oxima ion), whe e j is he eloci y o he wa e, which is assumed o be independen o he di ec ion o q. The ac o in ol ing he pola iza ion ec o s is eplaced wi h a cons an ac o a he alues ha minimize he p oduc 2 , z ,, q, q,, oj,(q)wj,,(q )" = =q/2" and he in eg al o e he B illouin zone is ex ended o in ini y. Wi h hese app oxima ions, he exp ession o second-o de TDS in ensi y o poin s su icien ly nea he ecip ocal- la ice poin s is ound o be, a high empe a u es [ E)(q) = KsT]: dI2(S = G-q)/dq = ( NVcS4 q3/16)(K8 T)2] F(G)I 2 × ~ ~ {[s. eLW(qj')s, eLW(qj")]2/ O],(q) o],,(q)}, J~cJae (4) whe e V~ is he uni -cell olume. Me hod o calcula ion We ha e pe o med ou calcula ions wi h monoclinic pheno hiazine and he condi ions o calcula ing he co ec ion ac o s o B agg in ensi ies due o TDS con ibu ion we e he same as hose adop ed o i s -o de calcula ions (C iado, Conde & M i quez, 1985a), assuming a B agg-peak symme ic scanning olume o pa allelepipedic shape cen ed on he ecip ocal-la ice poin s, each edge 7/25 o he co e- sponding basic ecip ocal ec o . Pola iza ion ec o s and wa e equencies ha e been ob ained om la ice dynamical calcula ions. In o de o s udy he in luence ha second-o de TDS in ensi y has o e densi y maps we ha e con- side ed he c ys al con igu a ion om which we ha e calcula ed la ice dynamics and B agg co ec ion ac- o s as he ue one. The in ensi ies measu ed in a eal expe imen would hen be ob ained by adding o he calcula ed B agg in ensi ies he TDS calcula ed con ibu ion a 300 K as I~xp(G) = IB~Gc(G){1 + a2(G)}, (5) whe e a2(G) = I2(TDS, G)/IBRAGG(G), (6) and he in luence on densi y maps would be ob ained by means o a di e ence Fou ie syn hesis wi h coe icien s Fexp(G)- Fca,(G). In his way we ha e calcula ed co ec ion ac o s and 'expe imen al' in ensi ies o 1026 independen e lec ions wi h sin 0/A <0.6 A,-'. A di e ence Fou ie syn hesis (p og am FOURR ; S ewa , Kundell & Baldwin, 1970) has been pe o med, whose $~" N," ;C : 1A (a) (b) Fig. 1. (a) Second-o de di e ence densi y map. Con ou s a e d awn a in e als o 0.012 e/1. -2. (b) Fi s -o de di e ence densi y map. Con ou s a e d awn a in e als o 0.06 e/~-2. p ojec ion o e a plane pe pendicula o b is shown in Fig. l(a), whe e only posi i e densi y egions a e ep esen ed, and he black poin s co espond o he ' ue' a omic posi ions. In o de o make he com- pa ison easie we ep oduce in Fig. 1 (b) he di e ence densi y map ob ained wi h he same e lec ions bu conside ing only he i s -o de con ibu ion (C iado, Conde & M~i quez, 1985a). Bo h maps p esen a zone o nega i e densi y su ounding each posi i e peak, a esul ha is ound o co espond o an unde alu- a ion o he mal c ys allog aphic pa ame e s (Bue ge , 1960). As in he i s -o de case, posi i e densi y peaks in he second-o de map a e si ua ed a ound he ' ue' a omic posi ions and he e o e i mus be expec ed ha he second-o de TDS con i- bu ion will no al e e y much he posi ional pa am- e e s ob ained in a s uc u al analysis. Ne e heless, dis o ions o he peaks a e g ea e han in he i s - o de case, especially nea he hea ie a om (S) indica ing ha he second-o de TDS e ec is mo e equally dis ibu ed be ween posi ional and he mal pa ame e s han in he i s -o de case. Whe eas in he i s -o de map peaks a ound H a oms a e p ac i- cally non-exis en , we ha e de ec ed small peaks nea H-a om posi ions, which may well co espond o an al e a ion o he H-a om densi y, al hough hey may be a esidual elec onic densi y as well. Leas -squa es e inemen To e i y he conclusions deduced abo e we ha e ca ded ou a leas -squa es p ocess (p og am CRYLSQ; S ewa , Kundell & Baldwin, 1970) wi h he 'expe imen al' in ensi ies a 300 K om he ' ue' s uc u e and calcula ed la ice dynamical he mal pa ame e s, keeping he he mal pa ame e s o H a oms ixed. Posi ional and he mal pa ame e s we e e ined oge he wi h K, a scale ac o de ined by E6{[Fexp(G)l-lFca,(G)l/K} 2, and h ee di e en A. CRIADO, A. CONDE AND R. M/~RQUEZ 493 Table 1. Resul s o leas -squa es e inemen s .a. ull angle; 1.o. low o de ; h.o. high o de . A and A U. a e he a ia ions in posi ional and he mal pa ame e s. Fi s o de Second o de ~(G)(% ) A *(X104/~) IA [(H)(xl0 3 A) -A Uu( xl04 A 2) --A U22(x104 A 2) --A U33(X104 A 2) R (%) g~ .a. 1.o. h.o. .a. 1.o. <80 <37 37-107 <40 <10 5-10 4-8 11-18 5-9 4-7 5-8 3-6 18-38 19-40 8-13 55-57 56-58 48-50 21-24 14-16 51-55 52-54 44-46 7-13 5-7 47-51 48-51 41-45 15-19 10-13 8.4-0.4 6.4-0.3 21.4-0.2 2.3-0.6 1-0-0.2 1.005 1.003 1.047 0.992 0.996 * A e age and maximum de ia ions. Ini ial and inal ag eemen ac o s. Final scale ac o . h.o. 10-77 9-18 60-180 56-58 37-41 47-52 1.2-0.7 0.904 anges ha e been chosen wi h 1026 e lec ions in each one: a ull-angle e inemen up o sin 0/A = 0.9 A -1, a low-o de e inemen up o sin 0/) = 0.6/~-1 and a high-o de one wi h 0.7<sin 0/h<l.0/~ -1. A simila p ocess has been pe o med wi h i s -o de con ibu ions and we compa e he main esul s in Table 1. Va ia ions in posi ional pa ame e s o non-H a oms a e o he same o de o accu acy in c ys allo- g aphic wo k and a e alike o i s and second o de in spi e o he smalle magni ude o second-o de co ec ion ac o s. On he con a y, a ia ions o H-a om coo dina es a e g ea e o second-o de e inemen s, a esul ha ag ees wi h he conclusions d awn om he di e ence Fou ie maps. The mal pa ame e s p esen a dec ease o bo h i s - and second-o de e inemen s. Fo i s o de he pa ame e s a e smalle a high angles because co ec ion ac o s a e la ge and 1 + a~(G) canno be adjus ed o a empe a u e ac o wi h exponen ial o m as well as in ull angle o low-o de cases. Maximum de ia ions o he mal pa ame e s a e e y close o a e age, indica ing ha he e ec is mo e impo an o he ansla ional igid-body enso T han he lib a ional L. This seems o con i m he p oposal om he ecen p ojec epo on he com- pa ison o s uc u al pa ame e s o oxalic acid dihy- d a e ob ained in a ious labo a o ies (Coppens 1984), which shows he main disc epancies a e in he mal pa ame e s, p incipally on T enso s, which a e p obably due o TDS con ibu ions. The non- ans e abili y o he mal pa ame e s om high-o de o low-o de e inemen mus be aken in o accoun when calcula ing he mal pa ame e s in accu a e elec- onic densi y s udies (Dam, Ha kema & Feil, 1983) om e lec ions a high alues o sin 0/A whe e he in luence o bonding e ec s o e he mal pa ame e s is minimized when we use a sphe ical-a om model. In he case o second-o de e inemen s we ob ain di e en alues o he mal pa ame e s depending on he chosen ange o sin 0/A because 1 + ce2(G) does no adjus a all o a empe a u e- ac o unc ional o m, e en a low angles; and he pa e n is he in e se o he i s -o de case: he dec ease is la ge in high- o de e inemen s. In he same way, a ia ions in he scale ac o a e opposi e in i s - and second-o de cases. Concluding ema ks Posi ional pa ame e s a e mo e sensi i e o second- han o i s -o de TDS in ensi y, especially hose conce ning hyd ogen a oms. The mal pa ame e s a e di e en o each ange when he second-o de TDS con ibu ion is no sub ac ed om measu emen s and, al hough i s in luence is small o low-angle e lec ions, i is compa able o ha o i s -o de o high-o de e inemen s. So expe imen al in ensi ies should be co ec ed o second-o de con ibu ions when calcula ing accu a e he mal pa ame e s om high-o de e inemen s, al hough he bes emedy is o wo k a as low a empe a u e as possible. Final ag eemen ac o s R a e equal o la ge o second-o de e inemen s, al hough ini ial ac o s a e much smalle han i s -o de ones, indica ing a poo e abso p ion o TDS con ibu ions in leas - squa es p ocesses. As a inal o e all conclusion we can say ha e en when he second-o de con ibu ion is smalle han he i s -o de one, i s e ec s may be compa able o i s -o de ones and a e mo e equally dis ibu ed among di e en a iable pa ame e s and a e no p e- dominan ly concen a ed on he mal pa ame e s, which is he case o he i s -o de con ibu ion. This wo k has been suppo ed in pa by he Spanish Go e nmen h ough he 'Comisi6n Aseso a de In es igaci6n Cien ica y T6cnica'. Re e ences BORN, N. & HUANG, K. (1968). Dynamical Theo y o C ys al La ices. Ox o d: Cla endon. BUERGER, M. J. (1960). C ys al-S uc u e Analysis. New Yo k: Wiley. 494 FIRST- AND SECOND-ORDER THERMAL DIFFUSE SCATTERING COCHRAN, W. (1963). Rep. P og. Phys. 26, 1-45. COCHRAN, W. & PAWLEY, G. S. (1964). P oc. IL Soc. London, 280, 1-22. COPPENS, P. (P ojec Repo e ) (1984). Ac a C ys . A40, 184-195. CRIADO, A., CONDE, A. & MARQUEZ, R. (1985a). Ac a C ys . A41, 158-163. CRIADO, A., CONDE, A. & M,~RQUEZ, R. (1985b). Ac a C ys . A41,316-320. DAM, J., HARKEMA, S. & FELL, D. (1983). Ac a C ys . B39, 760-768. HELMHOLDT, R. B. & VOS, A. (1977). Ac a C ys . A33, 38-45. MARADUDIN, A. A., MONTROLL, E. W., WEISS, G. H. & IPATOVA, I. P. (1971). Theo y o La ice Dynamics in he Ha - monic App oxima ion. New Yo k: Academic P ess. RAMACHANDRAN, G. N. & WOOSTER, W. A. (1951). Ac a C ys . 4, 335-344. STEWART, J. M., KUNDELL, F. A. & BALDWIN, J. C. (1970). The XRAY 70 sys em. Compu e Science Cen e , Uni . o Ma y- land, College Pa k, Ma yland. Ac a C ys . (1985). A41, 494-500 Calcula ion o Elas ic Cons an s by he Me hod o C ys al S a ic De o ma ion* BY MICHELE CATTI Dipa imen o di Chimica Fisica ed Ele ochimica, Uni e si d di Milano, ia Golgi 19, 20133 Milano, I aly (Recei ed 7 No embe 1984; accep ed 22 Ap il 1985) Abs ac The con ibu ion o homogeneous la ice de o ma- ions (neglec ing in e nal s ains) o elas ic p ope ies o c ys als wi h iclinic o highe symme y is examined. The de o med la ice cons an s a e exp essed as unc ions o he componen s o he ini e Lag angian s ain enso , and hei de i a i es a e calcula ed. Thus equa ions a e ob ained ha ela e he second-o de elas ic cons an s o i s and second pa ial de i a i es o he s a ic c ys al ene gy wi h espec o uni -cell pa ame e s. Wi h he assump ion o a wo-body Bo n- ype in e a omic po en ial, he ene gy de i a i es we e calcula ed analy ically, and a igid-body app oxima ion was in oduced o accoun o molecula g oups in he c ys al s uc u e. Tes compu a ions o elas ic cons an s we e pe - o med o MgF2 ( u ile- ype), benzene and naph- halene, using li e a u e po en ial pa ame e s op i- mized on s uc u al da a; esul s a e discussed wi h espec o adequacy o he po en ials and o he app oxima ions o he model used. In oduc ion The semi-empi ical modelling o in e a omic and in e molecula o ces in c ys als has been de eloped in ensi ely in ecen yea s, in o de o ep oduce and possibly p edic a ious chemical-physical p ope ies by compu e simula ions (Ca low & Mack od , 1982). In he pas , his wo k was mainly pe o med by i ing * A p elimina y accoun o pa o his wo k was p esen ed a he XIII h In e na ional Cong ess o C ys allog aphy, Hambu g, Fede al Republic o Ge many, 9-18 Augus 1984 (Ca i, 1984). 0108-7673/85/050494-07501.50 he po en ial pa ame e s o s uc u al p ope ies only, so ha he leas -ene gy a omic con igu a ion app oached he expe imen al one as closely as poss- ible (Busing, 1970; Ki aigo odskii, 1973; Williams, 1981); such po en ials we e hen p oposed o de e - mining unknown c ys al s uc u es by minimum- ene gy sea ch. In a emp s o ex end he modelling o o he physical p ope ies o c ys als, elas ic beha iou and ib a ional spec oscopic equencies a e usually conside ed, as hey a e ela ed o he slope changes o he ene gy hype su ace (in he space o a omic posi ion ec o s) a i s minimum poin . Howe e , ib a ional p ope ies a e accoun ed o by dynamical me hods only, whe eas he c ys al elas- ici y can be ela ed bo h o la ice dynamics and o he s a ics o equilib ium a omic con igu a ion. In he o me case a mic oscopic c ys al de o ma ion chang- ing wi h ime is examined, h ough a omic oscilla ions du ing he p opaga ion o an elas ic wa e (long- wa eleng h acous ic ib a ion mode); in he la e , a mac oscopic s a ic de o ma ion o he c ys al is assumed, implying a omic shi s om equilib ium posi ions ha a e cons an wi h ime. In bo h cases he elas ic p ope ies exp ess he co ela ion be ween c ys al s ain and applied s ess. The i s ull heo y on he subjec was de eloped by Bo n & Huang (1954). A p e ious pa ial app oach (Ca i, 1981) is ex en- ded and he calcula ion o elas ic cons an s by he me hod o c ys al s a ic de o ma ion is conside ed he e. The con ibu ion o ex e nal s ains will be aken in o accoun by de i ing equa ions ha ela e he elas ici y enso componen s o i s and second pa - ial de i a i es o he c ys al s a ic ene gy wi h espec o la ice cons an s, calcula ed a ze o s ain, o i- O 1985 In e na ional Union o C ys allog aphy