Kachhava, C M

Saxena, S C

English

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SIMPLE PROPERTIES OF CRYSTALS AND RESTSTRAIILEN FREQUENCYC. M. KACHHAVA a n d  S C. HAXKNAPh y s i c s  DKrAaxiviENT, E . A J A S T a \ N l h s i t v , .Ia u 'i i!{E^ice^ved Jammnj 22, l!)(Ui, lie-suhviiiUd J'Unc 2, HHi(i)ABSTRACT. Boi'ii-M.t.ycr liil-tico thoovy lui-, lim>n usml lo tlic himpR' LiilKproporLms of alkali hali(li>p b'oi tlLitonnuiing ihr ronhLaiitu ol llii^  ])o(iul'U»l lim« limi (lie HdsiHLralilon froquour-y iurLoikI of c'omjjroBfjibilits' or mnlciuliii tlalii liiif-' ft t-n uhoiI J’roiu'i(h'r which havo lioou computed an- : colic^ yivo tmergy, comprobMbjlitv, lluumiiUxpHiiKicjii, th uik im'h cousUiiif and olaaiic coustauhs In all ousi'S fho huccohs according to the pn)po,sc«l iinKcduio in oithoi of the Mimo order oi is bottci than I hat olit.mmd liy t bo com’i'iitioiml pvofcthiC'-sI N  T 1! O D V  V. 1^’ 11) NBorn-Mayer theory (1932) has been widely aiiphed and is now laiily well established for disenssing the various bulk proiiertios ol ionie crystals 1 he general appi’oaoli is to assume an analytjcal Torin for the interalTimio potnitial lunetmn and then exploit the two familiar Born-Mayej conditions (1932) to evaluate thf^  potential parameters. So far, in such appioachos, information regaiding thi>rmal expansion a, compressibility /?, pressure and temperature dei ivatives of (i are used. In the simplified approach based on this method often only /f is used The purpose of this article is to suggest the use of Ri^ Htstiuhleii fre.qneiiuy m ils place. Once the potential energy function is thus completely determmod, various hulk properties can bo computed according to known procedmes. Adopting this procedure we have computed the different propeities of tlu^  alkali halide ci ys- tals. The properties tiiscussed and compaied with the experimental pi^dietions are the cohesive energy W, compressibility /J, thermal expansion a, Chuneison constant y, and the different clastic constants Wt‘ cousidfT also tlie simple Born-Mayer potential in which only the coulomb and oveilap ims aie con. i The reasons for choosing this simplified jiotimlial are many. Ihsily, tlu. bui.ion of all of the other types ol forces to interaction potential is small and tie effect on the calculation of most of the above mentioned propertK^ s is mvana y negligible. Secondly, the comparison of the properties so compute a ' Values obtained w^ hen compressibihty data are used instead ol Ilislstia - quency is possible in a straightway manner 3 he former type o ca ■ has been reported earlier by the authors (19G4). Thirdly computational labour is worth for the underlying idea is to i us ra ( ment of JS by even in a sophisticated calculation without any ( ang pairing the accuracy.616674J N T 15 H A T O ]\I I C P O T E N T I A L  The simplified Born-Mayer potential (1932) which is used in this article(j){r) =  — -\-A exp i—rlf)).5 0 8  C, M , Kachhava and S. C. Saxena(1)H(jre (j){r) is the inteiaction energy per atom, a the Madclung constant, e the elec­tronic charge, r the interatomic distance, A  and p the two potential parametfys which in the conventional jirocedure are determined I'rom the Born-Mayer condi­tions (1932)- \r # W ]L dr ]r=rnand1 1 L dr^  ‘ 1^\.1,. (2)(3)Here is the volume of the unit cell, the equilibrium interatomic distance.Kellcrmann (1940), and Sharan and Tiwari (1904) have deduced expressions for for NaCl and CsCl-type structures respectively on the assumption that the overlap extends only uii to the nearest neighbours. The two relations can bo conveniently combined into a single expression viz..flVa(4)Here /i is the reduced mass appropriate to the two kinds of ions in the crystal, M  the coordination number, and A' and B' are defined as follows :and, _  2w„ r dv{r)1 _  20LVa . . .  (5)L dr J ’2wa r dH{r) 1 2va r 9va , 2ae® IMe^ J ■... (6)e“ L dr^  \r:A  exp {-rIp). ... (7)HereSubstitution of A' and B' from Eqs. (5) and (6) respectively in Eq. (4) leads to the following relation for :______  ... (8)/? =Simple Praperlics of Crystals and Reststrahhn Frequency f.fiOWhen /tf from Eq (8) is substituted in Eq (:j) one oblaius,V., (!))Ill the present work wo suggest to evaluate the constants A  and p of Va\ (1) Ironi Eqs. (2) and (9). Wo shall refer this as the “propoH(‘d procedure” and the on<> m which use is made of Eqs. (2) and (.‘1) as the “coiivcmtional procethire” . 'rhe prvfieiit endeavour aims at examimng how far a single characlenstic ho(pi<aicy is ciipahlc' of explaining crystal properties. The two pavanieterK A nnd p of Eq (I) eau thus be calculated from the following explicit (expressions obtained by applying condn tions given by Eqs. (2) and (9) to Eq. (1) ■e x p ( - r » ,andp3/«rg2a)o2+ae^( 10)( 11)In this calculation the experimental values of Og and r,, are needed (i>„ value's used here are those recorded in Table I, column 2, while y„ vahuis are those complied by us earlier (1963) Thus, having determined the potential comxdetely we proceed to calculate the various properties.C 0 1 r 15 S I  V  E 10 N 10 R G1 VCohesive energy per mole, W, is rather sim i^ly relati'.d with su(3h thatr  = -[JVs4(r„)+c„]. ... (12)Here N  is the Avogadro number and Cg the zcroqjoint ('iiergy xa*r mole Values fg have also been compiled by us (1963) earlier. Tnsc'rting Eq (1) into Eq (12) leads to the following working relation for W in terms of A and p 'W =  — — -1-^ ex p (-rg /p )|  +^;o] . (13)Computed values of the cohesive energy according to Eqs (10), (II) and (13) arc recorded in Table I, column 4.Experimental values arc shown in column 3 of this very Table Iho agreement is satisfactory. The average absolute dciviation is 3 3 percent. According to the conventional procedure used by us (1964) earlier, the average absolute deviation is 2.9. This should be regnrded as the fust encouraging confirmation for tho use of 6>y data as an alt(3 native to (i suggesting by it the5 7 0 C. M. Kachhava and S. C. Saxenavalidity of single frequency model for purposes of computing cohesive energy. A more* crucial check is presented in the next section by calculating compressibility itself fromTABLE ICalculated and experimental values of W, ft, a and y.o)o Tf (K Cnl/mr)l(>) P(10“12<lyi\p- icin’q a(10- odog-i) :fiiyrifnl (lot-LS-i; — —^(»)Expl.J ('!all^ d. Fxpil(l>)Cab-d Exptl(0Cali-d. Exptl\hi) \Cidf'H,hiF r, 73 242 2 245.3 1 43 1.39 34 0 27 9 1.99 . 1 48LiCl 3 8^1 201 5 193 0 3 17 3 60 44 0 42 2 1 51 1 50Li Hr 3 26 191 5 180.2 3 90 4.78 50 0 45 3 1 49Lil 2 71 180 0 164 7 5 30 fi 93 59.0 61.0 1.76Nal'^ 4 63 215 4 220 7 2 06 1 S3 30 0 31.7 1 57 1 79NaCl 3 09 181 7 182 2 3 97 3 87 40 0 41 1 1.43 ] 83NaBr 2 54 175 9 172 3 4 76 4 81 43 0 44.5 1 65 1 86Nal 2 20 166 3 159 5 6 21 6 47 48 3 50 0 1 59 1 88K F 3 02 190 9 193 3 3 14 3 02 30 7 38 2 1 48 1 88KCl 2.71 167 8 166 5 5 50 4 94 38 3 45 0 1 34 2 12KBr 2 18 101 2 159 0 0.45 5 77 40.0 49 1 1 43 2 ! 1KT 1 94 152 8 149 4 8 07 7 21 ■45.0 52 I 1 58 2 25RbF 3 01 185 0 1R2.8 3 66 5 04 33.33 42 0 1 28 1 65RbCl 2,24 163 6 159.4 6 10 5.73 36 0 46 9 1 25 2 16RbBr 1 09 158 0 153.3 7 38 6 55 38.0 49 9 1 27 2 23Rbl 1 41 149 7 142 5 9 00 H 25 43 0 41 0 1 60 1 77CsF 2 39 176 0 170 0 4 25 3 92 1 49 1 75CaCl 1 80 157 8 149 8 5 55 5.50 56 0 46 8 1 97 2 20CsBr 1.39 152 3 144 6 0 28 6 00 1 93 2 43CbI 1 17 145 4 136.0 7 83 7 35 2.00 2 53(a) Oompilod by Martin, D. H ., 1905, Adoances in  Phyi , 14, 39.(b) Compilod by Karlihava, C M. and Raxonfi, S 0 , 1964, Indian J. Phys  , 88, 3S8- (o) Compiloci by Kauhliava, 0 . M and SuXHiia, S O , 1965, Indian  ,7. Phys . 89, 145.(d) Compilod by Born, M. and Huanpt, K., 1956, Dynaimical Theory of Crystal Lititioe'', Clarondon Vi-ohh, Oxford, 85C O M P U C S S I B I L I T YCompressibility, ft, i.s related with Reststrahlen frequency, Wo, through tho relation of Eq. (8) We employ this relation to compute ft from the known values of the other quantities invaluod. The values so obtained are reproduced in column 0 of Table T in which are also listed in column 5 the experimental ft values for a direct comparison. It is encouraging to note that the agreement betweeniho experimental and calcnlated values is qmW satisfactary. Tlie avopQL^ e abso­lute deviation is 10.4 percent Tt is mteresliiig to compare this liguvo wiili 15.4 obtained in an earlier effort by ns (1964) when jnolecular data amltho rdatiou of Eq. (2) were used to determine the potential paramct-cis f^hi.s ooiiqjari.son in fact suggests a relative preference for the proeediuo suggested in 11ns paper and theieby emphasizes the importance of the ReststrahliMi froquiMicy and its capa­bility to predict the various crystal properties,T H E R M A L IG X  P A N S I () NTWe have (1965a I960) recently derived an expic.ssion for the coeilicient ol thermal expansion, a, in term.s of the derivatives ol «;6(r) ami other qiianthii's For the sake of brevity only the final result is (pioted here It i,sG y2Nr„ ’111 which Gy is the siiccific heat per mole at constant volimu', ^"'(r„) and f/i"(^ 'o) the third and second derivatives of 0(r) at t — lespictivcly . Equation (14) lias been derived by Kumar (1959) adopting a different apjiroach. lias been deriv(;d by Kumar (1959) adopting a different azpproach. ol Erp (J) this gets simplified to the following expression .Simple Properties of Crystals and Reststrakhn Frequency 571a — — — ; - (14)For th(! form For the form. - r f .  ( l - ‘ > ( ? - ) •(15)The values of a arc then readily calculated from Eep (15) as all the other quan­tities are known and these a values are lecordcd in column 8 of Table 1, while the experimental values are given in column 7 In the earlier calculation of a by us,(1966) based on Eq (14) but where wa.s us(h1 to determine tlu^  unknown para­meters of almost similar agreement between theory and experiment wasachieved inspite of the fact that the dispeision terms in ^(r) werti also considered. Oy values used in the present calculations arc the same as compiled m that paper (1966) The average absolute deviations m the present and previous calculations Iroin the oxjiermiental data are 13.4 and 14 3 percent rcspeetivcly. This is again an evidence for the suitability of the proposed method of fixing the interaction potential.a  R U  N E I S E X  0 O NT S T A N 1’Recently we (1965a) have suggested a method for cominitiiig Gruneisen constant, y, based on a single frequency model. Again we do not reproduce here the details of the derivation but quote only the final result which is,y =  - i ( 16)572 (7. M. Kachhava and S. G. SaxenaFor the form of of Eq. (1), Eq. (7) reduces tor  == , r; -) (17)In the calculation of y  therefore only p and values are required. When,/) values obtained from Eq. (11) arc employed we get the values of y  recorded in Table I, column 10, The agreement between theory and experiment is not vcy ifatisfac- tory and is also somewhat inferior to that obtained earlier by the authors (l9G5a) The reason for this lies partly in the use of a simplified potential (the di8^ )er,sion terms have been neglected) which will affect the computation of y  and a more than the other properties such as /?, W, etc. As our idea is more to demonstrate the suc­cess of the outlined procedure rather than computing the various properties accurately, we refrain from repeating calculations of a and y  on a more detailed sophisticated potential.l i J L A S T T C  C O N S T A N T SThe elastic constants can also be computed from the knowledge of the inter­atomic potential, Kachhava and Saxena (1965b). These calculations are confined to the Kellormann (1940), and Krishnan and Roy (1952) models Parameters were determined either using the compressibility or moloeular data. We re­produce here in brief only the important aspects of the calculation.Thus, for NaCl-type crystals on the Kellormann potential (1940),|-6«;(r), (18)the clastic constant is given byC „ = [ - 2 . 5 6 + f | | , .where B' is defined by Eq. (6) and is given byB' =  —6.5169. (19)According to Sharan and Tiwari (1964) the three elastic constants for the CsCl-type crystals on the Kellermann model are given by the following expressions0,1 =9e^32r« if 5.621+1 (B '+24')] . ... (20)Simple Properties of CrysMs and ReststraUen Frequency b73andC\(21)3X - ... (22)in which again the defining relations for J ' and E' are tln^  aaiiiu as gm n Uy E(js (5) and (6) rcsjDeniively. These load to finally3 ’and (2:t)V3e“TABLE ItThe calculated and experimental values of (7^ ] in mills of 10^  ^ dyne cm"dnlf'dCrystal Exptl. ' KL'llomiann KTislijiaumoclul and UoymttdollaF 11 36 (a) 1] 74 11.82LiCl 4 94 (a) 4 61 4 64LiBr 3 94 (a) 3 45 5.64Lil 2 85 (a) 2 37 2 35NaF 9 71 (a) 10 .55 10 83NaCl 4 93 (a) 5 19 5 21NdBr 4.02 (a) 4.22 4.23N al 3.035 (a) 3 16 3.17K F 6.58 (a) 6.77 6.93KCl 4.08 (a) 4 48 4 49KBr 3.49 (a) 3.69 3 71K I 2.776 (a) 3 16 3.17RbF 5.7 (a) 5.35 5 37RbCl 3.645 (a) 3.90 3 91RbBr 3.185 (a) 3.43 3 44R bl 2.685 (n) 2.74 2.75CsF 3 62 3 63CbCI 3 64 (^) 4.05 4.10CsBr 3.10 (b) 3.65 3.57Cal 2.45 {^) 2.74 2 90(a) Spongenborg, K . and Hauasuhl, S , 1967, Z. Kriftt., 109, 4-2.(b ) HausBuhl, S., 1960, Acta CrysL, 13, 687.5 74 C. M, Kachhava and S. G. SaxenaThe computed values of on the Kellermann model for all the all{:ali hahdea are listed in Tabic II column 3 alongwith the experimental values in column 2 . The average absolute deviation is 8 5% . This figure competes well with the one of 8.2 obtained when compressibility data are used in place of Wq.In Table 111, column 3, are shown the computed values of obtained from the use of Eq (21) togetJier with the experimental data in column 2. The aveiugi' absolute percentage deviation is 18.1 whicli is a marked improvement over the figure of 25.1 corresponding to the use of i the conventional proccdiiiri'Thevalucsof O44for the three CsCl-typo crystals obtained according to\Eq. (22) arc recorded in Table 111, column fi, along with the expiTimcmtal data in cojlunm 5 . Here the average absolute deviation of theory Irom experiment is only 7 0 ik‘r cent compared to 14 7 obtained from the use of /J.TABLE illCalculated and exiiei imental values of elastic constants 111 units oI‘ 10^  ^dyne cm ^Crystul(JbCICsBrChI("12 C4.Chr-.LExpU (ii)JvnllcjiimiruiiTKulolfirislui.'m n.]ud Rioy mocl('lBxptl (h)Kollo I'liiaim nn'idolK imkIjiihh Hjul Itoy mode']0 92 0 69 0 7J 0 80 0 090 0 7J0 84 0 70 0.7.7 0 7,5 0 094 0 750 71 0 02 0 04 0 02 0 61 0 04(a) HauBSuhl, S., 1900, Acta Ctyst , 13, 0S7.On the model of Krishnan and Roy (1952) for NaCl-t5rpe crystals the working Eqs. areO,, =  r2a(l +  d')-6yj ^and12Vwhile for CsCl-type these are... (24)... (25).. (20)andCj, =  C'„ =  L a '(2« -7 )+ 9 /] . ... (27)Hor« S =  rjp, a' =  1.018, X  3 U  and X ' == 1 08. Tho values of (7,, nsinK Kqs. (24) and (26) are shown in Table 1 1 , column 4 The average absolute per- centage deviation is 10.8 compared to 8.5 for the convontiojuil approach The values of obtained form Eq. (27) are rtiproducod in columns 1 and 6of Table 111. Eor the average absolute peiccntage deviation is 14 5which IS to be compared with the figure of 2  ^3 found from the conviait-ionnl procedure. Again for C44 tho average absolute deviation is 4 7 only wliich indi­cates great improvement over the figure 12.7 obtained previously. Thus wo infer that the elastic constants ai'o in general bettor rei>roduced tluough tiuj iiwi* of compared to /? data or in other words, the proposed procedure is better than tho conventional procedure.C O N C L U S I t ) N SWe now sum up the results obtained on the computation of diffiirent bulk crystal properties obtained on the proposed and conventional jirocediui'S in parti­cular reference to their success in reproducing thi'^  exiierimental data. The values of cohesive energy of alkali hahdes arc reijroduced with almost (iqnal success. In compressibility the proposed method leads to appreciably better results than tho conventional procedure. Thermal expansion coefficients are also bi4,ter re­produced on the basis of proposed method than the conventional apiiroach The Grimoisen constants on the other hand are v'ell reproduced by tho conventional method m comparison to the proposi'd one. On tlie other hand, the tdastic constants are. better reproduced by the proposed than the conventional method Tlie. only exception is where the conventional approach commands a rclativi' superi­ority over the proposed oneThus on tho whole wc find that the use of Reststrahlcn frequency instead ol y  in determining the potential parameters and then ui predicting the propoitics, is preferable. It leads to a happy conclusion viz the use of Reststrahlcn frequenc^ ’^ m interpreting tho bulk behaviour of cry.stals as encomjiasscd in the iliffiiront properties considered in this paper, incidently, this work also brings to light a Imk between lattice vibrations and the fundamental simple crystal properties through interatomic forceps and that too in a very simxdo way.K E F E R N C E NB o th , M., and Mayor, .T. E.. 1932, Z  Phys%k, 75, 1.ICaohhava, O. M. aird Saxona, S. C-, 19(i3, Phil. Mag , 8, 1429._______________________________________ 1964, Indian J Phys., 88, 388.______________ _ _________________ _ ____ lOGHa, Indian J. Phys (lu______________________________________ 19G5h, Indian J Pifre App. Phys , (lu Ti'crh).EoUermuiim, E. V/., 1940, Phil. Tnaus., A236, 613Krislman, K . S. and Roy, S. IC., 1902, Proc. Boy Soc.. A 2i0 . 481Kumar, S., 1059, Proc. Natl. Inst. {India), A23. 364Saxona, S. O. and Kachhava, 0 . M.. 1966, Ajrp {Netherlands), ( n uSharan, B . and Tawari, L. M., 1964, Phys Stat. SoUdi., 7. 39.Simple Properties o f Crystals amd Ecststrahlen Frequency 575

Simple Properties of Crystals and Reststrahlen Frequency

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Simple Properties of Crystals and Reststrahlen Frequency

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