REFERENCE INTERNATIONAL CENTRE FOR THEORETICAL PHYSICS IC/T2/99 ANHARMONIC UON-GAUSSIAN COKTRIBUTIOE TO THE DEBYE-WALLER FACTOR - II: KCS,

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1 REFERENCE IC/T2/99 INTERNATIONAL CENTRE FOR THEORETICAL PHYSICS ANHARMONIC UON-GAUSSIAN COKTRIBUTIOE TO THE DEBYE-WALLER FACTOR - II: KCS, G. Slt N.M. Butt and D.A. O'Cnnr INTERNATIONAL ATOMIC ENERGY AGENCY UNITED NATIONS EDUCATIONAL, SCIENTIFIC AND CULTURAL ORGANIZATION 1972 MIRAMARE-TRIESTE

3 IG/72/99 Internatinal Atmic Energy Agency and United Natins Educatinal Scientific and Cultural Organizatin INTERNATIONAL CENTRE FOR THEORETICAL PHYSICS ANHARMONIC NON-GAUSSIAN CONTRIBUTION TO THE DEBYE-WALLER FACTOR - II: KCZ * G. Slt **, M.M. Butt *** Internatinal Centre fr Theretical Physics, Trieste, Italy, and D.A. O'Cnnr Physics Department, Birmingham University, England. MIRAMARE - TRIESTE August 1972 * T "be submitted fr publicatin. ** On leave f absence frm Central Research Institute fr Physics, Budapest llu, : r B U9, Hungary. *** On leave f absence frm Physics Dept., Birmingham University, England. Permanent address: Pakistan Inst. f Nuclear Science and Technlgy, Islamabad, Pakistan.

5 ABSTRACT Jlssbauer gamma-ray diffractin data btained by the technique f O'Cnnr and. Butt (1963) are analysed t deduce the anharmnic fur-fchrder term in the Debye- aller factr fr KCfi, and t estimate the third-rder anharmnic cupling cnstant in this way. The theretical treatment assumes that the displacement crrelatin functin can be apprximated by its large-distance asympttic value. Fr the anharmnic cupling parameter ^ an estimate f -6,2x10^ erg/cm is fund, which differs by a factr f abut 6 frm that given previusly by Leibfried and Ludwig. 1-

6 INTRODUCTION Departure frm the Gaussian shape f the 13 ebye-waller factr as a functin f sinq/x is ne f the mst prminent effects f lattice,, anharmnicity since it shws the limits f even the "best quasiharmnic apprximatin. Such deviatin frm the Gaussian appears, fr cubic crystals, thrugh the furth (and higher) pwers f sin^/a in the expnent f the Debye-Waller factr. In a'previus paper (Butt and Slt, 1971) a simple theretical treatment was presented fr calculating the quartic nn-gaussian term, and it was applied t experimental results n NaCSL. It was shwn that by using accurate ela.sjn.c diffractin intensity data fr different scattering angles within a wide temperature range, direct infrmatin n the anharmnic cupling cnstants can be btained. Since a smewhat detailed discussin can be fund in the abve reference, here nly sme essential pints f the theretical mdel will be mentined. The gamma-ray diffractin peak intensity at mmentum transfer k ( kl» 47r«sin /\) is given by*) ; l(k,t) ~ I f(k,t) where the Debye-Waller factr f(kjt) defined thrugh the inic displacement vectr u can be expressed as f(k,t)- <exp{ik.u5> T. exp - <(k.u) 2 > T+^ J/(k.u) (1) T evaluate the quartic term the fllwing tw assumptins have been used: i) In the harmnic lattice, the displacement crrelatin functin fr tw ins at different sites is entirely determined "by the acustical sund wavesj difference between the Debye-tfaller factrs f the different kinds f ins is, fr the present, purpse, assumed t be negligible. This was seen t be irrelevant even fr UadJL ; thus fr KCJL the difference fr frm factrs can safely be ignred. The ntatin < ^~ means thermal average at temperature T. -2-

7 ii) Inharmnic frces can be represented "by nearest-neighbur central interactins. This, treatment is, in fact, a simplified versin f that given by Wlfe and Gdman (1969) thugh with the advantage that sme f the relevant physical parameters enter here in a mre explicit way. The experiments were perfrmed by the Mssbauer gamma-ray diffractin technique (O'Cnnr and Butt, 196*3) which allws measurement f the stritly elastic cmpnent f the diffractin intensity. The main result f this wrk is that the experimentally bserved quartic term fr KCJl shws, in fact, the expected behaviur as a funtin f temperature between K, thugh with a much higher amplitude than culd be expected n the basis f previus estimates fr» the anharmnic cupling cnstants. This result is thus cmpletely analgus t that btained in the freging wrk fr RESULTS The present analysis has been fcussed n the k. J term in the expnent f f(k,t), which is cmpletely absent when a harmnic r 1 quasihannni descriptin fr the lattie vibratins applies. In the present case, with k. pinting in the (100) directin, the quantity f interest is therefre Fr this, ne hab the apprximatin expressin (Butt and Slt, 1971) %3 D(T) 4*C 44 d (3) where T is the abslute temperature, u is the inic displacement in the (100) directin, d is the nearest-neighbur distance at T = 0, G.. is the cnventinal ntatin fr the shear mdulus, <f> and (f> L are the third and furth derivatives f the repulsive interactin ptential and A (b) and B (b) are simple lattice sums (Butt and Slt, 197l) determined by the structure and depending nly n the quantity -3

8 (nn) x u /_ where in the denminatr ne has the displacement crrelatin functin fr nearest-neighbur ins. Using the value f Buyers and Smith (1968) fr \ u x / ne EvtB b = 1.36 _+ 0,02, and this gives fr the lattice sums A and BXJC the values 0.37 and O.4O, respectively. Using xx then the data G 44 a * 63 x 1Q11 er^/ n3 (Leibfried, 1955) and 3.11A (Leibfried and Ludwig,196l) ne cmes up with the expressin I? x10" ]3 III where the quantities ^~~~ = x 10 erg/cm, 6 M *» 0.11 z 10' erg/cm and (EX = 257 K have been intrduced as apprpriate scaling units, being actually the calculated values f Leibfried and Ludwig (1961) -IV fr Lebye temperature and anharmnic cupling parameters. On the ther hand, the quantity t be measured is the strictly elastic part f the diffractin peak intensity l(k,t) at different temperatures T and different reflectins (klm). If ne has, e.g., data fr the k.. = (200) and (400) reflectins at rm temperature and at ther pints' T, by simple algebra ne gets fr the anharmnic part lg X lg I((400),T) I((2OO),T) T T lg - lg 32TT 4 I(( 4 OO),T O ) I((2OO),T O ) T determine the strictly elastic part f the peak intensity, the Mssbauer gamma ray diffratin technique (O'Cnnr and Butt, 1963,and Butt and O'Cnnr, 1967) seems t be extremely well adapted, since inelastic cntributins are nt present even in the rugh data. Sme (4) -4-

9 details f the Mss'bauer diffractin set-up used fr btaining the present data were given in the previus paper (Butt and Slt, 1971) and here nly the results fr KCiL are discussed. In Figs.l(a) and (b) the intensity l(k^t) is pltted at tw reflectins versus temperature, while Fig.2 shws the expnent f the Debye-Waller factr divided by f-rr:) =» ( In harmnic r quasiharmnic apprximatin the tw curves shuld cincide. bth giving I6n < Ax / - /u y _ J and the difference represents precisely the nn-gaussian behaviur f the Debye-Waller factr. (One ntices that, at these temperatures, the deviatin f the curves frm the straight line itself is an effect f anharmnicity, but this can be described at least partly within the frame f a quasiharmnic thery.) Finally, Fig.3 shws [D(T) - D(TQ)] as prcessed frm the data accrding-t (4) versus (T/O&P. It is difficult t check accurately the predicted linearity since the statistical uncertainties (f abut 1-Zf) are, especially fr the (200) reflectin, cnsiderably blcwn up by the small value f (ain'0/a) in the final result. (The same can als be seen in Fig.2.) Cmparisn with the thery can be made by determining the slpe f the line in Fig.3, which fr the abve reasns is bviusly difficult, and the apprximate value f *10" is rather uncertain. One can g still further and estimate the anharmnic frce cnstant p by assuming that (f> }(p and p j <p are f the same rder f magnitude and therefre which leads t <b lu ^( ) $ UI - - ( ) * erg/cm riv /~IV Due t the small numerical cefficient befre <p /<p, that rati is almst cmpletely uncertain and, in turn, it plays a much less imprtant rle in this anharmnic term. The final result is very similar t that btained fr ITaCi, where the estimate f 0 has given. 4.2 times the value fund by Leibfried and Ludwig (l96l). As t the accuracy f the abve values, ne shuld mentin that, in additin t the statistical errrs already discussed, n extinctin effects were accunted fr in the data prcessing. This was already nted in cnnectin with the results fr EaC.fi. (Butt and Slt, 1971) and the crrespnding errr shuld bviusly be greater here where n -5-

10 data fr the (600) reflectin were available and we had t use thse fr the (200) reflectin. Therefre, t check the abve values, experiments at higher-rder reflectins are required, the need fr these being indicated already by the (200) data fr IteC which seemed nt t "be accurate enugh fr a similar discussin. CONCLUSION The temperature dependence f the furth-rder anharmni term D(T) in the Debye-Waller factr fr KCi was theretically estimated, and was measured within the range K. The experimental data seem t shw th'e predicted trend, thugh the amplitude f the variatin with temperature is much larger than expected. As a cnsequence, the thirdrder anharmnic cupling cnstant btainable frm the present data turns ut - within a statistical accuracy f apprximately 25$ - t be abut six times as large aa that predicted by Leibfried and Ludwig. The disagreement, r at least part f it, may be cnnected with experimental uncertainties like statistical errrs and neglect f extinctin effects. Still, the very fact that, just as fr UaClL, thugh by using different reflectins," the rati <& /^ fr KCJL was als fund t be much larger than unity, indicates that the departure frm the results f Leibfried and Ludwig (1961) may be a real ne. If this is the case, ne has, f curse, t chse between the tw sets f values fr <f> and als explain why they differ frm each ther s greatly. In this cnnectin, besides emphasising again the need fr further experimental wrk t test accurately the values given abve, we may mentin that the present methd, principally, seems t be ne f the simplest and mst direct ways f measuring the anharmnic cupling cefficients. -6-

11 ACOOWLEDQffiKTS Tw f the authrs (BIB and GS) are indebted t Prfessr Abdus Salam, Prfessr P. Budini, the Internatinal Atmic Energy Agency and UNESCO fr hspitality at the Internatinal Centre fr Theretical Physics and t the Swedish Internatinal Develpment Authrity fr financial supprt. They are grateful t Prfessrs I. Waller and S. Lundqvist fr valuable discussins. MB takes pleasure in acknwledging the hspitality and kindness f Prfessr P.B. Mn during the experimental part f this vrk at Birmingham University. GS is grateful t the Central Research Institute fr Physics.f the Hungarian Academy f Sciences and MB t the Pakistan Atmic Energy Cmmissin fr leave f absence. KEFEKEECES Butt, N. M. and O'Cnnr, D. A. (19^7); Prc. Phys. Sc. 22, Butt, N. M. and Slt, & (1971); Acta Cryst. A27_, 238. Buyers, VT. J. L. and Smith, T. (1968); J. Phys. Chem. Slids, 12, 1051 Leibfried, G. (1955)J Encyclpaedia f Physics, Ed. by S. Plugge, Vl. VIII/1, p.197. Leibfried, &. and Ludwig, TST. (1961); Slid State Physics. Ed. by P. Seitz and D. Tumbull, Vl. 12, pp.276-^4. O'Cnnr, D. A. and Butt, N. II. (1963); Ehys. Letters 2, 233. Wlfe, &. A. and Gdman, B. (1969); Phys. Rev. 1^8,

12 FIGURE CAPTIONS Figs.l(a) and (b) Elastic (zer-phnn) scattered intensity l(k,t) frm KCJL at different scattering angles versus temperature: a) fr the (200) reflectins, t>) fr the (400) reflectins. Fig.2 The quantity lg(l(k,t) /l(k,t Q )) / (sin >//\) 2 versus temperature fr the (400) ancl (200) reflectins. Fr harmnic r q.uasiharmnic lattice vibratins bth curves give I611 I ^u y - /u /mi aj^ shuld therefre cincide. Deviatin frm the Gaussian frm f the Debye-Waller factr versus temperature as deduced frm the experiments. T Q «* 293 K Zer slpe wuld crrespnd t Gaussian behaviur. -8-

13 Kfe.T) COUNTS/min O 01 i I m 3 m CO

14 ;* KCje(400) c: CO 2: A u. \ 120 * TEMPERATURE K* it m>i 1*

15 FIG. 2

16 2.0 0 W 0 l CM. I -2.0 O a [(T/e 0 ) 3 -(T 0 /e 0 ) 3 ] " TEMPERATURE K*